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

Author

R. J. Mathar, Nov 08 2008

Keywords

• easy
• nonn

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).

Crossrefs

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

Programs

Magma

R:=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

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 *)

SageMath

def A147621_list(prec):
    P. = 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

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