A218143 -id:A218143 - cp's OEIS frontend

A218141 a(n) = Stirling2(n^2, n).

1, 1, 7, 3025, 171798901, 2436684974110751, 14204422416132896951197888, 50789872166903636182659702516635946082, 155440114706926165785630654089245708839702615196926765, 541500903058656141876322139677626107784896646583041951351456223689104719

Offset: 0

Paul D. Hanna, Oct 21 2012

nonn

			O.g.f.: A(x) = 1 + x + 7*x^2 + 3025*x^3 + 171798901*x^4 + 2436684974110751*x^5 +...

Paul D. Hanna, Table of n, a(n) for n = 0..25

Cf. A008277, A218142, A218143, A007820, A217913, A217914, A217915.

Table[StirlingS2[n^2, n],{n,0,10}] (* Vaclav Kotesovec, May 11 2014 *)

makelist(stirling2(n^2,n),n,0,30 ); /* Martin Ettl, Oct 21 2012 */

{a(n)=polcoeff(sum(k=0,n,(k^n)^k*exp(-k^n*x +x*O(x^n))*x^k/k!),n)}

{a(n)=polcoeff(1/prod(k=1, n, 1-k*x +x*O(x^(n^2+1))), n^2-n)}

{Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
{a(n) = Stirling2(n^2, n)}
for(n=0, 10, print1(a(n), ", "))

a(n) = [x^n] Sum_{k>=0} k^(n*k) * exp(-k^n*x) * x^k / k!.
a(n) = [x^(n^2-n)] 1 / Product_{k=1..n} (1-k*x).
a(n) ~ n^(n^2)/n!. - Vaclav Kotesovec, May 11 2014

A218142 a(n) = Stirling2(n^2+n, n).

Original entry on oeis.org

1, 1, 31, 86526, 45232115901, 7713000216608565075, 666480349285726891499539272955, 41929298560838945526242744414099901692285884, 2610516895723221966171633379256064857587637240616032299710417

Offset: 0

Paul D. Hanna, Oct 21 2012

nonn

			O.g.f.: A(x) = 1 + x + 31*x^2 + 86526*x^3 + 45232115901*x^4 +...

Paul D. Hanna, Table of n, a(n) for n = 0..25

Cf. A008277, A218141, A218143, A007820, A217913, A217914, A217915.

Table[StirlingS2[n^2+n, n],{n,0,10}] (* Vaclav Kotesovec, May 11 2014 *)

makelist(stirling2(n^2+n,n),n,0,30 ); /* Martin Ettl, Oct 21 2012 */

{a(n)=polcoeff(sum(k=0,n,(k^(n+1))^k*exp(-k^(n+1)*x +x*O(x^n))*x^k/k!),n)}

{a(n)=polcoeff(1/prod(k=1, n, 1-k*x +x*O(x^(n^2))), n^2)}

{Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
{a(n) = Stirling2(n^2+n, n)}
for(n=0, 10, print1(a(n), ", "))

a(n) = [x^n] Sum_{k>=0} k^((n+1)*k) * exp(-k^(n+1)*x) * x^k / k!.
a(n) = [x^(n^2)] 1 / Product_{k=1..n} (1-k*x).
a(n) ~ n^(n^2+n)/n!. - Vaclav Kotesovec, May 11 2014

cp's OEIS Frontend

A218141 a(n) = Stirling2(n^2, n).

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A218142 a(n) = Stirling2(n^2+n, n).

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Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Maxima

PARI

PARI

PARI

Formula