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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!).

%I A271204 #20 Feb 26 2018 05:53:48
%S A271204 1,2,1,40,50,14,1,2240,4240,2200,440,36,1,246400,608960,447200,141520,
%T A271204 22080,1760,68,1,44844800,134780800,125843200,53412800,12015360,
%U A271204 1538320,114800,4900,110,1,12197785600,42767648000,47935328000,25213686400,7308806400,1268761760,137790240,9523920,416000,11050,162,1
%N 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!).
%C A271204 The n-th row sum gives A271049(n): Sum_{k=0..2*n}S(n,k) = A271049(n)
%F A271204 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)))
%e A271204 Example: S(n,k) in table form for n=0..4;
%e A271204 1
%e A271204 2,1
%e A271204 40,50,14,1
%e A271204 2240, 4240, 2200, 440, 36, 1
%e A271204 246400, 608960, 447200, 141520, 22080, 1760, 68, 1.
%p A271204 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;
%p A271204 for n from 1 to 6 do seq(round(evalf(S(n,kk))),kk=1..2*n) end do;
%p A271204 # The above Maple program reproduces the data without the initial value 1.
%Y A271204 Cf. A271049.
%K A271204 nonn,tabf
%O A271204 0,2
%A A271204 _Karol A. Penson_, Apr 01 2016