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A061928 Array T(n,m) = 1/beta(n+1,m+1) read by antidiagonals. 6
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

beta(n+1,m+1) = integral x^n * (1-x)^m dx from 0 to 1 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 xrange(1, 11): print [T(m, n - m + 1) for m in xrange(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

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Last modified November 14 14:43 EST 2019. Contains 329126 sequences. (Running on oeis4.)