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, 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

G. C. Greubel, Antidiagonals n = 0..50, flattened

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

Adjacent sequences: A157047 A157048 A157049 * A157051 A157052 A157053

Keyword

nonn,tabl

Author

Roger L. Bagula, Feb 22 2009

Extensions

Edited by G. C. Greubel, Jan 23 2022

Status

approved