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 A052685 Expansion of e.g.f. (1-x^2)/(1-x-2*x^2+x^4). 1
 1, 1, 4, 24, 168, 1680, 18720, 252000, 3830400, 65681280, 1251936000, 26225337600, 599710003200, 14851444608000, 396138155212800, 11320537003776000, 345079573622784000, 11176410365632512000 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 LINKS G. C. Greubel, Table of n, a(n) for n = 0..350 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 633 FORMULA E.g.f.: (1 - x^2)/(1 - x - 2*x^2 + x^4). D-finite recurrence: a(0)=1, a(1)=1, a(2)=4, a(3)=24, a(n) = n*a(n-1) + 2*n*(n-1)*a(n-2) - n*(n-1)*(n-2)*(n-3)*a(n-4). a(n) = (n!/283)*Sum_{alpha=RootOf(1-z-2*Z^2+Z^4)} (27 + 112*alpha + 9*alpha^2 - 48*alpha^3)*alpha^(-1-n). a(n) = n!*A052535(n). - R. J. Mathar, Nov 27 2011 MAPLE spec := [S, {S=Sequence(Prod(Z, Union(Z, Sequence(Prod(Z, Z)))))}, labeled]: seq(combstruct[count](spec, size=n), n=0..20); MATHEMATICA b[n_]:= b[n]= If[n<4, n!-2*Boole[n==3], b[n-1] +2*b[n-2] -b[n-4]]; (* b=A052535 *) A052685[n_]:= n!*b[n]; Table[A052685[n], {n, 0, 40}] (* G. C. Greubel, Jun 03 2022 *) With[{nn=20}, CoefficientList[Series[(1-x^2)/(1-x-2 x^2+x^4), {x, 0, nn}], x] Range[0, nn]!] PROG (Magma) R:=PowerSeriesRing(Rationals(), 40); Coefficients(R!(Laplace( (1-x^2)/(1-x-2*x^2+x^4) ))); // G. C. Greubel, Jun 03 2022 (SageMath) @CachedFunction def b(n): # b = A052535 if (n<4): return factorial(n) - 2*bool(n==3) else: return b(n-1) + 2*b(n-2) - b(n-4) def A052685(n): return factorial(n)*b(n) [A052685(n) for n in (0..40)] # G. C. Greubel, Jun 03 2022 CROSSREFS Cf. A000142, A052535. Sequence in context: A246423 A188913 A364324 * A221088 A365225 A347651 Adjacent sequences: A052682 A052683 A052684 * A052686 A052687 A052688 KEYWORD easy,nonn AUTHOR encyclopedia(AT)pommard.inria.fr, Jan 25 2000 STATUS approved

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Last modified September 27 00:54 EDT 2023. Contains 365669 sequences. (Running on oeis4.)