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

Author

Frank Ellermann, May 22 2001

Keywords

• nonn
• tabl
• easy
• nice

Comments

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

Examples

			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
		

References

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

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).

Programs

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 *)

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

Formula

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