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 1-3 of 3 results.

A108459 Number of labeled partitions of (n,n) into pairs (i,j).

Original entry on oeis.org

1, 1, 5, 52, 855, 19921, 614866, 24040451, 1152972925, 66200911138, 4465023867757, 348383154017581, 31052765897026352, 3128792250765898965, 353179564583216567917, 44320731930172534543092, 6141797839043095806714667, 934330605640859569909566925
Offset: 0

Views

Author

Christian G. Bower, Jun 03 2005

Keywords

Comments

Partitions of n black objects labeled 1..n and n white objects labeled 1..n. Each partition must have at least one white object.
a(n) is also the number of elements of the partition monoid P_n with domain {1,...,n}. Elements of P_n are set partitions of {1,1',...,n,n'}, and the domain of such a partition is the set of all points in {1,...,n} that belong to a block containing a dashed element. - James East, Apr 10 2018

Links

Crossrefs

Main diagonal of A108458. Cf. A108461.
Cf. A048993 (Stirling2), A068424 (falling factorial).
Cf. A326600, A326270, A326271, A326288.
Bisection of A124421 (even part).

Programs

  • Maple
    b:= proc(n) option remember; expand(`if`(n=0, 1,
      x*add(b(n-j)*binomial(n-1, j-1), j=1..n)))
    end:
a:= n-> add(coeff(b(n), x, j)*j^n, j=0..n):
seq(a(n), n=0..21);  # Alois P. Heinz, Dec 02 2023
  • Mathematica
    a[n_] := If[n == 0, 1, Sum[k^n*StirlingS2[n, k], {k, 0, n}]];
Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Dec 10 2024 *)
  • PARI
    {a(n)=polcoeff(sum(m=0, n, m^m*x^m/prod(k=1, m, 1-m*k*x +x*O(x^n))), n)} \\ Paul D. Hanna, Sep 17 2013
  • PARI
    {a(n)=n!*polcoeff(sum(m=0, n, (exp(m*x+x*O(x^n))-1)^m/m!), n)} \\ Paul D. Hanna, Sep 17 2013

Formula

a(n) = Sum_{k=0..n} k^n*Stirling2(n,k). - Vladeta Jovovic, Aug 31 2006
E.g.f.: Sum_{n>=0} (exp(n*x)-1)^n / n!. - Vladeta Jovovic, Jul 12 2007
E.g.f.: Sum_{n>=0} exp(n^2*x) * exp( -exp(n*x) ) / n!. - Paul D. Hanna, Jun 28 2019
O.g.f.: Sum_{n>=0} n^n * x^n / Product_{k=1..n} (1 - n*k*x). - Paul D. Hanna, Sep 17 2013
a(n) = Sum_{k=0..n} Stirling2(n,k) * Sum_{l=k..n} Stirling2(n,l)*T(l,k). Here T(l,k) are the falling factorials. - James East, Apr 10 2018

A326271 E.g.f.: Sum_{n>=0} 3^n * (exp(n*x) - 1)^n / n!.

Original entry on oeis.org

1, 3, 39, 948, 34869, 1757163, 114320118, 9226773993, 897658726215, 103005144933870, 13705015429716807, 2085418048857405375, 358813807291982519184, 69146346672687725451039, 14803634157756603606592167, 3496440993213535696041009924, 905508769623362527769907535857, 255762146004426658313683324505247, 78413243604482526944814375388910526, 25983968388767783226046397856822603645
Offset: 0

Views

Author

Paul D. Hanna, Jun 28 2019

Keywords

Comments

More generally, the following sums are equal:
(1) Sum_{n>=0} (p + q^n)^n * r^n/n!,
(2) Sum_{n>=0} q^(n^2) * exp(p*q^n*r) * r^n/n!;
here, q = exp(x) with p = -1, r = 3.
In general, let F(x) be a formal power series in x such that F(0)=1, then
Sum_{n>=0} m^n * F(q^n*r)^b * log( F(q^n*r) )^n / n! =
Sum_{n>=0} r^n * [y^n] F(y)^(m*q^n + p);
here, F(x) = exp(x), q = exp(x), p = -1, r = 3, m = 1.

Examples

			E.g.f.: A(x) = 1 + 3*x + 39*x^2/2! + 948*x^3/3! + 34869*x^4/4! + 1757163*x^5/5! + 114320118*x^6/6! + 9226773993*x^7/7! + 897658726215*x^8/8! + 103005144933870*x^9/9! + ...
such that
A(x) = 1 + 3*(exp(x) - 1) + 3^2*(exp(2*x) - 1)^2/2! + 3^3*(exp(3*x) - 1)^3/3! + 3^4*(exp(4*x) - 1)^4/4! + 3^5*(exp(5*x) - 1)^5/5! + 3^6*(exp(6*x) - 1)^6/6! + ...
also
A(x) = exp(-3) + 3*exp(x)*exp(-3*exp(x)) + 3^2*exp(4*x)*exp(-3*exp(2*x))/2! + 3^3*exp(9*x)*exp(-3*exp(3*x))/3! + 3^4*exp(16*x)*exp(-3*exp(4*x))/4! + 3^5*exp(25*x)*exp(-3*exp(5*x))/5! + 3^6*exp(36*x)*exp(-3*exp(6*x))/6! + ...
ORDINARY GENERATING FUNCTION.
O.g.f.: B(x) = 1 + 3*x + 39*x^2 + 948*x^3 + 34869*x^4 + 1757163*x^5 + 114320118*x^6 + 9226773993*x^7 + 897658726215*x^8 + 103005144933870*x^9 + ...
such that
B(x) = 1 + 3*x/(1-x) + 3^2*2^2*x^2/((1-2*x)*(1-4*x)) + 3^3*3^3*x^3/((1-3*x)*(1-6*x)*(1-9*x)) + 3^4*4^4*x^4/((1-4*x)*(1-8*x)*(1-12*x)*(1-16*x)) + 3^5*5^5*x^5/((1-5*x)*(1-10*x)*(1-15*x)*(1-20*x)*(1-25*x)) + ...

Links

Crossrefs

Cf. A108459, A326270, A326288.

Programs

  • PARI
    {a(n) = sum(k=0, n, 3^k * k^n * stirling(n, k, 2) )}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    /* E.g.f.: Sum_{n>=0} 3^n * (exp(n*x) - 1)^n / n! */
{a(n) = n! * polcoeff(sum(m=0, n, 3^m * (exp(m*x +x*O(x^n)) - 1)^m / m!), n)}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    /* O.g.f.: Sum_{n>=0} 3^n * n^n * x^n / Product_{k=1..n} (1 - n*k*x) */
{a(n) = polcoeff(sum(m=0, n, 3^m * m^m * x^m / prod(k=1, m, 1-m*k*x +x*O(x^n))), n)}
for(n=0, 30, print1(a(n), ", "))

Formula

E.g.f.: Sum_{n>=0} 3^n * (exp(n*x) - 1)^n / n!.
E.g.f.: Sum_{n>=0} 3^n * exp(n^2*x) * exp( -3*exp(n*x) ) / n!.
O.g.f.: Sum_{n>=0} 3^n * n^n * x^n / Product_{k=1..n} (1 - n*k*x).
a(n) = Sum_{k=0..n} 3^k * k^n * Stirling2(n,k).

A326288 E.g.f.: Sum_{n>=0} 4^n * (exp(n*x) - 1)^n / n!.

Original entry on oeis.org

1, 4, 68, 2116, 98436, 6217924, 503491204, 50282169284, 6023071906180, 847321700204740, 137695169475601540, 25505309294030757316, 5326002105122774427524, 1242268006104279981404868, 321107726934189274515747460, 91359880704866957348006879172, 28441686041231472428045000672644, 9637951929231839144943126955386052, 3538621024404268912313596289954242692, 1401869934089183216934147248975602680260
Offset: 0

Views

Author

Paul D. Hanna, Jun 28 2019

Keywords

Comments

More generally, the following sums are equal:
(1) Sum_{n>=0} (p + q^n)^n * r^n/n!,
(2) Sum_{n>=0} q^(n^2) * exp(p*q^n*r) * r^n/n!;
here, q = exp(x) with p = -1, r = 4.
In general, let F(x) be a formal power series in x such that F(0)=1, then
Sum_{n>=0} m^n * F(q^n*r)^b * log( F(q^n*r) )^n / n! =
Sum_{n>=0} r^n * [y^n] F(y)^(m*q^n + p);
here, F(x) = exp(x), q = exp(x), p = -1, r = 4, m = 1.

Examples

			E.g.f.: A(x) = 1 + 4*x + 68*x^2/2! + 2116*x^3/3! + 98436*x^4/4! + 6217924*x^5/5! + 503491204*x^6/6! + 50282169284*x^7/7! + 6023071906180*x^8/8! + 847321700204740*x^9/9! + ...
such that
A(x) = 1 + 4*(exp(x) - 1) + 4^2*(exp(2*x) - 1)^2/2! + 4^3*(exp(3*x) - 1)^3/3! + 4^4*(exp(4*x) - 1)^4/4! + 4^5*(exp(5*x) - 1)^5/5! + 4^6*(exp(6*x) - 1)^6/6! + ...
also
A(x) = exp(-4) + 4*exp(x)*exp(-4*exp(x)) + 4^2*exp(4*x)*exp(-4*exp(2*x))/2! + 4^3*exp(9*x)*exp(-4*exp(3*x))/3! + 4^4*exp(16*x)*exp(-4*exp(4*x))/4! + 4^5*exp(25*x)*exp(-4*exp(5*x))/5! + 4^6*exp(36*x)*exp(-4*exp(6*x))/6! + ...
ORDINARY GENERATING FUNCTION.
O.g.f.: B(x) = 1 + 4*x + 68*x^2 + 2116*x^3 + 98436*x^4 + 6217924*x^5 + 503491204*x^6 + 50282169284*x^7 + 6023071906180*x^8 + 847321700204740*x^9 + ...
such that
B(x) = 1 + 4*x/(1-x) + 4^2*2^2*x^2/((1-2*x)*(1-4*x)) + 4^3*3^3*x^3/((1-3*x)*(1-6*x)*(1-9*x)) + 4^4*4^4*x^4/((1-4*x)*(1-8*x)*(1-12*x)*(1-16*x)) + 4^5*5^5*x^5/((1-5*x)*(1-10*x)*(1-15*x)*(1-20*x)*(1-25*x)) + ...

Links

Crossrefs

Cf. A108459, A326270, A326271.

Programs

  • PARI
    {a(n) = sum(k=0, n, 4^k * k^n * stirling(n, k, 2) )}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    /* E.g.f.: Sum_{n>=0} 4^n * (exp(n*x) - 1)^n / n! */
{a(n) = n! * polcoeff(sum(m=0, n, 4^m * (exp(m*x +x*O(x^n)) - 1)^m / m!), n)}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    /* O.g.f.: Sum_{n>=0} 4^n * n^n * x^n / Product_{k=1..n} (1 - n*k*x) */
{a(n) = polcoeff(sum(m=0, n, 4^m * m^m * x^m / prod(k=1, m, 1-m*k*x +x*O(x^n))), n)}
for(n=0, 30, print1(a(n), ", "))

Formula

E.g.f.: Sum_{n>=0} 4^n * (exp(n*x) - 1)^n / n!.
E.g.f.: Sum_{n>=0} 4^n * exp(n^2*x) * exp( -4*exp(n*x) ) / n!.
O.g.f.: Sum_{n>=0} 4^n * n^n * x^n / Product_{k=1..n} (1 - n*k*x).
a(n) = Sum_{k=0..n} 4^k * k^n * Stirling2(n,k).
Showing 1-3 of 3 results.

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