A165883 Irregular triangle T(n, k) = [x^k]( p(n, x) ), where p(n, x) = (1-x)^(2*n+3)*( Sum_{j >= 0} (2*j+ 1)^n*x^j )*( Sum_{j >= 0} j^(n+1)*x^j ), read by rows.

1, 1, 2, 1, 1, 10, 26, 10, 1, 1, 34, 287, 508, 287, 34, 1, 1, 102, 2272, 11098, 19134, 11098, 2272, 102, 1, 1, 294, 15493, 169432, 675706, 1042948, 675706, 169432, 15493, 294, 1, 1, 842, 98374, 2151026, 17138559, 55643460, 82178676, 55643460, 17138559, 2151026, 98374, 842, 1

Offset: 0

Author

Roger L. Bagula, Sep 29 2009

Keywords

• nonn
• tabf

Examples

			Irregular triangle begins as:
  1;
  1,   2,     1;
  1,  10,    26,     10,      1;
  1,  34,   287,    508,    287,      34,      1;
  1, 102,  2272,  11098,  19134,   11098,   2272,    102,     1;
  1, 294, 15493, 169432, 675706, 1042948, 675706, 169432, 15493, 294, 1;
		

Links

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

Crossrefs

Cf. A158782, A165889, A165890, A165891.

Programs

Mathematica

p[n_, x_]:= p[n, x]= (1/x)*(1-x)^(2*n+3)*Sum[(2*k+1)^n*x^k, {k,0,Infinity}]*Sum[k^(n+1)*x^k, {k,0,Infinity}];
Table[CoefficientList[p[n, x], x], {n,0,10}]//Flatten (* modified by G. C. Greubel, Mar 08 2022 *)

Sage

def p(n,x): return (1/x)*(1-x)^(2*n+3)*sum( (2*j+1)^n*x^j for j in (0..2*n+3) )*sum( j^(n+1)*x^j for j in (0..2*n+3) )
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)]) # G. C. Greubel, Mar 08 2022

Formula

T(n, k) = [x^k]( p(n, x) ), where p(n, x) = (1-x)^(2*n+3)*( Sum_{j >= 0} (2*j+ 1)^n*x^j )*( Sum_{j >= 0} j^(n+1)*x^j ).

T(n, k) = [x^k]( p(n, x) ), where p(n, x) = 2^n*(1-x)^(2*n+3)*LerchPhi(x, -n, 1/2)*PolyLog(-n-1, x)/x.

T(n, n-k) = T(n, k). - G. C. Greubel, Mar 08 2022

Extensions

Edited by G. C. Greubel, Mar 08 2022