|
| |
|
|
A129506
|
|
Number of partitions of a {2n-1}-set into n nonempty subsets.
|
|
9
|
|
|
|
1, 3, 25, 350, 6951, 179487, 5715424, 216627840, 9528822303, 477297033785, 26826851689001, 1672162773483930, 114485073343744260, 8541149231801585700, 689692892575539953400, 59932861644880019603520, 5576731051262006158950735, 553234633385550257808059085
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,2
|
|
|
COMMENTS
|
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
|
|
|
LINKS
|
|
|
|
FORMULA
|
Central Stirling numbers of the second kind: a(n) = A008277(2n-1,n) for n >= 1.
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
a(n) = 1/n! * Sum_{k=1..n} (-1)^(n-k) * binomial(n,k) * k^(2*n-1). - Paul D. Hanna, Oct 15 2012
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
a(n-1) = sum(j=0..n, binomial(2*n,j)*stirling2(2*n-j,n)). - Vladimir Kruchinin, Jun 14 2013
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
|
|
|
EXAMPLE
|
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
|
|
|
MAPLE
|
a:= n-> Stirling2(2*n-1, n):
|
|
|
MATHEMATICA
|
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 *)
|
|
|
PROG
|
(PARI) a(n)=polcoeff(1/prod(k=1, n, 1-k*x +x*O(x^n)), n-1)
(PARI) vector(66, n, abs( stirling(2*n-1, n, 2) ) ) /* Joerg Arndt, Jun 09 2012 */
(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
(PARI) {a(n)=polcoeff(sum(m=1, n, m^(2*m-1)*x^m*exp(-m^2*x+x*O(x^n))/m!), n)}
for(n=1, 20, print1(a(n), ", "))
(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);
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
