A156890 Triangle formed by coefficients of the expansion of p(x, n), where p(x,n) = (1+x-x^2)^(n+1)*Sum_{j >= 0} (j+1)^n*(-x + x^2)^j.

1, 1, 1, -1, 1, 1, -4, 5, -2, 1, 1, -11, 22, -23, 14, -3, 1, 1, -26, 92, -158, 145, -82, 32, -4, 1, 1, -57, 359, -906, 1265, -1135, 649, -238, 67, -5, 1, 1, -120, 1311, -4798, 9630, -12132, 10163, -5970, 2406, -620, 135, -6, 1, 1, -247, 4540, -24205, 66769, -113626, 131045, -106889, 62261, -26426, 8033, -1517, 268, -7, 1

Offset: 0

Author

Roger L. Bagula, Feb 17 2009

Keywords

• tabf
• sign

Comments

Row sums are equal to 1.

Examples

			Irregular triangle begins as:
  1;
  1;
  1,   -1,    1;
  1,   -4,    5,    -2,    1;
  1,  -11,   22,   -23,   14,     -3,     1;
  1,  -26,   92,  -158,  145,    -82,    32,    -4,    1;
  1,  -57,  359,  -906, 1265,  -1135,   649,  -238,   67,   -5,   1;
  1, -120, 1311, -4798, 9630, -12132, 10163, -5970, 2406, -620, 135, -6, 1;
		

Links

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

Crossrefs

Cf. A000295, A156896, A156901, A156918.

Programs

Mathematica

p[x_, n_]:= ((1+x-x^2)^(n+1))*Sum[(j+1)^n*(-x+x^2)^j, {j,0,Infinity}];
Table[CoefficientList[p[x, n], x], {n, 0, 10}]//Flatten

Sage

def T(n,k): return ( (1+x-x^2)^(n+1)*sum((j+1)^n*(x^2-x)^j for j in (0..2*n+1)) ).series(x, 2*n+3).list()[k]
[1]+flatten([[T(n,k) for k in (0..2*n-2)] for n in (0..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^2)^(n + 1)*Sum_{j >= 0} (j+1)^n*(-x + x^2)^j.

T(n, 1) = (-1)*A000295(n) for n >= 2. - G. C. Greubel, Jan 06 2022

Extensions

Edited by G. C. Greubel, Jan 06 2022