A326271 -id:A326271 - cp's OEIS frontend

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

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

Offset: 0

Christian G. Bower, Jun 03 2005

nonn

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

Seiichi Manyama, Table of n, a(n) for n = 0..200

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

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

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 *)

{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

{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

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

A326270 E.g.f.: Sum_{n>=0} 2^n * (exp(n*x) - 1)^n / n!.

Original entry on oeis.org

1, 2, 18, 314, 8434, 314362, 15278642, 928696442, 68509258098, 5995762219514, 611538502747826, 71656036268121978, 9532232740451770866, 1425414297318661354746, 237588200534263288095538, 43821269448954050939558522, 8887255081413035850889914994, 1970841722610600810208914571258, 475544555000142351430865220032434, 124299766720856839788225909600114042, 35056463298676734373530025799446104818

Offset: 0

Paul D. Hanna, Jun 28 2019

nonn

More generally, the following sums are equal:
Sum_{n>=0} (p + q^n)^n * r^n/n! =
Sum_{n>=0} q^(n^2) * exp(p*q^n*r) * r^n/n!;
here, q = exp(x) with p = -1, r = 2.
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 = 2, m = 1.

			E.g.f.: A(x) = 1 + 2*x + 18*x^2/2! + 314*x^3/3! + 8434*x^4/4! + 314362*x^5/5! + 15278642*x^6/6! + 928696442*x^7/7! + 68509258098*x^8/8! + 5995762219514*x^9/9! + 611538502747826*x^10/10! + ...
such that
A(x) = 1 + 2*(exp(x) - 1) + 2^2*(exp(2*x) - 1)^2/2! + 2^3*(exp(3*x) - 1)^3/3! + 2^4*(exp(4*x) - 1)^4/4! + 2^5*(exp(5*x) - 1)^5/5! + 2^6*(exp(6*x) - 1)^6/6! + ...
also
A(x) = exp(-2) + 2*exp(x)*exp(-2*exp(x)) + 2^2*exp(4*x)*exp(-2*exp(2*x))/2! + 2^3*exp(9*x)*exp(-2*exp(3*x))/3! + 2^4*exp(16*x)*exp(-2*exp(4*x))/4! + 2^5*exp(25*x)*exp(-2*exp(5*x))/5! + 2^6*exp(36*x)*exp(-2*exp(6*x))/6! + ...
ORDINARY GENERATING FUNCTION.
O.g.f.: B(x) = 1 + 2*x + 18*x^2 + 314*x^3 + 8434*x^4 + 314362*x^5 + 15278642*x^6 + 928696442*x^7 + 68509258098*x^8 + 5995762219514*x^9 + ...
such that
B(x) = 1 + 2*x/(1-x) + 2^2*2^2*x^2/((1-2*x)*(1-4*x)) + 2^3*3^3*x^3/((1-3*x)*(1-6*x)*(1-9*x)) + 2^4*4^4*x^4/((1-4*x)*(1-8*x)*(1-12*x)*(1-16*x)) + 2^5*5^5*x^5/((1-5*x)*(1-10*x)*(1-15*x)*(1-20*x)*(1-25*x)) + ...
RELATED SERIES.
Below we illustrate the following identity at specific values of x:
Sum_{n>=0} 2^n * (exp(n*x) - 1)^n / n!  =  Sum_{n>=0} 2^n * exp(n^2*x) * exp( -2*exp(n*x) ) / n!.
(1) At x = -1, the following sums are equal
S1 = Sum_{n>=0} (-2)^n * (1 - exp(-n))^n / n!,
S1 = Sum_{n>=0} 2^n * exp(-n^2) * exp( -2*exp(-n) ) / n!,
where S1 = 0.51596189603321982013621912500044621350106513780391377129738...
(2) At x = -2, the following sums are equal
S2 = Sum_{n>=0} (-2)^n * (1 - exp(-2*n))^n / n!,
S2 = Sum_{n>=0} 2^n * exp(-2*n^2) * exp( -2*exp(-2*n) ) / n!,
where S2 = 0.34246794778612083304129071190905516612972983097016819355092...
(3) At x = -log(2), the following sums are equal
S3 = Sum_{n>=0} 2^(-n*(n-1)) * (2^n - 1)^n * (-1)^n / n!,
S3 = Sum_{n>=0} 2^(-n*(n-1)) * exp( -1/2^(n-1) ) / n!,
where S3 = 0.58106816860114387883649557314841837351794236167582918403231...

Paul D. Hanna, Table of n, a(n) for n = 0..300

Cf. A108459, A326271, A326288.

Flatten[{1, Table[Sum[2^k * k^n * StirlingS2[n, k], {k, 0, n}], {n, 1, 20}]}] (* Vaclav Kotesovec, Jul 09 2019 *)

{a(n) = sum(k=0,n, 2^k * k^n * stirling(n,k,2) )}
for(n=0,30,print1(a(n),", "))

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

/* O.g.f.: Sum_{n>=0} 2^n * n^n * x^n / Product_{k=1..n} (1 - n*k*x) */
{a(n) = polcoeff(sum(m=0, n, 2^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),", "))

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

Paul D. Hanna, Jun 28 2019

nonn

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.

			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)) + ...

Paul D. Hanna, Table of n, a(n) for n = 0..300

Cf. A108459, A326270, A326271.

{a(n) = sum(k=0, n, 4^k * k^n * stirling(n, k, 2) )}
for(n=0, 30, print1(a(n), ", "))

/* 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), ", "))

/* 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), ", "))

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).

cp's OEIS Frontend

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

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A326270 E.g.f.: Sum_{n>=0} 2^n * (exp(n*x) - 1)^n / n!.

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

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