cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 11 results. Next

A245501 Number A(n,k) of endofunctions f on [n] such that f^k(i) = f(i) for all i in [n]; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 3, 27, 1, 1, 1, 4, 10, 256, 1, 1, 1, 3, 19, 41, 3125, 1, 1, 1, 4, 12, 110, 196, 46656, 1, 1, 1, 3, 19, 73, 751, 1057, 823543, 1, 1, 1, 4, 10, 116, 556, 5902, 6322, 16777216, 1, 1, 1, 3, 21, 41, 901, 4737, 52165, 41393, 387420489, 1
Offset: 0

Views

Author

Alois P. Heinz, Jul 24 2014

Keywords

Examples

			Square array A(n,k) begins:
  1,     1,    1,    1,    1,    1,    1, ...
  1,     1,    1,    1,    1,    1,    1, ...
  1,     4,    3,    4,    3,    4,    3, ...
  1,    27,   10,   19,   12,   19,   10, ...
  1,   256,   41,  110,   73,  116,   41, ...
  1,  3125,  196,  751,  556,  901,  220, ...
  1, 46656, 1057, 5902, 4737, 8422, 1921, ...

Links

Crossrefs

Column k=0-10 give: A000012, A000312, A000248, A060905, A060906, A060907, A245502, A245503, A245504, A245505, A245506.
Main diagonal gives A245507.

Programs

  • Maple
    with(numtheory):
A:= (n, k)-> `if`(k=0, 1, `if`(k=1, n^n, n! *coeff(series(
    exp(add((x*exp(x))^d/d, d=divisors(k-1))), x, n+1), x, n))):
seq(seq(A(n, d-n), n=0..d), d=0..12);
  • Mathematica
    A[0, 1] = 1; A[n_, k_] := If[k==0, 1, If[k==1, n^n, n!*SeriesCoefficient[ Exp[ DivisorSum[k-1, (x*Exp[x])^#/#&]], {x, 0, n}]]]; Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Mar 20 2017, translated from Maple *)

Formula

A(n,k) = n! * [x^n] exp(Sum_{d|(k-1)} (x*exp(x))^d/d) for k>1, A(n,0)=1, A(n,1)=n^n.

A060913 E.g.f.: exp(x*exp(x*exp(x*exp(x))) + 1/3*x^3*exp(x*exp(x*exp(x)))^3).

Original entry on oeis.org

1, 1, 3, 18, 157, 1656, 20727, 300784, 4955337, 91229616, 1853584651, 41147256624, 989990665677, 25647894553048, 711630284942319, 21049888453838136, 661180220075555473, 21976354057916680416
Offset: 0

Views

Author

Vladeta Jovovic, Apr 07 2001

Keywords

Comments

a(n) is the number of functions f:{1,2,...,n} -> {1,2,...,n} such that the functional digraphs have cycles of length 1 or 3 and no element is at a distance of more than 3 from a cycle. - Geoffrey Critzer, Sep 23 2012

References

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

Crossrefs

Cf. A000949-A000951, A060905-A060913.

Programs

  • Mathematica
    nn=20; a=x Exp[x]; b=x Exp[a]; c=x Exp[b]; t=Sum[n^(n-1)x^n/n!, {n, 1, nn}]; Range[0,nn]! CoefficientList[Series[Exp[c+c^3/3], {x, 0, nn}], x] (* Geoffrey Critzer, Sep 23 2012 *)

Formula

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 = 3.

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

Views

Author

Vladeta Jovovic, Apr 07 2001

Keywords

Comments

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

References

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

Links

Crossrefs

Cf. A000949 - A000951, A060905 - A060913.
Column k=4 of A245501.

Programs

  • Mathematica
    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 *)

Formula

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(x*exp(x) + 1/2*x^2*exp(x)^2 + 1/4*x^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

Views

Author

Vladeta Jovovic, Apr 07 2001

Keywords

Comments

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

References

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

Links

Crossrefs

Cf. A000949 - A000951, A060905 - A060913.
Column k=5 of A245501.

Programs

  • Maple
    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
  • Mathematica
    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 *)

Formula

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.

A189487 Expansion of e.g.f. exp(x*exp(x) + x^2*exp(2*x)).

Original entry on oeis.org

1, 1, 5, 28, 185, 1456, 13267, 135598, 1528193, 18805240, 250522451, 3585332554, 54774501025, 888739031116, 15249006695483, 275641537989766, 5231788966650113, 103968303762747472, 2157673505603964643, 46656574558459795522, 1049037051211541521121
Offset: 0

Views

Author

Vladimir Kruchinin, Apr 23 2011

Keywords

Links

Crossrefs

Cf. A060905.

Programs

  • Mathematica
    With[{nn=30},CoefficientList[Series[Exp[x Exp[x]+x^2 Exp[2x]],{x,0,nn}], x]Range[0,nn]!] (* Harvey P. Dale, Sep 22 2011 *)
  • Maxima
    a(n):=n!*sum(sum((k^(n-k)*binomial(m,k-m))/(n-k)!,k,m,n)/m!,m,1,n);
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x*exp(x)+x^2*exp(2*x)))) \\ Seiichi Manyama, Jul 17 2023

Formula

a(n) = n!*sum(m=1..n, sum(k=m..n, (k^(n-k)*binomial(m,k-m))/(n-k)!)/m!), n>0, a(0)=1.

Extensions

More terms from Harvey P. Dale, Sep 22 2011

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

Views

Author

Vladeta Jovovic, Apr 07 2001

Keywords

Comments

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

References

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

Crossrefs

Cf. A000949-A000951, A060905-A060913.

Programs

  • Mathematica
    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 *)

Formula

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(x*exp(x*exp(x)) + 1/3*x^3*exp(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

Views

Author

Vladeta Jovovic, Apr 07 2001

Keywords

References

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

Crossrefs

Cf. A000949 - A000951, A060905 - A060913.

Formula

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(x*exp(x*exp(x)) + 1/2*x^2*exp(x*exp(x))^2 + 1/4*x^4*exp(x*exp(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

Views

Author

Vladeta Jovovic, Apr 07 2001

Keywords

References

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

Crossrefs

Cf. A000949 - A000951, A060905 - A060913.

Formula

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(x*exp(x*exp(x*exp(x))) + 1/2*x^2*exp(x*exp(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

Views

Author

Vladeta Jovovic, Apr 07 2001

Keywords

References

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

Crossrefs

Cf. A000949 - A000951, A060905 - A060913.

Formula

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(x*exp(x*exp(x*exp(x*exp(x)))) + 1/2*x^2*exp(x*exp(x*exp(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

Views

Author

Vladeta Jovovic, Apr 07 2001

Keywords

References

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

Crossrefs

Cf. A000949 - A000951, A060905 - A060913.

Formula

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.
Showing 1-10 of 11 results. Next

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