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A093729 Square table T, read by antidiagonals, where T(n,k) gives the number of n-th generation descendents of a node labeled (k) in the tree of tournament sequences. 9
1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 7, 7, 3, 1, 0, 41, 41, 15, 4, 1, 0, 397, 397, 123, 26, 5, 1, 0, 6377, 6377, 1656, 274, 40, 6, 1, 0, 171886, 171886, 36987, 4721, 515, 57, 7, 1, 0, 7892642, 7892642, 1391106, 134899, 10810, 867, 77, 8, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

COMMENTS

Column 1 forms A008934 (number of tournament sequences). Row sums form A093730.

A tournament sequence is an increasing sequence of positive integers (t_1,t_2,...) such that t_1 = 1 and t_{i+1} <= 2*t_i, where integer k>1.

LINKS

Table of n, a(n) for n=0..54.

M. Cook and M. Kleber, Tournament sequences and Meeussen sequences, Electronic J. Comb. 7 (2000), #R44.

M. Somos, A functional power series equation, Mathematics StackExchange answer.

FORMULA

T(0, k)=1 for k>=0, T(n, 0)=0 for n>=1; else T(n, k)=T(n, k-1)-T(n-1, k)+T(n-1, 2*k-1)+T(n-1, 2*k) for k<=n; else T(n, k)=Sum_{j=1..n+1} (-1)^(j-1)*C(n+1, j)*T(n, k-j) for k>n (Cook-Kleber).

Column k of T equals column 0 of the matrix k-th power of triangle A097710, which satisfies the matrix recurrence: A097710(n, k) = [A097710^2](n-1, k-1) + [A097710^2](n-1, k) for n>k>=0.

EXAMPLE

Array begins:

[1,1,1,1,1,1,1,1,1,1,1,...],

[0,1,2,3,4,5,6,7,8,9,10,...],

[0,2,7,15,26,40,57,77,100,...],

[0,7,41,123,274,515,867,1351,1988,...],

[0,41,397,1656,4721,10810,21456,38507,64126,...],

[0,397,6377,36987,134899,376175,880032,1818607,3426722,...],

[0,6377,171886,1391106,6501536,...],

[0,171886,7892642,...],

[0,7892642,627340987,...],...

MATHEMATICA

t[n_?Negative, _] = 0; t[0, _] = 1; t[n_, k_] /; k <= n := t[n, k] = t[n, k - 1] - t[n-1, k] + t[n - 1, 2 k - 1] + t[n - 1, 2 k]; t[n_, k_] := t[n, k] = Sum[(-1)^(j - 1)*Binomial[n + 1, j]*t[n, k - j], {j, 1, n + 1}]; Flatten[Table[t[i - k, k - 1], {i, 10}, {k, i}]] (* Jean-Fran├žois Alcover, May 31 2011, after PARI prog. *)

PROG

(PARI) {T(n, k)=if(n<0, 0, if(n==0, 1, if(k==0, 0, if(k<=n, T(n, k-1)-T(n-1, k)+T(n-1, 2*k-1)+T(n-1, 2*k), sum(j=1, n+1, (-1)^(j-1)*binomial(n+1, j)*T(n, k-j))))))}

(PARI) {a(n, m) = my(A=1); for(k=1, n, A = (A - q^k * r * subst( subst(A, q, q^2), r, r^2)) / (1-q); subst(subst(A, r, q^(m-1)), q, 1)}; /* Michael Somos, Jun 19 2017 */

CROSSREFS

Cf. A008934, A093730.

Cf. A113080, A113081, A113092, A113103.

Sequence in context: A131182 A254883 A266599 * A113080 A174420 A266318

Adjacent sequences:  A093726 A093727 A093728 * A093730 A093731 A093732

KEYWORD

nonn,tabl

AUTHOR

Paul D. Hanna, Apr 14 2004; revised Oct 14 2005

STATUS

approved

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Last modified February 22 16:26 EST 2019. Contains 320399 sequences. (Running on oeis4.)