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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.
4
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.
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
KEYWORD
tabl,sign
AUTHOR
Roger L. Bagula, Jan 16 2009
EXTENSIONS
Edited by G. C. Greubel, Mar 11 2021
STATUS
approved