A060913 -id:A060913 - cp's OEIS frontend

A060905 Expansion of e.g.f. exp(xexp(x) + 1/2x^2*exp(x)^2).

1, 1, 4, 19, 110, 751, 5902, 52165, 509588, 5437729, 62828306, 780287839, 10351912276, 145944541159, 2176931651546, 34225419288421, 565282627986368, 9779830102138945, 176776613812205074, 3330780287838743575

Offset: 0

Vladeta Jovovic, Apr 07 2001

nonn

Number of functions f from a set of size n to itself such that f(f(f(x))) = f(x). - Joel B. Lewis, Dec 12 2011

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983.

Alois P. Heinz, Table of n, a(n) for n = 0..200
Vladimir Kruchinin, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.

Cf. A000949 - A000951, A060905 - A060913.

Column k=3 of A245501.

nn=20;a=x Exp[x];Range[0,nn]!CoefficientList[Series[Exp[a+a^2/2],{x,0,nn}],x]  (* Geoffrey Critzer, Sep 18 2012 *)

a(n):=sum(sum(k^(n-k)/(n-k)!*binomial(m,k-m)*(1/2)^(k-m),k,m,n)/m!,m,1,n); /* Vladimir Kruchinin, Aug 20 2010 */

E.g.f.: exp(Sum_{d|m} T_k^d/d), where T_k = x*exp(T_(k - 1)), k >= 1, T_0 = x; k = 1, m = 2.

a(n) = sum(sum(k^(n-k)/(n-k)!*binomial(m,k-m)*(1/2)^(k-m),k,m,n)/m!,m,1,n), n>0. - Vladimir Kruchinin, Aug 20 2010

A060906 E.g.f.: exp(xexp(x) + 1/3x^3*exp(x)^3).

Original entry on oeis.org

1, 1, 3, 12, 73, 556, 4737, 44122, 453441, 5186664, 65671201, 906052654, 13418086497, 211472682604, 3535616946513, 62621439810066, 1172370604136833, 23118679430573008, 478329265510033473, 10349724555927678934, 233633352312272612001, 5492655756487132979796

Offset: 0

Vladeta Jovovic, Apr 07 2001

nonn

The number of functions from {1,2,...,n} to itself such that f(x)=f^4(x). - Geoffrey Critzer, Sep 18 2012

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983.

Alois P. Heinz, Table of n, a(n) for n = 0..200

Cf. A000949 - A000951, A060905 - A060913.

Column k=4 of A245501.

nn=20;a=x Exp[x];Range[0,nn]!CoefficientList[Series[Exp[a+a^3/3],{x,0,nn}],x] (* Geoffrey Critzer, Sep 18 2012 *)

E.g.f.: exp(Sum_{d|m} T_k^d/d), where T_k = x*exp(T_(k - 1)), k >= 1, T_0 = x; k = 1, m = 3.

A060907 E.g.f.: exp(xexp(x) + 1/2x^2exp(x)^2 + 1/4x^4*exp(x)^4).

Original entry on oeis.org

1, 1, 4, 19, 116, 901, 8422, 89755, 1061048, 13746169, 193901066, 2965146559, 48946004956, 867463969789, 16405240966766, 329147315037811, 6973157545554128, 155446026607476145, 3636697161715448914, 89099916704329731895, 2281451214192505136516

Offset: 0

Vladeta Jovovic, Apr 07 2001

nonn

The number of functions from {1,2,...,n} into itself such that f(x) = f^5(x). - Geoffrey Critzer, Sep 18 2012

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983.

Alois P. Heinz, Table of n, a(n) for n = 0..200

Cf. A000949 - A000951, A060905 - A060913.

Column k=5 of A245501.

egf:= exp(x*exp(x)+x^2*exp(x)^2/2+x^4*exp(x)^4/4):
a:= n-> n!*coeff(series(egf, x, n+11), x, n):
seq(a(n), n=0..25);  # Alois P. Heinz, Jul 25 2014

nn=20;a=x Exp[x];Range[0,nn]!CoefficientList[Series[Exp[a+a^2/2+a^4/4],{x,0,nn}],x] (* Geoffrey Critzer, Sep 18 2012 *)

E.g.f.: exp(Sum_{d|m} T_k^d/d), where T_k = x*exp(T_(k - 1)), k >= 1, T_0 = x; k = 1, m = 4.

A060908 E.g.f.: exp(xexp(xexp(x)) + 1/2x^2exp(x*exp(x))^2).

Original entry on oeis.org

1, 1, 4, 25, 194, 1791, 19312, 237637, 3280524, 50136049, 839267936, 15255154179, 298936866736, 6277386102703, 140540145723720, 3339966073612921, 83936496568012208, 2223184658988286113, 61877234830148427808

Offset: 0

Vladeta Jovovic, Apr 07 2001

nonn

a(n) = the number of functions f:{1,2,...,n} -> {1,2,...,n} such that the functional digraphs have cycle lengths at most 2 and no element is at a distance of more than 2 form a cycle. - Geoffrey Critzer, Sep 23 2012

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983.

Cf. A000949-A000951, A060905-A060913.

nn=20; a=x Exp[x]; b=x Exp[a]; t=Sum[n^(n-1)x^n/n! ,{n, 1, nn}]; Range[0,nn]! CoefficientList[Series[Exp[b+b^2/2], {x, 0, nn}], x]  (* Geoffrey Critzer, Sep 23 2012 *)

E.g.f.: exp(Sum_{d|m} T_k^d/d), where T_k = x*exp(T_(k - 1)), k >= 1, T_0 = x; k = 2, m = 2.

A060909 E.g.f.: exp(xexp(xexp(x)) + 1/3x^3exp(x*exp(x))^3).

Original entry on oeis.org

1, 1, 3, 18, 133, 1236, 13767, 176674, 2547561, 40614408, 708601771, 13433957934, 275200324797, 6061423076476, 142868492357151, 3587417860571346, 95560989416582353, 2690066742390963216, 79752454967110250835

Offset: 0

Vladeta Jovovic, Apr 07 2001

nonn

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983.

Cf. A000949 - A000951, A060905 - A060913.

E.g.f.: exp(Sum_{d|m} T_k^d/d), where T_k = x*exp(T_(k - 1)), k >= 1, T_0 = x; k = 2, m = 3.

A060910 E.g.f.: exp(xexp(xexp(x)) + 1/2x^2exp(xexp(x))^2 + 1/4x^4exp(xexp(x))^4).

Original entry on oeis.org

1, 1, 4, 25, 200, 1941, 22552, 304207, 4660224, 79627609, 1496962736, 30645682299, 677868344056, 16102526543533, 408764126148120, 11042583947604871, 316299747976627808, 9574687031473970673

Offset: 0

Vladeta Jovovic, Apr 07 2001

nonn

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983.

Cf. A000949 - A000951, A060905 - A060913.

E.g.f.: exp(Sum_{d|m} T_k^d/d), where T_k = x*exp(T_(k - 1)), k >= 1, T_0 = x; k = 2, m = 4.

A060911 E.g.f.: exp(xexp(xexp(xexp(x))) + 1/2x^2exp(xexp(x*exp(x)))^2).

Original entry on oeis.org

1, 1, 4, 25, 218, 2331, 29152, 417607, 6746700, 121312441, 2401341056, 51857779689, 1212621122176, 30509979042115, 821524617293304, 23563369209520711, 717014609781379568, 23064363484845390513

Offset: 0

Vladeta Jovovic, Apr 07 2001

nonn

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983.

Cf. A000949 - A000951, A060905 - A060913.

E.g.f.: exp(Sum_{d|m} T_k^d/d), where T_k = x*exp(T_(k - 1)), k >= 1, T_0 = x; k = 3, m = 2.

A060912 E.g.f.: exp(xexp(xexp(xexp(xexp(x)))) + 1/2x^2exp(xexp(xexp(x*exp(x))))^2).

Original entry on oeis.org

1, 1, 4, 25, 218, 2451, 33112, 516727, 9117180, 179330905, 3890434256, 92271385449, 2374775505016, 65900749176835, 1961009596461840, 62275489622799751, 2101798757669917328, 75111617959762807473

Offset: 0

Vladeta Jovovic, Apr 07 2001

nonn

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983.

Cf. A000949 - A000951, A060905 - A060913.

E.g.f.: exp(Sum_{d|m} T_k^d/d), where T_k = x*exp(T_(k - 1)), k >= 1, T_0 = x; k = 4, m = 2.

cp's OEIS Frontend

A060905 Expansion of e.g.f. exp(x*exp(x) + 1/2*x^2*exp(x)^2).

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Maxima

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A060906 E.g.f.: exp(x*exp(x) + 1/3*x^3*exp(x)^3).

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A060907 E.g.f.: exp(x*exp(x) + 1/2*x^2*exp(x)^2 + 1/4*x^4*exp(x)^4).

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A060908 E.g.f.: exp(x*exp(x*exp(x)) + 1/2*x^2*exp(x*exp(x))^2).

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A060909 E.g.f.: exp(x*exp(x*exp(x)) + 1/3*x^3*exp(x*exp(x))^3).

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A060910 E.g.f.: exp(x*exp(x*exp(x)) + 1/2*x^2*exp(x*exp(x))^2 + 1/4*x^4*exp(x*exp(x))^4).

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A060911 E.g.f.: exp(x*exp(x*exp(x*exp(x))) + 1/2*x^2*exp(x*exp(x*exp(x)))^2).

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A060912 E.g.f.: exp(x*exp(x*exp(x*exp(x*exp(x)))) + 1/2*x^2*exp(x*exp(x*exp(x*exp(x))))^2).

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A060905 Expansion of e.g.f. exp(xexp(x) + 1/2x^2*exp(x)^2).

A060906 E.g.f.: exp(xexp(x) + 1/3x^3*exp(x)^3).

A060907 E.g.f.: exp(xexp(x) + 1/2x^2exp(x)^2 + 1/4x^4*exp(x)^4).

A060908 E.g.f.: exp(xexp(xexp(x)) + 1/2x^2exp(x*exp(x))^2).

A060909 E.g.f.: exp(xexp(xexp(x)) + 1/3x^3exp(x*exp(x))^3).

A060910 E.g.f.: exp(xexp(xexp(x)) + 1/2x^2exp(xexp(x))^2 + 1/4x^4exp(xexp(x))^4).

A060911 E.g.f.: exp(xexp(xexp(xexp(x))) + 1/2x^2exp(xexp(x*exp(x)))^2).

A060912 E.g.f.: exp(xexp(xexp(xexp(xexp(x)))) + 1/2x^2exp(xexp(xexp(x*exp(x))))^2).