A061928 Array T(n,m) = 1/beta(n+1,m+1) read by antidiagonals.

6, 12, 12, 20, 30, 20, 30, 60, 60, 30, 42, 105, 140, 105, 42, 56, 168, 280, 280, 168, 56, 72, 252, 504, 630, 504, 252, 72, 90, 360, 840, 1260, 1260, 840, 360, 90, 110, 495, 1320, 2310, 2772, 2310, 1320, 495, 110, 132, 660, 1980, 3960, 5544, 5544, 3960

Offset

1,1

Comments

beta(n+1,m+1) = Integral_{x=0..1} x^n * (1-x)^m dx for real n, m.

References

G. Boole, A Treatise On The Calculus of Finite Differences, Dover, 1960, p. 26.

Links

Table of n, a(n) for n=1..52.

Formula

beta(n+1, m+1) = gamma(n+1)*gamma(m+1)/gamma(n+m+2) = n!*m!/(n+m+1)!.

Example

Antidiagonals:
   6,
  12, 12,
  20, 30, 20,
  30, 60, 60, 30,
  ...
Array:
   6  12  20   30   42
  12  30  60  105  168
  20  60 140  280  504
  30 105 280  630 1260
  42 168 504 1260 2772

Mathematica

t[n_, m_] := 1/Beta[n+1, m+1]; Take[ Flatten[ Table[ t[n+1-m, m], {n, 1, 10}, {m, 1, n}]], 52] (* Jean-François Alcover, Oct 11 2011 *)

Prog

(PARI) A(i, j)=if(i<1||j<1, 0, 1/subst(intformal(x^i*(1-x)^j), x, 1)) /* Michael Somos, Feb 05 2004 */
(PARI) A(i, j)=if(i<1||j<1, 0, 1/sum(k=0, i, (-1)^k*binomial(i, k)/(j+1+k))) /* Michael Somos, Feb 05 2004 */
(Python)
from sympy import factorial as f
def T(n, m): return f(n + m + 1)/(f(n)*f(m))
for n in range(1, 11): print([T(m, n - m + 1) for m in range(1, n + 1)]) # Indranil Ghosh, Apr 29 2017

Crossrefs

Rows: 1/b(n, 2): A002378, 1/b(n, 3): A027480, 1/b(n, 4): A033488. Diagonals: 1/b(n, n): A002457, 1/b(n, n+1) A005430, 1/b(n, n+2): A000917.

T(i, j)=A003506(i+1, j+1).

Sequence in context: A315582 A315583 A129858 * A315584 A315585 A256257

Adjacent sequences: A061925 A061926 A061927 * A061929 A061930 A061931

Keyword

nonn,tabl,easy,nice

Author

Frank Ellermann, May 22 2001

Status

approved