A052697 Expansion of e.g.f. 1/(1-x^3-x^4).

1, 0, 0, 6, 24, 0, 720, 10080, 40320, 362880, 10886400, 119750400, 958003200, 24908083200, 523069747200, 6538371840000, 125536739328000, 3556874280960000, 70426110763008000, 1338096104497152000

Offset: 0

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

• easy
• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..430
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 646

Crossrefs

Cf. A000142, A017817.

Programs

Magma

[Factorial(n)*(&+[Binomial(k,n-3*k): k in [0..Floor(n/3)]]): n in [0..30]]; // G. C. Greubel, May 31 2022

Maple

spec := [S,{S=Sequence(Prod(Z,Z,Union(Z,Prod(Z,Z))))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

Mathematica

With[{nn=20},CoefficientList[Series[1/(1-x^3-x^4),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Mar 06 2016 *)

SageMath

[factorial(n)*sum(binomial(k, n-3*k) for k in (0..n//3)) for n in (0..30)] # G. C. Greubel, May 31 2022

Formula

E.g.f.: 1/(1 - x^3 - x^4).

D-finite recurrence: a(0)=1, a(1)=0, a(2)=0, a(3)=6, a(n+4) = (24 + 26*n + 9*n^2 + n^3)*a(n+1) + (24 + 50*n + 35*n^2 + 10*n^3 + n^4)*a(n).

a(n) = (n!/283) * Sum_{alpha=RootOf(-1 + Z^3 + Z^4)} (- 16 - 73*alpha + 3*alpha^2 + 12*alpha^3)*alpha^(-1-n).

a(n) = n!*A017817(n). - R. J. Mathar, Nov 27 2011