A000949 -id:A000949 - cp's OEIS frontend

A362273 Expansion of e.g.f. 1/(1 - x * exp(-x * exp(-x))).

1, 1, 0, 3, 8, -15, 264, -35, -1968, 87633, -499600, 2375901, 48964200, -830424023, 9884072184, -11730111315, -1407884197216, 36601422429345, -416600839315872, 191233500832189, 136472124267672120, -3513232740127917639, 46653752740647748520

Offset: 0

Seiichi Manyama, Apr 13 2023

sign

Winston de Greef, Table of n, a(n) for n = 0..455

Cf. A000949, A362274, A362275.

Cf. A362244.

my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-x*exp(-x*exp(-x)))))

a(n) = n! * Sum_{i=0..n} (-1)^(n-i) * Sum_{j=0..n-i} i^j * j^(n-i-j) / (j! * (n-i-j)!).

A362274 Expansion of e.g.f. 1/(1-xexp(xexp(x))).

Original entry on oeis.org

1, 1, 4, 27, 232, 2465, 31416, 467173, 7940080, 151824177, 3225664720, 75386002901, 1921987809336, 53085088574713, 1578989028324904, 50320957240554645, 1710595789688360416, 61783832256587799137, 2362794061464082513440, 95380148553556615501717

Offset: 0

Seiichi Manyama, Apr 13 2023

nonn

Winston de Greef, Table of n, a(n) for n = 0..398

Cf. A000949, A362273, A362275.

Cf. A362245.

my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-x*exp(x*exp(x)))))

a(n) = n! * Sum_{i=0..n} Sum_{j=0..n-i} i^j * j^(n-i-j) / (j! * (n-i-j)!).

A362275 Expansion of e.g.f. exp(xexp(-xexp(-x))).

Original entry on oeis.org

1, 1, -1, 4, -3, -44, 595, -5214, 36953, -176840, -498969, 31239550, -605916971, 9001842708, -110410388101, 998140500826, -1114748107215, -247879285927184, 8410011463871695, -201620611378992906, 4024515940445770541, -66832538504513547380

Offset: 0

Seiichi Manyama, Apr 13 2023

sign

Winston de Greef, Table of n, a(n) for n = 0..460

Cf. A000949, A362273, A362274.

Cf. A362247.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x*exp(-x*exp(-x)))))

a(n) = Sum_{i=0..n} (-1)^(n-i) * binomial(n,i) * Sum_{j=0..n-i} i^j * j^(n-i-j) * binomial(n-i,j).

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.

A239752 Number of pairs of functions (f,g) from a set of n elements into itself satisfying f(f(x)) = g(f(f(x))).

Original entry on oeis.org

1, 1, 6, 123, 4792, 294645, 26351856, 3213829339, 511432765824, 102813166760265, 25450790212460800, 7599894406225438911, 2691706949197641133056, 1114869818722491048119773, 533573397145124307890731008, 292063395009538745067415219875, 181221082305680372426427865071616

Offset: 0

Chad Brewbaker, Mar 26 2014

nonn

Also pairs of functions where f(f(f(x))) = g(f(f(x))).

Alois P. Heinz, Table of n, a(n) for n = 0..160

Cf. A000248, A000949.

t1[n_, d_, c_] := StirlingS2[c + d, d] Sum[ StirlingS2[b, c]/(b! (n - b - c - d)!) d^(n - b - c - d), {b, c, n - c - d}]
a[n_] := If[n == 0, 1, n! Sum[n^(n - d) Sum[t1[n, d, c], {c, 0, Floor[(n - d)/2]}], {d, 1, n}]]
Table[a[n], {n, 0, 20}] (* David Einstein, Oct 23 2016 *)

a(n) = n! Sum_{d = 0..n} n^(n-d) Sum_{c = 0..(n-d)/2} S2(c+d,d) Sum_{b = c..n-c-d} S2(b,c)/(b! (n-b-c-d)!) d^(n-b-c-d). - David Einstein, Oct 23 2016

a(6)-a(7) from Giovanni Resta, Mar 28 2014
a(8)-a(10) from Paul Boddington, Feb 23 2015
a(11)-a(16) from David Einstein, Oct 23 2016

A239759 Number of pairs of functions (f,g) from a set of n elements into itself satisfying f(g(x)) = g(f(g(x))).

Original entry on oeis.org

1, 1, 8, 147, 5032, 274765, 21702096, 2327543743, 324317092352, 56816985879801, 12201268360672000, 3147485927264980411, 959356469360927222784, 340785570064053310187173, 139449094593887763170361344, 65080820095941557895304404375

Offset: 0

Chad Brewbaker, Mar 26 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..241

Cf. A000248, A000949.

Unprotect[Power]; 0^0 = 1;
a[n_] := If[n == 0, 1, Sum[Binomial[n, k] k^k Sum[Binomial[n - k, j] k^j (j + k - 1)^j ((n - k)(j + k))^(n - j - k), {j, 0, n - k}], {k, 1, n}]];
a /@ Range[0, 15] (* Jean-François Alcover, Oct 04 2019 *)

a(n) = if(n==0,1,sum(k=1,n, binomial(n,k) * k^k * (sum(j=0,n-k, binomial(n-k,j)*k^j*(j+k-1)^j*((n-k)*(j+k))^(n-j-k))))) \\ Joerg Arndt, Oct 13 2016

a(n) = Sum_{k=0..n} C(n,k) * k^k * (Sum_{j=0..n-k} C(n-k,j)*k^j*(j+k-1)^j*((n-k)*(j+k))^(n-j-k)). - David Einstein, Oct 11 2016

a(6)-a(7) from Giovanni Resta, Mar 28 2014

a(8)-a(15) from David Einstein, Oct 11 2016

A239760 Number of pairs of functions (f,g) from a set of n elements into itself satisfying f(x) = f(f(g(x))).

Original entry on oeis.org

1, 1, 10, 123, 1984, 38765, 899856, 24135055, 734381824, 24985937241, 939674886400, 38697245287811, 1731506682516480, 83631998873272837, 4336190248653942784, 240186479159794548375, 14153996066789439471616, 884109266328681103701425, 58346867827408911965945856

Offset: 0

Chad Brewbaker, Mar 26 2014

nonn

David Einstein, Table of n, a(n) for n = 0..100
David Einstein, Sage notebook for A239760

Cf. A000248, A000949.

a(6)-a(7) from Giovanni Resta, Mar 28 2014

More terms from David Einstein, Nov 15 2016

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.

cp's OEIS Frontend

A362273 Expansion of e.g.f. 1/(1 - x * exp(-x * exp(-x))).

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A362274 Expansion of e.g.f. 1/(1-x*exp(x*exp(x))).

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A362275 Expansion of e.g.f. exp(x*exp(-x*exp(-x))).

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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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A239752 Number of pairs of functions (f,g) from a set of n elements into itself satisfying f(f(x)) = g(f(f(x))).

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Mathematica

Formula

Extensions

A239759 Number of pairs of functions (f,g) from a set of n elements into itself satisfying f(g(x)) = g(f(g(x))).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A239760 Number of pairs of functions (f,g) from a set of n elements into itself satisfying f(x) = f(f(g(x))).

Views

Author

Keywords

Links

Crossrefs

Extensions

A362274 Expansion of e.g.f. 1/(1-xexp(xexp(x))).

A362275 Expansion of e.g.f. exp(xexp(-xexp(-x))).

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).