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A174264 Irregular triangle T(n, k) = [x^k]( p(n, x) ), where p(n, x) = (1/x)*(1-x)^(3*n+1)*Sum_{k >= 0} (k*(k+1)*(2*k+1)/6)^n*x^k and p(0, x) = 1, read by rows.

%I A174264 #6 Mar 25 2022 23:15:31
%S A174264 1,1,1,1,18,42,18,1,1,115,1539,5065,5065,1539,115,1,1,612,30369,
%T A174264 359056,1439038,2255448,1439038,359056,30369,612,1,1,3109,487944,
%U A174264 16069256,177275075,808273143,1688579472,1688579472,808273143,177275075,16069256,487944,3109,1
%N A174264 Irregular triangle T(n, k) = [x^k]( p(n, x) ), where p(n, x) = (1/x)*(1-x)^(3*n+1)*Sum_{k >= 0} (k*(k+1)*(2*k+1)/6)^n*x^k and p(0, x) = 1, read by rows.
%H A174264 G. C. Greubel, <a href="/A174264/b174264.txt">Rows n = 0..50 of the irregular triangle, flattened</a>
%F A174264 T(n, k) = [x^k]( p(n, x) ), where p(n, x) = (1/x)*(1-x)^(3*n+1)*Sum_{k >= 0} (k*(k+1)*(2*k+1)/6)^n*x^k and p(0, x) = 1.
%F A174264 T(n, k) = Sum_{j=0..k+1} (-1)^(k-j+1)*binomial(3*n+1, k-j+1)*(j*(1+j)*(1+2*j)/6)^n, with T(0, k) = T(1, k) = 1. - _G. C. Greubel_, Mar 25 2022
%e A174264 Irregular triangle begins as:
%e A174264   1;
%e A174264   1,   1;
%e A174264   1,  18,    42,     18,       1;
%e A174264   1, 115,  1539,   5065,    5065,    1539,     115,      1;
%e A174264   1, 612, 30369, 359056, 1439038, 2255448, 1439038, 359056, 30369, 612, 1;
%t A174264 (* First program *)
%t A174264 p[n_, x_]:= p[n,x]= If[n==0, 1, (1-x)^(3*n+1)*Sum[(k*(k+1)*(2*k+1)/6)^n*x^k, {k, 0, Infinity}]/x];
%t A174264 Table[CoefficientList[p[n, x], x], {n,0,10}]//Flatten
%t A174264 (* Second program *)
%t A174264 T[n_, k_]:= T[n, k]= If[n<2, Binomial[n, k], Sum[(-1)^(k-j+1)*Binomial[3*n+1, k-j +1]*(j*(1+j)*(1+2*j)/6)^n, {j,0,k+1}]];
%t A174264 Join[{1}, Table[T[n, k], {n,0,10}, {k,0,3*n-2}]//Flatten] (* _G. C. Greubel_, Mar 25 2022 *)
%o A174264 (Sage)
%o A174264 @CachedFunction
%o A174264 def T(n,k):
%o A174264     if (n<2): return binomial(n,k)
%o A174264     else: return sum( (-1)^(k-j+1)*binomial(3*n+1, k-j+1)*(j*(1+j)*(1+2*j)/6)^n for j in (0..k+1) )
%o A174264 [1]+flatten([[T(n,k) for k in (0..3*n-2)] for n in (0..10)]) # _G. C. Greubel_, Mar 25 2022
%Y A174264 Cf. A060187, A154283.
%K A174264 nonn,tabf
%O A174264 0,5
%A A174264 _Roger L. Bagula_, Mar 14 2010
%E A174264 Edited by _G. C. Greubel_, Mar 25 2022