OFFSET
0,5
COMMENTS
Antidiagonal sums are 1, 2, 28, 200, 1112, 5572, 26540, 122768, 556912, 2490188, ... = 4^d*(d+1/2)-2*d(d+1), d > 0.
LINKS
Robert Israel, Table of n, a(n) for n = 0..10000
FORMULA
T(n,m) = 1/Beta(2*n+1, 2*m+1) - 2*n - 2*m where Beta(a,b) = Gamma(a)*Gamma(b)/Gamma(a+b).
EXAMPLE
The table starts in row n=0, column m=0 as:
1, 1, 1, 1, 1, 1, 1, 1,
1, 26, 99, 244, 485, 846, 1351, 2024,
1, 99, 622, 2300, 6423, 15001, 30924, 58122,
1, 244, 2300, 12000, 45031, 136120, 352698, 813940,
1, 485, 6423, 45031, 218774, 831384, 2645350, 7354688,
1, 846, 15001, 136120, 831384, 3879856, 14872836, 49031376,
1, 1351, 30924, 352698, 2645350, 14872836, 67603876, 260757874,
1, 2024, 58122, 813940, 7354688, 49031376, 260757874,1163381372,
MAPLE
T:= (m, n) -> (2*n+1)*binomial(2*m+1+2*n, 2*m)-2*n-2*m:
seq(seq(T(m, s-m), m=0..s), s=0..10); # Robert Israel, Jul 06 2017
MATHEMATICA
t[n_, m_] = 1/Beta[2*n + 1, 2*m + 1] - 2*n - 2*m;
a = Table[Table[t[n, m], {m, 0, 10}], {n, 0, 10}];
Table[Table[a[[m, n - m + 1]], {m, 1, n}], {n, 1, 10}];
Flatten[%]
PROG
(Python)
from sympy import binomial
def T(m, n): return (2*n + 1)*binomial(2*m + 1 + 2*n, 2*m) - 2*n - 2*m
for n in range(11): print([T(m, n - m) for m in range(n + 1)]) # Indranil Ghosh, Jul 06 2017
CROSSREFS
AUTHOR
Roger L. Bagula, May 16 2010
EXTENSIONS
Definition rewritten with A177944, examples brought into normal form, closed sum formula - The Assoc. Eds. of the OEIS, Nov 03 2010
STATUS
approved