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

%I A007820 #78 Jun 09 2025 11:29:30
%S A007820 1,1,7,90,1701,42525,1323652,49329280,2141764053,106175395755,
%T A007820 5917584964655,366282500870286,24930204590758260,1850568574253550060,
%U A007820 148782988064375309400,12879868072770626040000,1194461517469807833782085,118144018577011378596484455
%N A007820 Stirling numbers of second kind S(2n,n).
%C A007820 Chan and Manna prove that a(n) is odd if and only if n is in A003714. - _Jason Kimberley_, Sep 14 2009
%C A007820 The number of ways to partition a set of 2*n elements into n disjoint subsets. - _Vladimir Reshetnikov_, Oct 10 2016
%C A007820 Conjecture: a(2*n+1) is divisible by (2*n + 1)^2. - _Peter Bala_, Mar 30 2025
%D A007820 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.
%H A007820 Alois P. Heinz, <a href="/A007820/b007820.txt">Table of n, a(n) for n = 0..345</a> (terms n = 1..200 from Vincenzo Librandi)
%H A007820 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H A007820 O-Y. Chan and D. V. Manna, <a href="http://www.oyeat.com/papers/stirling9.pdf">Divisibility properties of Stirling numbers of the second kind</a> [From _Jason Kimberley_, Sep 14 2009]
%F A007820 a(n) = A048993(2n,n). - _R. J. Mathar_, Mar 15 2011
%F A007820 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
%F A007820 a(n) = 1/n! * Sum_{k = 0..n} binomial(n,k)*(-1)^k*(n-k)^(2*n). - _Emanuele Munarini_, Jul 01 2011
%F A007820 a(n) = [x^n] 1 / Product_{k=1..n} (1-k*x). - _Paul D. Hanna_, Oct 17 2012
%F A007820 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
%F A007820 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
%F A007820 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
%e A007820 G.f.: A(x) = 1 + x + 7*x^2 + 90*x^3 + 1701*x^4 + 42525*x^5 +...,
%e A007820 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
%p A007820 A007820 := proc(n) Stirling2(2*n,n) ; end proc:
%p A007820 seq(A007820(n),n=0..20) ; # _R. J. Mathar_, Mar 15 2011
%t A007820 Table[StirlingS2[2n, n], {n, 1, 12}] (* _Emanuele Munarini_, Mar 12 2011 *)
%o A007820 (Sage) [stirling_number2(2*i,i) for i in range(1,20)] # _Zerinvary Lajos_, Jun 26 2008
%o A007820 (Maxima) makelist(stirling2(2*n,n),n,0,12); /* _Emanuele Munarini_, Mar 12 2011 */
%o A007820 (PARI) a(n)=stirling(2*n,n,2); /* _Joerg Arndt_, Jul 01 2011 */
%o A007820 (PARI) {a(n)=polcoeff(1/prod(k=1, n, 1-k*x +x*O(x^(2*n))), n)} \\ _Paul D. Hanna_, Oct 17 2012
%o A007820 (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
%o A007820 (Python)
%o A007820 from sympy.functions.combinatorial.numbers import stirling
%o A007820 def A007820(n): return stirling(n<<1,n) # _Chai Wah Wu_, Jun 09 2025
%Y A007820 Cf. A002465, A008277, A187646, A191236, A217913, A217914, A217915, A247238.
%Y A007820 Cf. A350376, A383862, A384060.
%K A007820 nonn,easy
%O A007820 0,3
%A A007820 kemp(AT)sads.informatik.uni-frankfurt.de (Rainer Kemp)
%E A007820 Typo in Mathematica program fixed by _Vincenzo Librandi_, May 04 2013
%E A007820 a(0)=1 prepended by _Alois P. Heinz_, Feb 01 2018