A186451 E.g.f. A(x) satisfies A(x)=exp(x*A(x))*(1+x*A(x))/(1-x*A(x)-x^2*A(x)^2).

1, 3, 29, 514, 13521, 475176, 20967901, 1115481312, 69530059521, 4971803518720, 401273421456381, 36089072534460672, 3579320890641355921, 388132854150472295424, 45685483585216535296125

Offset

0,2

Links

Table of n, a(n) for n=0..14.

Vladimir Kruchinin and D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.

Formula

For n>0, a(n-1) = n!/n^2 * Sum_{m=0..n-1} (Sum_{i=0..m} (binomial(i+n,m-i)*binomial(i+n-1,n-1))*(n^(n-m-1)/(n-m-1)!)). [corrected by Vaclav Kotesovec, Dec 02 2017]

a(n) ~ sqrt(s*(3 - r*s - 6*r^2*s^2 - 2*r^3*s^3 + 2*r^4*s^4 + r^5*s^5) / (r*(11 + 11*r*s - 3*r^2*s^2 - 3*r^3*s^3 + r^4*s^4 + r^5*s^5))) * n^(n-1) / (exp(n) * r^n), where r = 0.1070858770219294378019065333027469181859804667559... and s = 2.341958504012575760306935528206207057409882206827... are real roots of the system of equations exp(r*s)*(1 + r*s) = s*(1 - r*s - r^2*s^2), (-1 + r*s + r^2*s^2)^2 + exp(r*s)*r*(-3 - 2*r*s + r^2*s^2 + r^3*s^3) = 0. - Vaclav Kotesovec, Dec 02 2017

Mathematica

Table[n!/n^2 * Sum[Sum[Binomial[i + n, m - i]*Binomial[i + n - 1, n - 1], {i, 0, m}]*(n^(n - m - 1)/(n - m - 1)!), {m, 0, n - 1}], {n, 1, 20}] (* Vaclav Kotesovec, Dec 02 2017 *)

Prog

(Maxima)
a(n):=(n+1)!/(n+1)^2*sum(sum(binomial(i+n+1, m-i)*binomial(i+n, n), i, 0, m)*((n+1)^(n-m)/(n-m)!), m, 0, n);
makelist(a(n), n, 0, 20); /* fixed by Vaclav Kotesovec, Dec 02 2017 */

Crossrefs

Sequence in context: A366005 A326433 A113871 * A248828 A210827 A092251

Adjacent sequences: A186448 A186449 A186450 * A186452 A186453 A186454

Keyword

nonn

Author

Vladimir Kruchinin, Feb 22 2011

Status

approved