A147618 The 3rd Witt transform of A000217.

0, 0, 0, 0, 3, 15, 54, 165, 429, 999, 2145, 4290, 8100, 14586, 25194, 41985, 67830, 106590, 163431, 245157, 360525, 520749, 740025, 1036035, 1430703, 1950975, 2629575, 3506085, 4628052, 6052068, 7845255, 10086780, 12869340, 16301142

Offset: 0

Author

R. J. Mathar, Nov 08 2008

Keywords

• easy
• nonn

Comments

The 2nd Witt transform of A000217 is essentially in A032092.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Pieter Moree, The formal series Witt transform, Discr. Math. no. 295 vol. 1-3 (2005) 143-160.
• Index entries for linear recurrences with constant coefficients, signature (6,-15,23,-33,51,-64,63,-63,64,-51,33,-23,15,-6,1).

Crossrefs

Cf. A000217, A032092, A049347, A099254, A128504.

Programs

Magma

R:=PowerSeriesRing(Integers(), 30); [0,0,0,0] cat Coefficients(R!( 3*x^4*(1-x+3*x^2-x^3+x^4)/((1-x)^9*(1+x+x^2)^3) )); // G. C. Greubel, Oct 24 2022

Mathematica

CoefficientList[Series[3*x^4*(1-x+3*x^2-x^3+x^4)/((1-x)^9*(1+x+x^2)^3), {x,0,40}], x] (* Vincenzo Librandi, Dec 13 2012 *)

SageMath

def A147618_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 3*x^4*(1-x+3*x^2-x^3+x^4)/((1-x)^9*(1+x+x^2)^3) ).list()
A147618_list(30) # G. C. Greubel, Oct 24 2022

Formula

G.f.: 3*x^4*(1-x+3*x^2-x^3+x^4)/((1-x)^9*(1+x+x^2)^3).

a(n) = (1/81)*(n*(n+3)*(3*n^6 +27*n^5 +45*n^4 -135*n^3 -288*n^2 +108*n -2000)/4480 +2*A049347(n) +A049347(n-1) +(-1)^n*(A099254(n) -2*A099254(n- 1)) -3*(-1)^n*(A128504(n) -2*A128504(n-1))). - G. C. Greubel, Oct 24 2022