A154853 Triangle of coefficients of p(x,n) = (1-x)^(n+1)*Sum_{m >= 0} ((3*m+2)^n - (3*m+1)^n)*x^m, read by rows.

0, 1, -1, 3, 0, -3, 7, 33, -33, -7, 15, 294, 0, -294, -15, 31, 1915, 3820, -3820, -1915, -31, 63, 11088, 65115, 0, -65115, -11088, -63, 127, 60725, 783237, 1019935, -1019935, -783237, -60725, -127, 255, 322794, 8095794, 26928930, 0, -26928930, -8095794, -322794, -255

Offset

0,4

Comments

Row sums are zero.

Links

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

Formula

Rows are coefficients of p(x,n) = (1-x)^(n+1)*Sum_{m >= 0} ((3*m+2)^n - (3*m+1)^n)*x^m.

Example

Triangle begins as:
    0;
    1,     -1;
    3,      0,      -3;
    7,     33,     -33,       -7;
   15,    294,       0,     -294,      -15;
   31,   1915,    3820,    -3820,    -1915,       -31;
   63,  11088,   65115,        0,   -65115,    -11088,      -63;
  127,  60725,  783237,  1019935, -1019935,   -783237,   -60725,    -127;
  255, 322794, 8095794, 26928930,        0, -26928930, -8095794, -322794, -255;

Mathematica

T[n_, k_, p_, q_, r_, t_]:= SeriesCoefficient[(1/p)*(1-x)^(n+1)*Sum[((q*j+r)^n - (q*j+t)^n )*x^j, {j, 0, n}], {x, 0, k}];
Table[T[n, k, 1, 3, 2, 1], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Mar 11 2021 *)

Prog

(Sage)
def f(n, p, q, r, t, x) : return (1/p)*(1-x)^(n+1)*sum( ((q*j+r)^n - (q*j+t)^n)*x^j for j in (0..n))
[[( f(n, 1, 3, 2, 1, x) ).series(x, n+1).list()[k] for k in (0..n)] for n in (0..12)] # G. C. Greubel, Mar 11 2021

Crossrefs

Cf. A154852, A154854, A154855.

Sequence in context: A074694 A127803 A021771 * A139214 A010030 A372339

Adjacent sequences: A154850 A154851 A154852 * A154854 A154855 A154856

Keyword

tabl,sign

Author

Roger L. Bagula, Jan 16 2009

Extensions

Edited by G. C. Greubel, Mar 11 2021

Status

approved