A053532 -id:A053532 - cp's OEIS frontend

A053533 Expansion of e.g.f.: (1-x)^(-1/2)*exp(-x/2 -x^2/4 -x^3/6 -x^4/8).

1, 0, 0, 0, 0, 12, 60, 360, 2520, 20160, 199584, 2147040, 25043040, 315485280, 4274281440, 62237343168, 968728662720, 16046598597120, 281802435747840, 5229395457937920, 102253297006250496, 2101387824575550720, 45281611027331723520

Offset: 0

N. J. A. Sloane, Jan 16 2000

nonn

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.15(a), k=5.

G. C. Greubel, Table of n, a(n) for n = 0..445

Cf. A001205, A053532.

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( (1-x)^(-1/2)*Exp(-x/2 -x^2/4 -x^3/6 -x^4/8) )); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, May 15 2019

With[{m = 30}, CoefficientList[Series[(1-x)^(-1/2)*Exp[-x/2 -x^2/4 -x^3/6 -x^4/8], {x, 0, m}], x]*Range[0, m]!] (* G. C. Greubel, May 15 2019 *)

my(x='x+O('x^30)); Vec(serlaplace( (1-x)^(-1/2)*exp(-x/2 -x^2/4 -x^3/6 -x^4/8) )) \\ G. C. Greubel, May 15 2019

m = 30; T = taylor((1-x)^(-1/2)*exp(-x/2 -x^2/4 -x^3/6 -x^4/8), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # G. C. Greubel, May 15 2019

a(n) ~ sqrt(2) * n^n / exp(n+25/24). - Vaclav Kotesovec, Aug 04 2014

A202065 The number of simple labeled graphs on 2n nodes whose connected components are even length cycles.

Original entry on oeis.org

1, 0, 3, 60, 2835, 219240, 25519725, 4169185020, 910363278825, 256123949281200, 90240816705714675, 38923077574032151500, 20174526711617730727275, 12373285262231460281715000, 8863077725980930704895768125, 7332455066541096999983523547500

Offset: 0

Geoffrey Critzer, Dec 10 2011

nonn

Robert Israel, Table of n, a(n) for n = 0..225

Cf. A001205, A053532, A053533.

f:= gfun:-rectoproc({(4*n^3-n)*a(n-1) + (4*n^2+2*n)*a(n) - a(n+1)=0,a(0)=1,a(1)=0},a(n),remember):
map(f, [$0..30]); # Robert Israel, Mar 02 2017

nn = 30; a = Log[1/(1 - x^2)^(1/4)] - x^2/4; Table[i, {i, 0, nn, 2}]! CoefficientList[Series[Exp[a], {x, 0, nn}], x][[Table[i, {i, 1, nn+1, 2}]]]
Table[((2 n)!/n!) HypergeometricPFQ[{1/4, -n}, {}, 4] (-1/4)^n, {n, 0, 15}] (* Benedict W. J. Irwin, May 24 2016 *)

E.g.f. for aerated sequence: exp(-x^2/4)/(1-x^2)^(1/4).
a(n) ~ (2*n)! * 2^(1/4)*exp(-1/4)*Gamma(3/4)/((2*n)^(3/4)*Pi). - Vaclav Kotesovec, Sep 24 2013
a(n) = ((2n)!/n!)*2F0(1/4,-n;;4)*(-1/4)^n. - Benedict W. J. Irwin, May 24 2016
(4n^3-n)a(n-1) + (4n^2+2n)a(n) - a(n+1) = 0. - Robert Israel, Mar 02 2017

a(14) and e.g.f. corrected by Robert Israel, Mar 02 2017

cp's OEIS Frontend

A053533 Expansion of e.g.f.: (1-x)^(-1/2)*exp(-x/2 -x^2/4 -x^3/6 -x^4/8).

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