OFFSET
1,3
COMMENTS
For n>1, a(n) is the number of set partitions of [2*n-2] into n blocks, i.e., Stirling2(2*n-2, n). E.g., a(3) = 6: [12|3|4, 13|2|4, 1|23|4, 14|2|3, 1|24|3, 1|2|34]. - Yuchun Ji, Jan 12 2021
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..250
FORMULA
a(n) = (1/n!) * Sum_{k=1..n} (-1)^(n-k) * binomial(n,k) * (k^2)^(n-1).
a(n) = [x^n] x + x^2/Product_{k=1..n} (1-k*x).
a(n) = [x^n] x + x^2*(1+x)^(2*n-3) / Product_{k=1..n-1} (1-k*x).
a(n) = Sum_{j=0..n-1} binomial(2*n-1,j)*Stirling2(2*n-j-1,n). - Vladimir Kruchinin, Jun 14 2013
a(n) ~ 2^(2*n-5/2) * n^(n-5/2) / (sqrt(Pi*(1-c)) * exp(n) * c^n *(2-c)^(n-2)), where c = -LambertW(-2*exp(-2)) = 0.4063757399599599... . - Vaclav Kotesovec, May 20 2014
EXAMPLE
O.g.f.: A(x) = x + x^2 + 6*x^3 + 65*x^4 + 1050*x^5 + 22827*x^6 + 627396*x^7 + ... where A(x) = 1^0*x*exp(-1*x) + 2^2*exp(-2^2*x)*x^2/2! + 3^4*exp(-3^2*x)*x^3/3! + 4^6*exp(-4^2*x)*x^4/4! + 5^8*exp(-5^2*x)*x^5/5! + ... simplifies to a power series in x with integer coefficients.
MATHEMATICA
a[n_] := Sum[ Binomial[2*n - 3, j]*StirlingS2[2*n - j - 3, n-1], {j, 0, n-2}]; a[1] = 1; Table[a[n], {n, 1, 19}] (* Jean-François Alcover, Jun 14 2013, after Vladimir Kruchinin *)
PROG
(PARI) {a(n)=polcoeff(sum(m=1, n, (m^2)^(m-1)*x^m*exp(-m^2*x+x*O(x^n))/m!), n)}
(PARI) {a(n)=1/n!*sum(k=1, n, (-1)^(n-k)*binomial(n, k)*(k^2)^(n-1))}
(PARI) {a(n)=polcoeff(x+x^2/prod(k=1, n, 1-k*x +x*O(x^n)), n)}
(PARI) {a(n)=polcoeff(x+x^2*(1+x)^(2*n-3)/prod(k=0, n-1, 1-k*x +x*O(x^n)), n)}
for(n=1, 20, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Oct 14 2012
STATUS
approved