A121355 -id:A121355 - cp's OEIS frontend

A005133 Number of index n subgroups of modular group PSL_2(Z).

1, 1, 4, 8, 5, 22, 42, 40, 120, 265, 286, 764, 1729, 2198, 5168, 12144, 17034, 37702, 88958, 136584, 288270, 682572, 1118996, 2306464, 5428800, 9409517, 19103988, 44701696, 80904113, 163344502, 379249288, 711598944, 1434840718, 3308997062, 6391673638, 12921383032, 29611074174, 58602591708, 119001063028, 271331133136, 547872065136, 1119204224666, 2541384297716, 5219606253184, 10733985041978, 24300914061436, 50635071045768, 104875736986272, 236934212877684, 499877970985660

Offset: 1

Simon Plouffe

Equivalently, the number of isomorphism class of transitive PSL_2(Z) actions on a finite dotted (i.e., having a distinguished element) set of size n. Also the number of different connected dotted trivalent diagrams of size n. - Samuel A. Vidal, Jul 23 2006

Connected and dotted version of A121352. Dotted version of A121350. Unlabeled version of A121356. Unlabeled and dotted version of A121355. - Samuel A. Vidal, Jul 23 2006

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Morris Newman, Classification of Normal Subgroups of the Modular Group, Transactions of the American Mathematical Society 126 (1967), no. 2, 267-277.
Morris Newman, Asymptotic formulas related to free products of cyclic groups, Math. Comp. 30 (1976), no. 136, 838-846.
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], 2007.
Index entries for sequences related to modular groups

Cf. A121357.

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) : logexs23:=sort(convert(taylor(log(exs23),t,N+1),polynom),t, ascending) : sort(add(op(n,logexs23)*n,n=1..N),t, ascending) ; # Samuel A. Vidal, Jul 23 2006

m = 50; 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}]; logexs23 = Series[ Log[exs23], {t, 0, m+1}] // Normal; CoefficientList[ Sum[ logexs23[[n]]*n, {n, 1, m}], t] // Rest (* Jean-François Alcover, Dec 05 2012, translated from Maple *)

N=50; x='x+O('x^(N+1));
A121357_ser = serconvol(serlaplace(exp(x+x^2/2)), serlaplace(exp(x+x^3/3)));
Vec(x*log(serconvol(A121357_ser, exp(x)))') \\ Gheorghe Coserea, May 10 2017

a(n) = A121355(n)/(n-1)!, a(n) = A121356(n)/n!. - Samuel A. Vidal, Jul 23 2006

If A(z) = g.f. of a(n) and B(z) = g.f. of A121356 then A(z) is the Borel transform of B(z). - Samuel A. Vidal, Jul 23 2006

More terms from Samuel A. Vidal, Jul 23 2006

Entry revised by N. J. A. Sloane, Jul 25 2006

A121350 Number of conjugacy class of index n subgroups in PSL_2 (ZZ).

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 8, 6, 7, 14, 27, 26, 80, 133, 170, 348, 765, 1002, 2176, 4682, 6931, 13740, 31085, 48652, 96682, 217152, 362779, 707590, 1597130, 2789797, 5449439, 12233848, 22245655, 43480188, 97330468, 182619250, 358968639, 800299302, 1542254973, 3051310056, 6783358130

Offset: 0

Samuel A. Vidal, Jul 23 2006

nonn

Equivalently, the number of isomorphism class of transitive PSL_2(ZZ) actions on a finite set of size n.
Also the number of different connected trivalent diagrams of size n.
Also the number of (r,s) pair of permutations in S_n, up to simultaneous conjugation, which generate a transitive action and for which r is involutive i.e. r^2 = id and s is of weak order three i.e. s^3 = id.

Andrew Howroyd, Table of n, a(n) for n = 0..1000
S. A. Vidal, Sur la Classification et le Dénombrement des Sous-groupes du Groupe Modulaire et de leurs Classes de Conjugaison (in French), arXiv:0702223 [math.CO], 2006.
Qiaochu Yuan, Drawing subgroups of the modular group, Annoying Precision, Blog, November 29, 2015.

Connected version of A121352.
Unlabeled version of A121355.
Cf. also A005133, A121356, A121357.

with(numtheory,mobius) : mu := k -> `if`( k mod 2 = 0, 2/k, 1/k ) : nu := k -> `if`( k mod 3 = 0, 3/k, 1/k ) : u := (k,n) -> add(mu(k)^(n-2*k2)/(n-2*k2)!/k2!/(2*k)^k2,k2=0..floor(n/ 2)) ; v := (k,n) -> add(nu(k)^(n-3*k3)/(n-3*k3)!/k3!/(3*k)^k3,k3=0..floor(n/ 3)) ; N := 100 # For example. add(convert(taylor(log(add(n!*k^n*u(k,n)*v(k,n)*t^(k*n), n = 0..floor (N/k))),t=0,N+1),polynom),k=1..N) : lZF := sort (%,t, ascending) : add(mobius(k)/k*rem(subs(t=t^k,lZF),t^(N+1),t),k=1..N) : sort (%,t, ascending);

max = 37; mu[k_] := If[Mod[k, 2] == 0, 2/k, 1/k]; nu[k_] := If[Mod[k, 3] == 0, 3/k, 1/k]; u[k_, n_] := Sum[ mu[k]^(n - 2*k2) / (((n - 2*k2)!*k2!)*(2*k)^k2), {k2, 0, Floor[n/2]}]; v[k_, n_] := Sum[ nu[k]^(n - 3*k3) / (((n - 3*k3)!*k3!)*(3*k)^k3), {k3, 0, Floor[n/3]}]; lZF[t_] = Sum[ Normal[ Series[ Log[ Sum[n!*k^n*u[k, n]*v[k, n]*t^(k*n), {n, 0, Floor[max/k]}]], {t, 0, max + 1}]], {k, 1, max}]; Rest[ CoefficientList[ Sum[ (MoebiusMu[k]*PolynomialMod[lZF[t^k], t^(max + 1)])/k, {k, 1, max}], t]] (* Jean-François Alcover, Dec 05 2012, translated from Samuel Vidal's Maple program *)

If A(z) = g.f. of a(n) and B(z) = g.f. of A121352 then A(z) = sum_{k > 0} mu(k)/k log(B(z^k)) (Moebius inversion formula).

a(0)=1 prepended and a(38) onwards from Andrew Howroyd, Jan 29 2025

A121352 Number of different, not necessarily connected, unlabeled trivalent diagrams of size n.

Original entry on oeis.org

1, 1, 2, 4, 7, 10, 24, 37, 63, 112, 200, 318, 607, 1058, 1814, 3247, 6004, 10316, 19048, 35478, 63496, 117023, 223822, 408121, 766661, 1484363, 2775201, 5270079, 10357605, 19714259, 37970066, 75439670, 146103241, 284719527, 571706625, 1123396477, 2214903209

Offset: 0

Samuel A. Vidal, Jul 23 2006

nonn

Equivalently, the number of isomorphism class of PSL_2(ZZ) actions on finite sets of size n.

Also the number of (r,s) pair of permutations up to simultaneous conjugation, 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.

Andrew Howroyd, Table of n, a(n) for n = 0..1000
S. A. Vidal, Sur la Classification et le Dénombrement des Sous-groupes du Groupe Modulaire et de leurs Classes de Conjugaison (in French), arXiv:0702223 [math.CO], 2006.

Not necessarily connected version of A121350.
Unlabeled version of A121357.
Cf. also A005133, A121355, A121356.

mu := k -> `if`( k mod 2 = 0, 2/k, 1/k ) : nu := k -> `if`( k mod 3 = 0, 3/k, 1/k ) : u := (k,n) -> add(mu(k)^(n-2*k2)/(n-2*k2)!/k2!/(2*k)^k2,k2=0..floor(n/ 2)) : v := (k,n) -> add(nu(k)^(n-3*k3)/(n-3*k3)!/k3!/(3*k)^k3,k3=0..floor(n/ 3)) : N := 100 : ZF := 1 : for k from N to 1 by -1 do ZF := rem(ZF * add(n!*k^n*u(k,n)*v(k,n)*t^(k*n), n = 0..floor(N/ k)),t^(N+1),t) ; end do : sort(ZF,t, ascending);

max = 34; mu[k_] := If[Mod[k, 2] == 0, 2/k, 1/k]; nu[k_] := If[Mod[k, 3] == 0, 3/k, 1/k]; u[k_, n_] := Sum[ mu[k]^(n - 2*k2) / (((n - 2*k2)!*k2!)*(2*k)^k2), {k2, 0, Floor[n/2]}]; v[k_, n_] := Sum[ nu[k]^(n - 3*k3) / (((n - 3*k3)!*k3!)*(3*k)^k3), {k3, 0, Floor[n/3]}]; ZF = 1; For[k = max, k >= 1, k--, ZF = PolynomialMod[ ZF*Sum[ n!*k^n*u[k, n]*v[k, n]*t^(k*n), {n, 0, Floor[max/k]}], t^(max + 1)]]; CoefficientList[ZF, t](* Jean-François Alcover, Dec 05 2012, translated from Samuel Vidal's Maple program *)

D(m,k)={my(g=gcd(m,k)); sumdiv(g, d, my(j=m/d); x^j*eulerphi(d)*k^(j-1)/j)}
seq(n)={Vec(prod(k=1, n, my(A=O(x^(n\k+1)), p=serconvol(exp(A + D(1,k) + D(3,k)), exp(A + D(1,k) + D(2,k)))); sum(r=0, n\k, r!*polcoef(p,r)/(k^r)*x^(k*r), O(x*x^n)) ))} \\ Andrew Howroyd, Jan 29 2025

Euler transform of A121350. - Andrew Howroyd, Jan 29 2025

a(35) onwards from Andrew Howroyd, Jan 29 2025

A121356 Number of transitive PSL_2(ZZ) actions on a finite dotted and labeled set of size n.

Original entry on oeis.org

1, 2, 24, 192, 600, 15840, 211680, 1612800, 43545600, 961632000, 11416204800, 365957222400, 10766518963200, 191617884057600, 6758061133824000, 254086360399872000, 6058779650187264000, 241382293453357056000

Offset: 1

Samuel A. Vidal, Jul 23 2006

nonn

"Dotted" means having a distinguished element. - N. J. A. Sloane, Feb 06 2012

Equivalently, the number of different connected, dotted and labeled trivalent diagrams of size n.

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

Labeled version of A005133.
Labeled and dotted version of A121350.
Dotted version of A121355.
Connected and dotted version of A121357.
Connected, labeled and dotted version of A121352.

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) : logexs23:=sort(convert(taylor(log(exs23),t,N+1),polynom),t, ascending) : sort(add(op(n,logexs23)*n!*n,n=1..N),t, ascending);

m = 18;
s2 = Exp[t + t^2/2] + O[t]^(m+1) // Normal;
s3 = Exp[t + t^3/3] + O[t]^(m+1) // Normal;
s = Sum[s2[[n+1]] s3[[n+1]]/(t^n/n!), {n, 0, m}];
CoefficientList[Log[s] + O[t]^(m+1), t] Range[0, m]! Range[0, m] // Rest (* Jean-François Alcover, Sep 02 2018, from Maple *)

N=18; x='x+O('x^(N+1));
A121357_ser = serconvol(serlaplace(exp(x+x^2/2)), serlaplace(exp(x+x^3/3)));
A121355_ser = serlaplace(log(serconvol(A121357_ser, exp(x))));
Vec(x*A121355_ser') \\ Gheorghe Coserea, May 10 2017

a(n) = A121355(n)*n.

If A(z) = g.f. of a(n) and B(z) = g.f. of A121355 then A(z) = z d/dz B(z) (Euler operator).

A121357 Number of different, not necessarily connected, labeled trivalent diagrams of size n.

Original entry on oeis.org

1, 1, 2, 12, 90, 546, 6156, 81432, 942012, 15114780, 294765336, 5069224656, 108842183352, 2770895886552, 64609245619920, 1742542175582496, 55074355772360976, 1626315165597840912, 53331321825434963232

Offset: 0

Samuel A. Vidal, Jul 23 2006

nonn

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.

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.

Unconnected version of A121355.
Labeled version of A121352.
Labeled, unconnected version of A121350.
Cf. also A005133, A121356.

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);

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-François Alcover, Dec 05 2012, translated from Samuel Vidal's Maple program *)

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

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

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A005133 Number of index n subgroups of modular group PSL_2(Z).

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A121350 Number of conjugacy class of index n subgroups in PSL_2 (ZZ).

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A121352 Number of different, not necessarily connected, unlabeled trivalent diagrams of size n.

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A121356 Number of transitive PSL_2(ZZ) actions on a finite dotted and labeled set of size n.

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A121357 Number of different, not necessarily connected, labeled trivalent diagrams of size n.

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