A147621 The 3rd Witt transform of A000292.

0, 0, 0, 0, 4, 26, 120, 455, 1456, 4122, 10608, 25194, 55980, 117572, 235144, 450681, 832048, 1485800, 2575368, 4345965, 7158060, 11532402, 18209100, 28224105, 43008120, 64512240, 95365920, 139075245, 200268432, 284997384, 401107356

Offset

0,5

Comments

The 2nd Witt transform is essentially in A032094.

Links

G. C. Greubel, 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 (8,-28,60,-102,168,-258,336,-393,452,-484,452,-393, 336,-258,168,-102,60,-28,8,-1).

Formula

G.f.: x^4*(2-x+2*x^2)*(2-2*x+9*x^2-2*x^3+2*x^4)/((1-x)^12*(1+x+x^2)^4).

a(n) = (1/729)*(b(n) + c(n)), where b(n) = n*(n+3)*(n+6)*(3*n^8 +72*n^7 +618*n^6 + 2052*n^5 +207*n^4 -11772*n^3 -14268*n^2 +9648*n -232960)/492800 and c(n) = 9*A049347(n) +5*A049347(n-1) +9*(-1)^n*(A099254(n) -A099254(n-1)) -18(-1)^n*A128504(n) +27*(-1)^n*Sum_{k=0..n} A099254(n-k)*A099254(k-1). - G. C. Greubel, Oct 24 2022

Mathematica

CoefficientList[Series[x^4(2*x^2 - x + 2)(2*x^4 - 2*x^3 + 9*x^2 - 2*x+2)/((1-x)^12 * (1 + x + x^2)^4), {x, 0, 40}],  x] (* Vincenzo Librandi  Dec 13 2012 *)

Prog

(Magma) R<x>:=PowerSeriesRing(Integers(), 40); [0, 0, 0, 0] cat Coefficients(R!( x^4*(2-x+2*x^2)*(2-2*x+9*x^2-2*x^3+2*x^4)/((1-x)^12*(1+x+x^2)^4) )); // G. C. Greubel, Oct 24 2022
(SageMath)
def A147621_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    return P( x^4*(2-x+2*x^2)*(2-2*x+9*x^2-2*x^3+2*x^4)/((1-x)^12*(1+x+x^2)^4) ).list()
A147621_list(40) # G. C. Greubel, Oct 24 2022

Crossrefs

Cf. A000292, A032094, A049347, A099254, A128504.

Sequence in context: A222663 A267675 A140804 * A246854 A267847 A211166

Adjacent sequences: A147618 A147619 A147620 * A147622 A147623 A147624

Keyword

easy,nonn

Author

R. J. Mathar, Nov 08 2008

Status

approved