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A129506 Number of partitions of a {2n-1}-set into n nonempty subsets.

%I A129506 #72 Aug 01 2025 14:47:15
%S A129506 1,3,25,350,6951,179487,5715424,216627840,9528822303,477297033785,
%T A129506 26826851689001,1672162773483930,114485073343744260,
%U A129506 8541149231801585700,689692892575539953400,59932861644880019603520,5576731051262006158950735,553234633385550257808059085
%N A129506 Number of partitions of a {2n-1}-set into n nonempty subsets.
%C A129506 B^{-1}(x) = Sum_{n>0} a(n)/(2*n-1)!*(n-1)! x^n is inverse function for x*B(x), where B(x) is g.f. for Bernoulli number (see A027641). - _Vladimir Kruchinin_, Jan 19 2012
%H A129506 Alois P. Heinz, <a href="/A129506/b129506.txt">Table of n, a(n) for n = 1..200</a>
%H A129506 D. Kruchinin and V. Kruchinin, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Kruchinin/kruchinin5.html">A Method for Obtaining Generating Function for Central Coefficients of Triangles</a>, Journal of Integer Sequences, Vol. 15 (2012), article 12.9.3.
%F A129506 Central Stirling numbers of the second kind: a(n) = A008277(2n-1,n) for n >= 1.
%F A129506 G.f.: Sum_{n>=1} n^(2*n-1) * exp(-n^2*x) * x^n / n!, an integer series. - _Paul D. Hanna_, Oct 15 2012
%F A129506 a(n) = 1/n! * Sum_{k=1..n} (-1)^(n-k) * binomial(n,k) * k^(2*n-1). - _Paul D. Hanna_, Oct 15 2012
%F A129506 a(n) = ((2*n-1)*((sum(i=1..n-2, (stirling2(2*i-1,i)*C(2*n-2,2*i-1)*stirling2(2*(n-i)-1,n-i-1))/((n-i-1)*C(n-1,i))))+(n-1)*stirling2(2*n-3,n-1) +stirling2(2*n-2,n-1)))/n. - _Vladimir Kruchinin_, Feb 28 2013
%F A129506 a(n-1) = sum(j=0..n, binomial(2*n,j)*stirling2(2*n-j,n)). - _Vladimir Kruchinin_, Jun 14 2013
%F A129506 a(n) ~ 2^(2*n-3/2) * n^(n-3/2) * (2-c)^(1-n) / (sqrt(Pi*(1-c)) * exp(n) * c^n), where c = -LambertW(-2*exp(-2)) = 0.4063757399599599... = 2*A106533. - _Vaclav Kotesovec_, Dec 15 2013
%F A129506 a(n) = A258170(2*n-1,n). - _Alois P. Heinz_, Mar 16 2018
%e A129506 G.f.: A(x) = x + 3*x^2 + 25*x^3 + 350*x^4 + 6951*x^5 + 179487*x^6 + ... where A(x) = 1^1*x*exp(-1^2*x) + 2^3*exp(-2^2*x)*x^2/2! + 3^5*exp(-3^2*x)*x^3/3! + 4^7*exp(-4^2*x)*x^4/4! + 5^9*exp(-5^2*x)*x^5/5! + ... forms a power series in x with integer coefficients. - _Paul D. Hanna_, Oct 15 2012
%p A129506 a:= n-> Stirling2(2*n-1, n):
%p A129506 seq(a(n), n=1..25);  # _Alois P. Heinz_, Dec 15 2013
%t A129506 a[n_] := Sum[ Binomial[2*n - 2, j]*StirlingS2[2*n - j - 2, n-1], {j, 0, n-1}]; Table[a[n], {n, 1, 18}] (* _Jean-François Alcover_, Jun 14 2013, after _Vladimir Kruchinin_ *)
%t A129506 Table[StirlingS2[2*n-1,n], {n, 1, 20}] (* _Vaclav Kotesovec_, Dec 15 2013 *)
%o A129506 (PARI) a(n)=polcoeff(1/prod(k=1,n,1-k*x +x*O(x^n)),n-1)
%o A129506 (PARI) vector(66, n, abs( stirling(2*n-1, n, 2) ) ) /* _Joerg Arndt_, Jun 09 2012 */
%o A129506 (PARI) {a(n)=1/n!*sum(k=0,n, (-1)^(n-k)*binomial(n,k)*k^(2*n-1))} \\ _Paul D. Hanna_, Oct 15 2012
%o A129506 (PARI) {a(n)=polcoeff(sum(m=1,n,m^(2*m-1)*x^m*exp(-m^2*x+x*O(x^n))/m!),n)}
%o A129506 for(n=1,20,print1(a(n),", "))
%o A129506 (Maxima) a(n):=((2*n-1)*((sum((stirling2(2*i-1, i)*binomial(2*n-2, 2*i-1)*stirling2(2*(n-i)-1, n-i-1))/((n-i-1)*binomial(n-1, i)), i, 1, n-2))+(n-1)* stirling2(2*n-3, n-1)+stirling2(2*n-2, n-1)))/(n);
%o A129506   makelist(a(n),n,1,10); /* _Vladimir Kruchinin_, Feb 28 2013 */
%Y A129506 Cf. A008277, A129505, A217900, A217910, A217913, A258170.
%K A129506 nonn
%O A129506 1,2
%A A129506 _Paul D. Hanna_, Apr 18 2007