A271204 Triangle of generalized Stirling numbers of the second kind S(n,k) associated with the generalized Bell numbers A271049(n); S(n,k) = Sum_{j=0..k} (-1)^(k-j)*binomial(k,j)*Gamma((j+2)/3)*Gamma(3*n+j-1)/(Gamma(j)*Gamma(n+(j-1)/3))/(3^(n-1)*k!).

1, 2, 1, 40, 50, 14, 1, 2240, 4240, 2200, 440, 36, 1, 246400, 608960, 447200, 141520, 22080, 1760, 68, 1, 44844800, 134780800, 125843200, 53412800, 12015360, 1538320, 114800, 4900, 110, 1, 12197785600, 42767648000, 47935328000, 25213686400, 7308806400, 1268761760, 137790240, 9523920, 416000, 11050, 162, 1

Offset: 0

Author

Karol A. Penson, Apr 01 2016

Keywords

• nonn
• tabf

Comments

The n-th row sum gives A271049(n): Sum_{k=0..2*n}S(n,k) = A271049(n)

Examples

			Example: S(n,k) in table form for n=0..4;
1
2,1
40,50,14,1
2240, 4240, 2200, 440, 36, 1
246400, 608960, 447200, 141520, 22080, 1760, 68, 1.
		

Crossrefs

Cf. A271049.

Programs

Maple

S:=proc(n,k) (((-1)^k*(3^(-n))*k/(4*(k!))) *(-12*GAMMA(3*n)*hypergeom([1/3-k/3,2/3-k/3,1-k/3,n+1/3,n+2/3],[1/3,2/3,2/3,4/3],1)/GAMMA(n)+6*(k-1)*GAMMA(4/3)*GAMMA(1+3*n)*hypergeom([2/3-k/3,1-k/3,4/3-k/3,2/3+n,n+1],[2/3,1,4/3,5/3],1)/GAMMA(n+1/3)-(k-2)*(k-1)*GAMMA(5/3)*GAMMA(3*n+2)*hypergeom([1-k/3,4/3-k/3,5/3-k/3,n+1,n+4/3],[4/3,4/3,5/3,2],1)/GAMMA(n+2/3)));end;
for n from 1 to 6 do seq(round(evalf(S(n,kk))),kk=1..2*n) end do;
# The above Maple program reproduces the data without the initial value 1.

Formula

Special values of generalized hypergeometric functions of type 5F4, in Maple notation: S(n,k) = (((-1)^k*(3^(-n))*k/(4*(k!))) *(-12*GAMMA(3*n)*hypergeom([1/3-k/3,2/3-k/3,1-k/3,n+1/3,n+2/3],[1/3,2/3,2/3,4/3],1)/GAMMA(n)+6*(k-1)*GAMMA(4/3)*GAMMA(1+3*n)*hypergeom([2/3-k/3,1-k/3,4/3-k/3,2/3+n,n+1],[2/3,1,4/3,5/3],1)/GAMMA(n+1/3)-(k-2)*(k-1)*GAMMA(5/3)*GAMMA(3*n+2)*hypergeom([1-k/3,4/3-k/3,5/3-k/3,n+1,n+4/3],[4/3,4/3,5/3,2],1)/GAMMA(n+2/3)))