A156896 Triangle formed by coefficients of the expansion of p(x, n), where p(x,n) = (1+x-x^3)^(n+1)*Sum_{j >= 0} (j+1)^n*(-x + x^3)^j.

1, 1, 1, -1, 0, 1, 1, -4, 1, 4, -2, 0, 1, 1, -11, 11, 10, -22, 3, 11, -3, 0, 1, 1, -26, 66, 0, -131, 78, 62, -78, 6, 26, -4, 0, 1, 1, -57, 302, -245, -547, 905, 74, -901, 342, 292, -228, 10, 57, -5, 0, 1, 1, -120, 1191, -2296, -1191, 7128, -3572, -6648, 7140, 1216, -4749, 1200, 1171, -600, 15, 120, -6, 0, 1

Offset: 0

Author

Roger L. Bagula, Feb 17 2009

Keywords

• tabf
• sign

Comments

Row sums are one.

Examples

			Irregular triangle begins as:
  1;
  1;
  1,  -1,   0,    1;
  1,  -4,   1,    4,   -2,   0,  1;
  1, -11,  11,   10,  -22,   3, 11,   -3,   0,   1;
  1, -26,  66,    0, -131,  78, 62,  -78,   6,  26,   -4,  0,  1;
  1, -57, 302, -245, -547, 905, 74, -901, 342, 292, -228, 10, 57, -5, 0, 1;
		

Links

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

Crossrefs

Cf. A000295, A156890, A156901, A156918.

Programs

Mathematica

p[x_, n_] = (1+x-x^3)^(n+1)*Sum[(j+1)^n*(-x+x^3)^j, {j,0,Infinity}];
Table[CoefficientList[p[x, n], x], {n, 0, 10}]//Flatten

Sage

def T(n,k): return ( (1+x-x^3)^(n+1)*sum((j+1)^n*(x^3-x)^j for j in (0..3*n+1)) ).series(x, 3*n+3).list()[k]
flatten([1]+[[T(n,k) for k in (0..3*n-3)] for n in (1..12)]) # G. C. Greubel, Jan 06 2022

Formula

T(n, k) = coefficients of the expansion of p(x, n), where p(x,n) = (1+x-x^3)^(n + 1)*Sum_{j >= 0} (j+1)^n*(-x + x^3)^j.

T(n, 1) = (-1)*A000295(n) for n >= 2.

Extensions

Edited by G. C. Greubel, Jan 06 2022