A165890 Irregular triangle T(n, k) = [x^k]( p(n, x) ), where p(n, x) = ( (1-x)^(n+1) * Sum_{k >= 0} (2*k+1)^(n-1)*x^k )^2, read by rows.

1, 1, -2, 1, 1, 0, -2, 0, 1, 1, 10, 15, -52, 15, 10, 1, 1, 44, 484, -44, -970, -44, 484, 44, 1, 1, 150, 5933, 22792, 466, -58684, 466, 22792, 5933, 150, 1, 1, 472, 58586, 682040, 2085135, -682512, -4287444, -682512, 2085135, 682040, 58586, 472, 1

Offset: 0

Author

Roger L. Bagula, Sep 29 2009

Keywords

• sign
• tabf

Examples

			Irregular triangle begins as:
  1;
  1,  -2,    1;
  1,   0,   -2,     0,    1;
  1,  10,   15,   -52,   15,     10,   1;
  1,  44,  484,   -44, -970,    -44, 484,    44,    1;
  1, 150, 5933, 22792,  466, -58684, 466, 22792, 5933, 150, 1;
		

Links

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

Crossrefs

Cf. A000007 (row sums), A158782, A165883, A165889, A165891.

Programs

Mathematica

p[n_, x_]:= p[n, x]= If[n==0, 1, (2^(n-1)*(1-x)^(n+1)*LerchPhi[x, -n+1, 1/2])^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*n+2)*sum( (2*j+1)^(n-1)*x^j for j in (0..2*n+2) )^2
def T(n,k): return ( p(n,x) ).series(x, 2*n+2).list()[k]
flatten([[T(n,k) for k in (0..2*n)] for n in (0..12)]) # G. C. Greubel, Mar 09 2022

Formula

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

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

Sum_{k=0..n} T(n, k) = 0^n.

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

Extensions

Edited by G. C. Greubel, Mar 09 2022