A007820 Stirling numbers of second kind S(2n,n).

1, 1, 7, 90, 1701, 42525, 1323652, 49329280, 2141764053, 106175395755, 5917584964655, 366282500870286, 24930204590758260, 1850568574253550060, 148782988064375309400, 12879868072770626040000, 1194461517469807833782085, 118144018577011378596484455

Offset

0,3

Comments

Chan and Manna prove that a(n) is odd if and only if n is in A003714. - Jason Kimberley, Sep 14 2009

The number of ways to partition a set of 2*n elements into n disjoint subsets. - Vladimir Reshetnikov, Oct 10 2016

References

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 835.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..345 (terms n = 1..200 from Vincenzo Librandi)

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

O-Y. Chan and D. V. Manna, Divisibility properties of Stirling numbers of the second kind [From Jason Kimberley, Sep 14 2009]

Formula

a(n) = A048993(2n,n). - R. J. Mathar, Mar 15 2011

Asymptotic: a(n) ~ (4*n/(e*z*(2-z)))^n/sqrt(2*Pi*n*(z-1)), where z = A256500 = 1.59362426... is a root of the equation exp(z)*(2-z)=2. - Vaclav Kotesovec, May 30 2011

a(n) = sum(binomial(n,k)*(-1)^k*(n-k)^(2*n),k,0,n)/n!. - Emanuele Munarini, Jul 01 2011

a(n) = [x^n] 1 / Product_{k=1..n} (1-k*x). - Paul D. Hanna, Oct 17 2012

O.g.f.: Sum_{n>=1} (n^2)^n * exp(-n^2*x) * x^n/n! = Sum_{n>=1} S2(2*n,n)*x^n. - Paul D. Hanna, Oct 17 2012

G.f.: sum(n>0, a(n)*n!/(2*n)!) = x*B'(x)/B(x)-1, where B(x) satisfies B(x)^2 = x*(exp(B(x))-1). - Vladimir Kruchinin, Mar 13 2013

a(n) = sum(j=0..n, (-1)^(n-j)*n^j*binomial(2*n,j)*stirling2(2*n-j,n)). - Vladimir Kruchinin, Jun 14 2013

Example

G.f.: A(x) = 1 + x + 7*x^2 + 90*x^3 + 1701*x^4 + 42525*x^5 +...
where A(x) = 1 + 1^2*x*exp(-1*x) + 2^4*exp(-2^2*x)*x^2/2! + 3^6*exp(-3^2*x)*x^3/3! + 4^8*exp(-4^2*x)*x^4/4! + 5^10*exp(-5^2*x)*x^5/5! +... - Paul D. Hanna, Oct 17 2012

Maple

A007820 := proc(n) Stirling2(2*n, n) ; end proc:
seq(A007820(n), n=0..20) ; # R. J. Mathar, Mar 15 2011

Mathematica

Table[StirlingS2[2n, n], {n, 1, 12}] (* Emanuele Munarini, Mar 12 2011 *)

Prog

(Sage) [stirling_number2(2*i, i) for i in range(1, 20)] # Zerinvary Lajos, Jun 26 2008
(Maxima) makelist(stirling2(2*n, n), n, 0, 12); /* Emanuele Munarini, Mar 12 2011 */
(PARI) a(n)=stirling(2*n, n, 2); /* Joerg Arndt, Jul 01 2011 */
(PARI) {a(n)=polcoeff(1/prod(k=1, n, 1-k*x +x*O(x^(2*n))), n)} \\ Paul D. Hanna, Oct 17 2012
(PARI) {a(n)=polcoeff(sum(m=1, n, (m^2)^m*exp(-m^2*x+x*O(x^n))*x^m/m!), n)} \\ Paul D. Hanna, Oct 17 2012

Crossrefs

Cf. A187646, A002465, A191236, A217913, A217914, A217915.

Sequence in context: A360914 A321164 A243699 * A306137 A226624 A266236

Adjacent sequences: A007817 A007818 A007819 * A007821 A007822 A007823

Keyword

nonn,easy

Author

kemp(AT)sads.informatik.uni-frankfurt.de (Rainer Kemp)

Extensions

Typo in Mathematica program fixed by Vincenzo Librandi, May 04 2013

a(0)=1 prepended by Alois P. Heinz, Feb 01 2018

Status

approved