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URL: https://oeis.org/A154854

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A154854
Triangle of coefficients of p(x,n) = (1/2)*(1-x)^(n+1)*Sum_{m >= 0} ((4*m+3)^n - (4*m+1)^n)*x^m, read by rows.
4
0, 1, -1, 4, 0, -4, 13, 57, -57, -13, 40, 688, 0, -688, -40, 121, 6115, 11770, -11770, -6115, -121, 364, 48464, 270620, 0, -270620, -48464, -364, 1093, 363965, 4401033, 5613265, -5613265, -4401033, -363965, -1093, 3280, 2657568, 61590368, 199134880, 0, -199134880, -61590368, -2657568, -3280
OFFSET
0,4
COMMENTS
Row sums are zero.
FORMULA
Rows are coefficients of p(x,n) = (1/2)*(1-x)^(n+1)*Sum_{m >= 0} ((4*m+3)^n - (4*m+1)^n)*x^m.
EXAMPLE
Triangle begins as:
0;
1, -1;
4, 0, -4;
13, 57, -57, -13;
40, 688, 0, -688, -40;
121, 6115, 11770, -11770, -6115, -121;
364, 48464, 270620, 0, -270620, -48464, -364;
1093, 363965, 4401033, 5613265, -5613265, -4401033, -363965, -1093;
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, 2, 4, 3, 1], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Mar 11 2021 *)
PROG
(SageMath)
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, 2, 4, 3, 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
sign,tabl
AUTHOR
Roger L. Bagula, Jan 16 2009
EXTENSIONS
Edited by G. C. Greubel, Mar 11 2021
STATUS
approved