A165889 Irregular triangle T(n, k) = [x^k]( p(n, x) ), where p(n, x) = (1-x)^(2*n+4)*( Sum_{j >= 0} j^(n+1)*x^j )^2/x^2, read by rows.

1, 1, 2, 1, 1, 8, 18, 8, 1, 1, 22, 143, 244, 143, 22, 1, 1, 52, 808, 3484, 5710, 3484, 808, 52, 1, 1, 114, 3853, 35032, 125746, 188908, 125746, 35032, 3853, 114, 1, 1, 240, 16782, 290672, 2000703, 6040992, 8702820, 6040992, 2000703, 290672, 16782, 240, 1

Offset: 0

Author

Roger L. Bagula, Sep 29 2009

Keywords

• nonn
• tabf

Examples

			Irregular triangle begins as:
  1;
  1,   2,    1;
  1,   8,   18,     8,      1;
  1,  22,  143,   244,    143,     22,      1;
  1,  52,  808,  3484,   5710,   3484,    808,    52,    1;
  1, 114, 3853, 35032, 125746, 188908, 125746, 35032, 3853, 114, 1;
		

Links

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

Crossrefs

Cf. A165883, A165890, A165891.

Programs

Mathematica

p[n_, x_]:= p[n, x]= (1/x^2)*(1-x)^(2*n+4)*Sum[k^(n+1)*x^k, {k, 0, Infinity}]^2;
Table[CoefficientList[p[n, x], x], {n,0,12}]//Flatten (* modified by G. C. Greubel, Mar 09 2022 *)

Sage

def p(n,x): return (1/x^2)*(1-x)^(2*n+4)*sum( j^(n+1)*x^j for j in (0..2*n+4) )^2
def T(n,k): return ( p(n,x) ).series(x, 2*n+1).list()[k]
flatten([[T(n,k) for k in (0..2*n)] for n in (0..12)])

Formula

T(n, k) = [x^k]( p(n, x) ), where p(n, x) = (1-x)^(2*n+4)*( Sum_{j >= 0} j^(n+1)*x^j )^2/x^2.

T(n, k) = [x^k]( p(n, x) ), where p(n, x) = (1-x)^(2*n+4)*( PolyLog(-n-1, x)/x)^2.

T(n, n-k) = T(n, k). - G. C. Greubel, Mar 09 2022

Extensions

Edited by G. C. Greubel, Mar 09 2022