N. J. A. Sloane. A Handbook of Integer Sequences. Academic Press, 1973.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... will be the generating function for the ordered sequences of blocks, the leftmost ones being underlined, the rightmost ones being non underlined, so that (1 Gamma (e Gamma 1) j 2 1 The sequences (b r ) and (b r ) do not appear (yet ) in the Sloane integral sequence basis [Sl94]. However our young colleague Jiang Zeng drew our attention to the paper by Knuth [Kn92] who himself pointed out that the generating function (2 Gamma e already appeared in Cayley (Collected Math. Papers, vol. 4, p. 112 115) for enumerating a special class of trees. According to Knuth the ....

N.J.A. Sloane, "A Handbook of Integer Sequences." New York, Academic Press, 1973.

....with respect to their number, since it gives this set a tree structure. Algorithm 1 acheives this. We will now show that T b ( can be described recursively, which allows us to give a new recursive formula for jR b (n)j. In order to do this, we will use a series known as the b ary carry sequence [Slo73]: c b (n) k if b k divides n but b k 1 does not. Notice that this function is defined only for n 0 (or one can consider that c b (0) These series appear in many contexts, and have many equivalent definitions . Here, we will mainly use the fact that the first n such that c b (n) k ....

N.J.A. Sloane. A Handbook of Integer Sequences. Academic Press, 1973. On-line version at http://www.research.att.com/%7Enjas/.

....properties listed above. This is often hard to do, especially by hand. Gosper s algorithm leads to a systematic procedure for constructing such R(n, k) To conclude this section, we mention that a useful resource when investigating sequences arising in combinatorial settings is the book of Sloane [345, 346], which lists several thousand sequences and gives references for them. Section 17 mentions some software systems that are useful in asymptotics. 11 4. Basic estimates: factorials and binomial coe#cients No functions in combinatorial enumeration are as ubiquitous and important as the factorials ....

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973. A revised and expanded edition is in press.

....210066388901; Note that um is the number of binary search trees with height at most m Gamma 1 and (12) has been studied from this point of view. While no closed form solution to (12) is known, one can show that um = bK 2 m c = bK n 1 c where K is approximately 1.502837. See Aho and Sloane (1973) for a discussion of this and other nonlinear recurrences of the form x n 1 = x 2 n g n , where g n is a slowly growing function of n. 3. One approach to computing D(S; T ; Delta) begins by constructing tables of ancestry relations for S and T . It is easy to see how to construct such ....

.... Gamma p 1 Gamma 4z 2z : From (14) it follows that the generating function for the number of derangement trees is A(z) B(z) 1 zB(z) 1 B(z) 2 z ; 16) and the result follows by computing the coefficient of z n . Remarks: 1. Values of ff n up through n = 21 can be found in Sloane (1973), sequence number 635. The first 10 numbers, starting with ff 1 , are: 0, 1, 2, 6, 18, 57, 186, 622, 2120, 7338. 2. The sequence (ff n ) has arisen in the context of Fine s (1970) work on closeness relations. We shall not go into detail about the connections. In 20 short, ff n is the number of ....

Sloane, N. J. A. (1973). A Handbook of Integer Sequences, Academic Press, New York.

....for any n in linear time with respect to their number, since it gives this set a tree structure. We will now show that T b (1) can be described recursively, which allows us to give a new recursive formula for jR b (n)j. In order to do this, we will use a series known as the b ary carry sequence [Slo73]: c b (n) k if b k divides n but b k 1 does not. Notice that this function is de ned only for n 0 (or one can consider that c b (0) 1) These series appear in many contexts, and have many equivalent de nitions 4 . Here, we will mainly use the fact that the rst n such that c b (n) k ....

N.J.A. Sloane. A Handbook of Integer Sequences. Academic Press, 1973. On-line version at http://www.research.att.com/%7Enjas/.

....5403223, 7314662, 11265825, 15258443, 23363143, an = 0, 1, 1, 1, 3, 1, 6, 1, 10, 4, 12, 1, 33, 1, 29, 13, 64, 1, 100, 1, 156, 30, 187, 1, 443, 10, 476, 78, 877, 1, 1326, 1, 2098, 188, 2745, 36, 5203, 1, 6408, 477, 11084. 99) These sequences do not appear in the 1973 edition of Sloane s book [32] but they do in the new edition [33] The analytical properties of the functions (97) should prove useful for the asymptotic analysis of the numbers m n ,forn ##. ....

N. J. A. Sloane, A Handbook of Integer Sequences. Academic Press, 1973.

.... sequences of blocks, the leftmost ones being underlined, the rightmost ones being non underlined, so that X r0 b r u r r = i 1 (1 Gamma (e u Gamma 1) j 2 = 1 (2 Gamma e u ) 2 : The sequences (b 0 r ) and (b r ) do not appear (yet ) in the Sloane integral sequence basis [Sl94]. However our young colleague Jiang Zeng drew our attention to the paper by Knuth [Kn92] who himself pointed out that the generating function (2 Gamma e u ) Gamma1 already appeared in Cayley (Collected Math. Papers, vol. 4, p. 112 115) for enumerating a special class of trees. According to ....

N.J.A. Sloane, "A Handbook of Integer Sequences." New York, Academic Press, 1973.

....less than 2:5 10 6 , namely 383. 1 Introduction The sequence we are interested in has the ID number A000522 in Sloane s On Line Encyclopedia of Integer Sequences (http: www.research.att.com e njas sequences) Former identi cation numbers of this sequence were M1497 in [SP] and N0589 in [Sl]. The sequence A000522 has many faces (see, e.g. Ga] Si] or [Ri] The most accessible one is its combinatorial interpretation: De nition 1 For n 2 N = f0; 1; 2; g let seq 1 1 (n) denote the number of one toone sequences these are sequences without repetitions we can build with n ....

N. J. A. Sloane: \A Handbook of Integer Sequences." Academic Press, New York (1973).

....5403223, 7314662, 11265825, 15258443, 23363143, a n = 0, 1, 1, 1, 3, 1, 6, 1, 10, 4, 12, 1, 33, 1, 29, 13, 64, 1, 100, 1, 156, 30, 187, 1, 443, 10, 476, 78, 877, 1, 1326, 1, 2098, 188, 2745, 36, 5203, 1, 6408, 477, 11084. 99) These sequences do not appear in the 1973 edition of Sloane s book [32] but they do in the new edition [33] The analytical properties of the functions (97) should prove useful for the asymptotic analysis of the numbers m n , for n 1. ....

N. J. A. Sloane, A Handbook of Integer Sequences. Academic Press, 1973.

....: 2) We now simplify by nondimensionalizing. Put v = hu and = t=h, and then dv d = B(v)v 2 (3) and v( 1) v( v 2 ( 4) A simple Maple program was written to compute more terms in the series for B(v) Once a few more terms in the series were computed, the series was recognized [26, 27]. It turns out that this problem has already been solved, in [20] in the domain of formal power series. The series for B(v) can be constructed recursively as follows. If B(v) c 1 c 2 v c 3 v 2 Delta Delta Delta (5) in a purely formal sense) then c 1 = 1 and c n = Gamma 1 n ....

I. Sloane, Handbook of Integer Sequences, 2nd ed.

....a k triangulation. A proper interval supergraph G(V; E[F ) of a graph G(V; E) with jF j k is called a k proper interval supergraph of G. 2.1 A linear algorithm for fixed k. Denote by c l the l th Catalan number. i.e. c l = Gamma 2l l Delta 1 l 1 . Note that c l 4 l . Lemma 2. 1 (cf. [29]) The number of minimal triangulations of a cycle with l vertices is c l Gamma2 . Each such triangulation contains l Gamma 3 chords. The algorithm will traverse part of a search tree in which each node corresponds to a supergraph of G. This search tree is defined as follows: The graph G itself ....

N. J. A. Sloane. A handbook of integer sequences. Academic Press, New York, 1973.

....of the partition lattices. 1991 Mathematics Subject Classification. Primary 05A15, 52B30; Secondary 05A18, 05B35, 06A07, 11B73, 11B83, 15A15, 20F55 Introduction When one looks up the sequence 1, 6, 31, 160, 856, 4802, 28 337, 175 896, in one of Sloane s integer sequence identifiers [HIS, EIS, OIS], one learns that these numbers are the numbers of driving point impedances of an n terminal network for n = 2, 3, 4, 5, 6, 7, 8, 9, as described in an old article by Riordan [Ri] In combinatorics there are two common ways of generalizing classical enumerative facts. One such ....

N. J. A. Sloane, A handbook of integer sequences, Academic Press, New York 1973. MR 50:9760

....than h, including for convenience the empty one with no nodes and no edges. Then p 1 = q 1 = 1, while for all h 1, q h 1 = q h p h (1) p h 1 = # # # 2 1 p h # # # p h q h (2) Note that the numbers p h were known to Etherington [2] they are sequence number 718 in Sloane s book, [6]. To justify (2) we observe that a planted tree of height h 1 has two major subtrees, one of height h and the other of height h or less. For both to have height h, there are # # # 2 1 p h # # # possibilities since we need to select two trees (which may be isomorphic) from among the p ....

N. J. A. SLOANE, "A Handbook of Integer Sequences" Academic, New York, 1973.

....infinite set and assume that there is a bijective function B : seq 1 1 (S) Gamma P(S) We use this function to construct an sequence 1 1 in S. Let n (n ) be the cardinality of seq 1 1 (n) Then 0 = 1; 1 = 2; 2 = 5; 16 = 56; 874; 039; 553; 217; see [Sl], No. 589) and, in general n = n X i=0 n i We begin by choosing four distinct elements of S; S 4 : fs 0 ; s 1 ; s 2 ; s 3 g and use these elements to construct a 4 sequence 1 1 hs 0 ; s 1 ; s 2 ; s 3 i 4 in S. This sequence will give us an order on the set seq 1 1 (S 4 ) e.g. we ....

N.J.A.Sloane, A Handbook of Integer Sequences, Academic Press, New York, 1973

....to Gian Carlo Rota for his inspiration and his friendship. 1. Incidence Algebras and Generating Functions 1.1. Summary of [DRS 1972] The idea of a generating function is a powerful tool by which one may calculating the many sequences (even tables) of numbers which arise in combinatorics. [Sloane 1973] For example, let F (x) P 1 n=0 fn x n where fn denotes the nth Fibonacci number. F (x) is called the (ordinary) generating function of f n . Using this generating function, the recurrence relation f n = fn Gamma1 f n Gamma2 and initial conditions f 0 = f 1 = 1 can be expressed (1 Gamma x ....

N. J. A. Sloane, "A Handbook of Integer Sequences," Academic Press (1973). The second edition will appear under the title "The New Book of Integer Sequences."

No context found.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, NY, 1973.

....Single Deletion Correcting Codes N. J. A. Sloane Dedicated to Dijen Ray Chaudhuri on the occasion of his 65th birthday Abstract. A survey of binary single deletion correcting codes. The VarshamovTenengolts codes appear to be optimal, but many interesting unsolved problems remain. The connections with shift register sequences also remain somewhat mysterious. 1991 Mathematics Subject Classification: ....

....Introduction The possibility of packet loss on internet transmissions has renewed interest in deletion correcting codes. Of course there are many other applications of such codes, including magnetic recording, although in that case there are usually additional conditions that must be satisfied. This paper considers the very simplest family of such codes, binary block codes capable of correcting single deletions. Even for these codes there remain several apparently unsolved problems. It is surprising, but these codes do not appear to be surveyed in any of the usual references ( MS77] PH98] ....

[Article contains additional citation context not shown here]

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, New York 1973.

No context found.

N. J. A. Sloane. A Handbook of Integer Sequences. Academic Press, 1973.

No context found.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973.

No context found.

N. J. A. Sloane. A Handbook of Integer Sequences. Academic Press, 1973.

No context found.

N.J.A. Sloane (1973). A Handbook of Integer Sequences. Academic Press.

No context found.

N.J.Sloane, \A handbook of integer sequences", Academic Press, New York, 1979.

No context found.

N.J.A. Sloane, A Handbook of Integer Sequences, Academic Press, New York and London, 1973. 6

No context found.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, NY, 1973.

No context found.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, New York, 1973.

First 50 documents

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC