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A121357 Number of different, not necessarily connected, labeled trivalent diagrams of size n. 5
1, 1, 2, 12, 90, 546, 6156, 81432, 942012, 15114780, 294765336, 5069224656, 108842183352, 2770895886552, 64609245619920, 1742542175582496, 55074355772360976, 1626315165597840912, 53331321825434963232 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Equivalently, the number of PSL_2(ZZ) actions on a finite labeled set of size n.

Also the number of (r,s) pair of permutations in S_n for which r is involutive i.e. r^2 = id and s is of weak order three i.e. s^3 = id.

LINKS

Table of n, a(n) for n=0..18.

S. A. Vidal, Sur la Classification et le Denombrement des Sous-groupes du Groupe Modulaire et de leurs Classes de Conjugaison, (in French), arXiv:math/0702223 [math.CO] 2006.

FORMULA

If A(z) = g.f. of a(n) and B(z) = g.f. of A121355 then A(z) = exp(B(z)).

Six term linear recurrence : (n^3 + 12*n^2 + 47*n + 61)*a(n + 6) = (29040 + 239224*n^2 + 127628*n + 20715*n^6 + 252267*n^3 + 166304*n^4 + 71889*n^5 + 33*n^9 + 3943*n^7 + 476*n^8 + n^10)*a(n) + (441*n^4 + 3*n^6 + 2160 + 57*n^5 + 4572*n + 3948*n^2 + 1779*n^3)*a(n + 1) + (34920 + 61314*n + 45886*n^2 + 18989*n^3 + 4697*n^4 + 695*n^5 + 57*n^6 + 2*n^7)*a(n + 2) + (19640 + 79*n^5 + 3*n^6 + 861*n^4 + 27598*n + 16084*n^2 + 4975*n^3)*a(n + 3) + (17*n^3 + 425 + n^4 + 350*n + 113*n^2)*a(n + 4) + (1 + 20*n + 9*n^2 + n^3)*a(n + 5) with n = 0, 1, ...

a(n) = A000085(n) * A001470(n)  - Mark van Hoeij, May 13 2013

MAPLE

N := 100 : exs2:=sort(convert(taylor(exp(t+t^2/2), t, N+1), polynom), t, ascending) : exs3:=sort(convert(taylor(exp(t+t^3/3), t, N+1), polynom), t, ascending) : exs23:=sort(add(op(n+1, exs2)*op(n+1, exs3)/(t^n/ n!), n=0..N), t, ascending) : sort(add(op(n+1, exs23)*n!, n=0..N), t, ascending);

MATHEMATICA

m = 18; exs2 = Series[Exp[t + t^2/2], {t, 0, m + 1}] // Normal; exs3 = Series[Exp[t + t^3/3], {t, 0, m + 1}] // Normal; exs23 = Sum[exs2[[n + 1]]*exs3[[n + 1]]/(t^n/n!), {n, 0, m}]; CoefficientList[ Sum[exs23[[n + 1]]*n!, {n, 0, m}], t] (* Jean-Fran├žois Alcover, Dec 05 2012, translated from Samuel Vidal's Maple program *)

PROG

(PARI) N=19; x='x+O('x^N);

Vec(serconvol(serlaplace(exp(x+x^2/2)), serlaplace(exp(x+x^3/3)))) \\ Gheorghe Coserea, May 10 2017

CROSSREFS

Unconnected version of A121355.

Labeled version of A121352.

Labeled, unconnected version of A121350.

Cf. also A005133, A121356.

Sequence in context: A079858 A224152 A174356 * A098926 A074610 A250130

Adjacent sequences:  A121354 A121355 A121356 * A121358 A121359 A121360

KEYWORD

nonn

AUTHOR

Samuel A. Vidal, Jul 23 2006

STATUS

approved

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Last modified November 16 00:11 EST 2018. Contains 317252 sequences. (Running on oeis4.)