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.

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.

Links

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

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

(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, 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

Cf. A154852, A154853, A154855.

Sequence in context: A281297 A058536 A371377 * A151672 A058493 A112149

Adjacent sequences: A154851 A154852 A154853 * A154855 A154856 A154857

Keyword

sign,tabl

Author

Roger L. Bagula, Jan 16 2009

Extensions

Edited by G. C. Greubel, Mar 11 2021

Status

approved