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.

1, 1, 1, 1, 18, 42, 18, 1, 1, 115, 1539, 5065, 5065, 1539, 115, 1, 1, 612, 30369, 359056, 1439038, 2255448, 1439038, 359056, 30369, 612, 1, 1, 3109, 487944, 16069256, 177275075, 808273143, 1688579472, 1688579472, 808273143, 177275075, 16069256, 487944, 3109, 1

Offset: 0

Author

Roger L. Bagula, Mar 14 2010

Keywords

• nonn
• tabf

Examples

			Irregular triangle begins as:
  1;
  1,   1;
  1,  18,    42,     18,       1;
  1, 115,  1539,   5065,    5065,    1539,     115,      1;
  1, 612, 30369, 359056, 1439038, 2255448, 1439038, 359056, 30369, 612, 1;
		

Links

• G. C. Greubel, Rows n = 0..50 of the irregular triangle, flattened

Crossrefs

Cf. A060187, A154283.

Programs

Mathematica

(* First program *)
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];
Table[CoefficientList[p[n, x], x], {n,0,10}]//Flatten
(* Second program *)
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}]];
Join[{1}, Table[T[n, k], {n,0,10}, {k,0,3*n-2}]//Flatten] (* G. C. Greubel, Mar 25 2022 *)

Sage

@CachedFunction
def T(n,k):
    if (n<2): return binomial(n,k)
    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) )
[1]+flatten([[T(n,k) for k in (0..3*n-2)] for n in (0..10)]) # G. C. Greubel, Mar 25 2022

Formula

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.

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

Extensions

Edited by G. C. Greubel, Mar 25 2022