A156918 Triangle formed by coefficients of the expansion of p(x,n) = (1+x-x^2)^(n+1)*Sum_{j >= 0} (2*j+1)^n*(-x + x^2)^j.

1, 1, -1, 1, 1, -6, 7, -2, 1, 1, -23, 46, -47, 26, -3, 1, 1, -76, 306, -536, 459, -232, 82, -4, 1, 1, -237, 1919, -5046, 6965, -5995, 3109, -958, 247, -5, 1, 1, -722, 11265, -44634, 91730, -113538, 90417, -49398, 17778, -3630, 737, -6, 1, 1, -2179, 62836, -381037, 1099549, -1878718, 2123525, -1658537, 898985, -346886, 93377, -13109, 2200, -7, 1

Offset: 0

Author

Roger L. Bagula, Feb 18 2009

Keywords

• sign
• tabf

Comments

Row sums are one.

Examples

			Irregular triangle begins as:
  1;
  1,   -1,     1;
  1,   -6,     7,     -2,     1;
  1,  -23,    46,    -47,    26,      -3,     1;
  1,  -76,   306,   -536,   459,    -232,    82,     -4,     1;
  1, -237,  1919,  -5046,  6965,   -5995,  3109,   -958,   247,    -5,   1;
  1, -722, 11265, -44634, 91730, -113538, 90417, -49398, 17778, -3630, 737, -6, 1;
		

Links

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

Crossrefs

Cf. A060188, A156896, A156890, A156901.

Programs

Mathematica

p[x_, n_] = (1+x-x^2)^(n+1)*Sum[(2*k+1)^n*(-x+x^2)^k, {k, 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((2*j+1)^n*(x^2-x)^j for j in (0..2*n+1)) ).series(x, 2*n+2).list()[k]
flatten([1]+[[T(n, k) for k in (0..2*n)] for n in (1..12)]) # G. C. Greubel, Jan 07 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} (2*j+1)^n*(-x + x^2)^j.

T(n, 1) = (-1)*A060188(n), for n >= 2. - G. C. Greubel, Jan 07 2022

Extensions

Edited by G. C. Greubel, Jan 07 2022