A177970 Array T(n,m) = A177944(2*n,2*m) read by antidiagonals.

1, 1, 1, 1, 26, 1, 1, 99, 99, 1, 1, 244, 622, 244, 1, 1, 485, 2300, 2300, 485, 1, 1, 846, 6423, 12000, 6423, 846, 1, 1, 1351, 15001, 45031, 45031, 15001, 1351, 1, 1, 2024, 30924, 136120, 218774, 136120, 30924, 2024, 1, 1, 2889, 58122, 352698, 831384

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

Sequence in context: A040677 A040676 A225532 * A225483 A183065 A157630

Adjacent sequences: A177967 A177968 A177969 * A177971 A177972 A177973

Keyword

nonn,easy,tabl,look

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