A174264 - OEIS

%I #6 Mar 25 2022 23:15:31

%S 1,1,1,1,18,42,18,1,1,115,1539,5065,5065,1539,115,1,1,612,30369,

%T 359056,1439038,2255448,1439038,359056,30369,612,1,1,3109,487944,

%U 16069256,177275075,808273143,1688579472,1688579472,808273143,177275075,16069256,487944,3109,1

%N 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 G. C. Greubel, <a href="/A174264/b174264.txt">Rows n = 0..50 of the irregular triangle, flattened</a>

%F 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 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 Irregular triangle begins as:

%e   1;

%e   1,   1;

%e   1,  18,    42,     18,       1;

%e   1, 115,  1539,   5065,    5065,    1539,     115,      1;

%e   1, 612, 30369, 359056, 1439038, 2255448, 1439038, 359056, 30369, 612, 1;

%t (* First program *)

%t 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 Table[CoefficientList[p[n, x], x], {n,0,10}]//Flatten

%t (* Second program *)

%t 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 Join[{1}, Table[T[n, k], {n,0,10}, {k,0,3*n-2}]//Flatten] (* _G. C. Greubel_, Mar 25 2022 *)

%o (Sage)

%o @CachedFunction

%o def T(n,k):

%o     if (n<2): return binomial(n,k)

%o     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 [1]+flatten([[T(n,k) for k in (0..3*n-2)] for n in (0..10)]) # _G. C. Greubel_, Mar 25 2022

%Y Cf. A060187, A154283.

%K nonn,tabf

%O 0,5

%A _Roger L. Bagula_, Mar 14 2010

%E Edited by _G. C. Greubel_, Mar 25 2022