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A154852
Triangle of coefficients of p(x,n) = (1/4)*(1-x)^(n+1)*Sum_{m >= 0} ((2*m- 1)^n - (2*m+3)^n)*x^m, read by rows.
4
0, -1, 1, -2, 0, 2, -7, -3, 3, 7, -20, -56, 0, 56, 20, -61, -415, -370, 370, 415, 61, -182, -2632, -5710, 0, 5710, 2632, 182, -547, -15155, -64407, -49735, 49735, 64407, 15155, 547, -1640, -82896, -619696, -1085840, 0, 1085840, 619696, 82896, 1640
OFFSET
0,4
COMMENTS
Row sums are zero.
FORMULA
Rows are coefficients of p(x,n) = (1/4)*(1-x)^(n+1)*Sum_{m >= 0} ((2*m-1)^n - (2*m+3)^n)*x^m.
EXAMPLE
Triangle begins as:
0;
-1, 1;
-2, 0, 2;
-7, -3, 3, 7;
-20, -56, 0, 56, 20;
-61, -415, -370, 370, 415, 61;
-182, -2632, -5710, 0, 5710, 2632, 182;
-547, -15155, -64407, -49735, 49735, 64407, 15155, 547;
-1640, -82896, -619696, -1085840, 0, 1085840, 619696, 82896, 1640;
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, 4, 2, -1, 3], {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, 4, 2, -1, 3, 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