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.

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, 0, 627340987, 627340987, 89574978, 6501536, 376175, 21456, 1351, 100, 9, 1

Offset: 0

Author

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

Keywords

• nonn
• tabl

Comments

Column 1, of array T and antidiagonals, equals A008934, which is the number of tournament sequences.

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.

Examples

			Array begins:
  1,      1,       1,       1,       1,      1,      1,     1,     1, ...],
  0,      1,       2,       3,       4,      5,      6,     7,     8, ...],
  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, .................],
  0,   6377,  171886, 1391106, 6501536, ...],
  0, 171886, 7892642, .....................];
Antidiagonals begin as:
  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;
		

Links

• G. C. Greubel, Antidiagonals n = 0..50, flattened
• M. Cook and M. Kleber, Tournament sequences and Meeussen sequences, Electronic J. Comb. 7 (2000), #R44.
• Michael Somos, A functional power series equation, Mathematics StackExchange answer.

Crossrefs

Cf. A008934, A097710, A113080, A113081, A113092, A113103.

Cf. A008934 (column k=1 of array and antidiagonals), A093730 (antidiagonal row sums).

Programs

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-François Alcover, May 31 2011, after PARI 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 */

SageMath

@CachedFunction
def T(n, k):
    if n<0: return 0
    elif n==0: return 1
    elif k==0: return 0
    elif kA093729(n,k): return T(n-k,k)
flatten([[A093729(n,k) for k in range(n+1)] for n in range(16)]) # G. C. Greubel, Feb 22 2024

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.

Sum_{k=0..n} T(n-k, k) = A093730(n) (antidiagonal row sums).