OFFSET
0,8
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.
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.
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).
EXAMPLE
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;
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. *)
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 k<n+1: return T(n, k-1) - T(n-1, k) + T(n-1, 2*k-1) + T(n-1, 2*k)
else: return sum((-1)^(j-1)*binomial(n+1, j)*T(n, k-j) for j in range(1, n+2))
def A093729(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
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Apr 14 2004; revised Oct 14 2005
STATUS
approved