%I #15 Jan 24 2022 07:08:29
%S 1,3,3,20,12,20,140,75,75,140,1449,588,495,588,1449,15939,5859,3780,
%T 3780,5859,15939,226512,68904,38880,30240,38880,68904,226512,3397680,
%U 953667,449955,306180,306180,449955,953667,3397680
%N Array A(n, k) = (1/2)*(2*n+1)!!*(2*k+1)!!*Integral_{x=-1..1} (1-x^(n+1))*(1-x^(k+1))/(1-x)^2 dx, read by antidiagonals.
%H G. C. Greubel, <a href="/A157050/b157050.txt">Antidiagonals n = 0..50, flattened</a>
%F A(n, k) = (1/2)*(2*n+1)!!*(2*k+1)!!*Integral_{x=-1..1} (1-x^(n+1))*(1-x^(k+1))/(1-x)^2 dx.
%F T(n, k) = A(k, n-k).
%F From _G. C. Greubel_, Jan 23 2022: (Start)
%F A(n, k) = ((2*n+1)!*(2*k+1)!/(2^(n+k+1)*n!*k!))*( (n+k+2)*psi(n+k+3) - (n+1)*psi(n+1) - (k+1)*psi(k+1) - Sum_{j=0..n} Sum_{i=0..k} (-1)^(j+i+1)/(i+j+1) ).
%F T(n, k) = (((2*(n-k)+1)!*(2*k+1)!)/(2^(n+1)*n!*(n-k)!)*( (n+2)*psi(n+3) - (k+1)*psi(k+2) - (n-k+1)*psi(n-k+2) - Sum_{j=0..k} Sum_{i=0..n-k} (-1)^(j+i+1)/(j+i+1) ), where psi(x) = digamma(x).
%F T(n, n-k) = T(n, k). (End)
%e Array begins as:
%e 1, 3, 20, 140, 1449, 15939, 226512 ...;
%e 3, 12, 75, 588, 5859, 68904, 953667 ...;
%e 20, 75, 495, 3780, 38880, 449955, 6364215 ...;
%e 140, 588, 3780, 30240, 306180, 3645180, 50990940 ...;
%e 1449, 5859, 38880, 306180, 3161025, 37266075, 528500700 ...;
%e 15939, 68904, 449955, 3645180, 37266075, 447192900, 6297966675 ...;
%e 226512, 953667, 6364215, 50990940, 528500700, 6297966675, 89576261775 ...;
%e Triangle begins as:
%e 1;
%e 3, 3;
%e 20, 12, 20;
%e 140, 75, 75, 140;
%e 1449, 588, 495, 588, 1449;
%e 15939, 5859, 3780, 3780, 5859, 15939;
%e 226512, 68904, 38880, 30240, 38880, 68904, 226512;
%e 3397680, 953667, 449955, 306180, 306180, 449955, 953667, 3397680;
%t A[n_, k_]:= (1/2)*(2*n+1)!!*(2*k+1)!!*Integrate[(1-x^(n+1))*(1-x^(k+1))/(1-x)^2, {x, -1, 1}];
%t T[n_, k_]:= A[k,n-k];
%t Table[T[n,k], {n,0,10}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Jan 23 2022 *)
%o (Magma)
%o F:=Factorial;
%o S:= func< n,k | (&+[ (&+[ (-1)^(j+i+1)/(j+i+1): i in [0..k]]): j in [0..n]]) >;
%o A:= func< n,k | Round((F(2*n+1)*F(2*k+1)/(2^(n+k+1)*F(n)*F(k)))*( (n+k+2)*Psi(n+k+3) - (n+1)*Psi(n+2) - (k+1)*Psi(k+2) - S(n,k) )) >;
%o [[A(k, n-k): k in [0..n]]: n in [0..10]]; // _G. C. Greubel_, Jan 23 2022
%o (Sage)
%o f=factorial
%o def s(n,k): return sum(sum( (-1)^(j+i+1)/(j+i+1) for i in (0..k)) for j in (0..n))
%o def A157050(n,k): return round((f(2*n+1)*f(2*k+1)/(2^(n+k+1)*f(n)*f(k)))*( (n+k+2)*psi(n+k+3) - (n+1)*psi(n+2) - (k+1)*psi(k+2) - s(n,k) ))
%o [[A157050(k,n-k) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Jan 23 2022
%Y Cf. A001147.
%K nonn,tabl
%O 0,2
%A _Roger L. Bagula_, Feb 22 2009
%E Edited by _G. C. Greubel_, Jan 23 2022
|