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A157050
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.
1
1, 3, 3, 20, 12, 20, 140, 75, 75, 140, 1449, 588, 495, 588, 1449, 15939, 5859, 3780, 3780, 5859, 15939, 226512, 68904, 38880, 30240, 38880, 68904, 226512, 3397680, 953667, 449955, 306180, 306180, 449955, 953667, 3397680
OFFSET
0,2
LINKS
FORMULA
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.
T(n, k) = A(k, n-k).
From G. C. Greubel, Jan 23 2022: (Start)
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) ).
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).
T(n, n-k) = T(n, k). (End)
EXAMPLE
Array begins as:
1, 3, 20, 140, 1449, 15939, 226512 ...;
3, 12, 75, 588, 5859, 68904, 953667 ...;
20, 75, 495, 3780, 38880, 449955, 6364215 ...;
140, 588, 3780, 30240, 306180, 3645180, 50990940 ...;
1449, 5859, 38880, 306180, 3161025, 37266075, 528500700 ...;
15939, 68904, 449955, 3645180, 37266075, 447192900, 6297966675 ...;
226512, 953667, 6364215, 50990940, 528500700, 6297966675, 89576261775 ...;
Triangle begins as:
1;
3, 3;
20, 12, 20;
140, 75, 75, 140;
1449, 588, 495, 588, 1449;
15939, 5859, 3780, 3780, 5859, 15939;
226512, 68904, 38880, 30240, 38880, 68904, 226512;
3397680, 953667, 449955, 306180, 306180, 449955, 953667, 3397680;
MATHEMATICA
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[n_, k_]:= A[k, n-k];
Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Jan 23 2022 *)
PROG
(Magma)
F:=Factorial;
S:= func< n, k | (&+[ (&+[ (-1)^(j+i+1)/(j+i+1): i in [0..k]]): j in [0..n]]) >;
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) )) >;
[[A(k, n-k): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Jan 23 2022
(Sage)
f=factorial
def s(n, k): return sum(sum( (-1)^(j+i+1)/(j+i+1) for i in (0..k)) for j in (0..n))
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) ))
[[A157050(k, n-k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Jan 23 2022
CROSSREFS
Cf. A001147.
Sequence in context: A101617 A131443 A304562 * A059368 A090694 A078431
KEYWORD
nonn,tabl
AUTHOR
Roger L. Bagula, Feb 22 2009
EXTENSIONS
Edited by G. C. Greubel, Jan 23 2022
STATUS
approved