A156901 Triangle formed by coefficients of the expansion of p(x, n), where p(x,n) = (1 + 2*x - x^2)^(n + 1)*Sum_{j >= 0} (j+1)^n*(-2*x + x^2)^j.

1, 1, 1, -2, 1, 1, -8, 8, -4, 1, 1, -22, 55, -52, 23, -6, 1, 1, -52, 290, -472, 394, -188, 50, -8, 1, 1, -114, 1265, -3624, 4838, -3668, 1750, -536, 97, -10, 1, 1, -240, 4884, -24092, 49239, -56448, 40664, -19320, 6231, -1360, 180, -12, 1, 1, -494, 17419, -142124, 441625, -730898, 749723, -515944, 247067, -83122, 19673, -3244, 331, -14, 1

Offset

0,4

Links

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

Formula

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

T(n, 1) = (-1)*A005803(n) for n >= 2.

Example

Irregular triangle begins as:
  1;
  1;
  1,   -2,    1;
  1,   -8,    8,     -4,     1;
  1,  -22,   55,    -52,    23,     -6,     1;
  1,  -52,  290,   -472,   394,   -188,    50,     -8,    1;
  1, -114, 1265,  -3624,  4838,  -3668,  1750,   -536,   97,   -10,   1;
  1, -240, 4884, -24092, 49239, -56448, 40664, -19320, 6231, -1360, 180, -12, 1;

Mathematica

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

Prog

(Sage)
def T(n, k): return ( (1+2*x-x^2)^(n+1)*sum((j+1)^n*(x^2-2*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-2)] for n in (1..12)]) # G. C. Greubel, Jan 07 2022

Crossrefs

Cf. A005803, A156890, A156896, A156918.

Sequence in context: A051428 A176698 A129276 * A167400 A225848 A165889

Adjacent sequences: A156898 A156899 A156900 * A156902 A156903 A156904

Keyword

sign,tabf

Author

Roger L. Bagula, Feb 17 2009

Extensions

Edited by G. C. Greubel, Jan 07 2022

Status

approved