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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 (list; table; graph; refs; listen; history; text; internal format)
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) = (1/4)*(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

Cf. A154853, A154854, A154855.

Sequence in context: A161014 A344768 A235712 * A088996 A211888 A293783

Adjacent sequences:  A154849 A154850 A154851 * A154853 A154854 A154855

KEYWORD

tabl,sign

AUTHOR

Roger L. Bagula, Jan 16 2009

EXTENSIONS

Edited by G. C. Greubel, Mar 11 2021

STATUS

approved

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Last modified September 25 20:01 EDT 2021. Contains 347659 sequences. (Running on oeis4.)