|
%I #24 Apr 16 2020 22:52:07
%S 6,12,12,20,30,20,30,60,60,30,42,105,140,105,42,56,168,280,280,168,56,
%T 72,252,504,630,504,252,72,90,360,840,1260,1260,840,360,90,110,495,
%U 1320,2310,2772,2310,1320,495,110,132,660,1980,3960,5544,5544,3960
%N Array T(n,m) = 1/beta(n+1,m+1) read by antidiagonals.
%C beta(n+1,m+1) = Integral_{x=0..1} x^n * (1-x)^m dx for real n, m.
%D G. Boole, A Treatise On The Calculus of Finite Differences, Dover, 1960, p. 26.
%F beta(n+1, m+1) = gamma(n+1)*gamma(m+1)/gamma(n+m+2) = n!*m!/(n+m+1)!.
%e Antidiagonals:
%e 6,
%e 12, 12,
%e 20, 30, 20,
%e 30, 60, 60, 30,
%e ...
%e Array:
%e 6 12 20 30 42
%e 12 30 60 105 168
%e 20 60 140 280 504
%e 30 105 280 630 1260
%e 42 168 504 1260 2772
%t 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 *)
%o (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 */
%o (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 */
%o (Python)
%o from sympy import factorial as f
%o def T(n, m): return f(n + m + 1)/(f(n)*f(m))
%o for n in range(1, 11): print([T(m, n - m + 1) for m in range(1, n + 1)]) # _Indranil Ghosh_, Apr 29 2017
%Y 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.
%Y T(i, j)=A003506(i+1, j+1).
%K nonn,tabl,easy,nice
%O 1,1
%A _Frank Ellermann_, May 22 2001
|
