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
Conjecture: a(2*n+1) is divisible by (2*n + 1)^2. - Peter Bala, Mar 30 2025
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.
Adelberg, Arnold, and Tamás Lengyel. "New Results on the 2-Adic Valuation of the Central Stirling Numbers S(2k, k)." The Fibonacci Quarterly 63.2 (2025): 146-162.
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) = 1/n! * Sum_{k = 0..n} binomial(n,k)*(-1)^k*(n-k)^(2*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^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
(SageMath) [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
(Python)
from sympy.functions.combinatorial.numbers import stirling
def A007820(n): return stirling(n<<1, n) # Chai Wah Wu, Jun 09 2025
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Rainer Kemp (kemp(AT)sads.informatik.uni-frankfurt.de)
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
