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A363744
E.g.f. satisfies A(x) = exp(x * (1 + x * A(x))^2).
2
1, 1, 5, 31, 313, 3981, 63841, 1223419, 27378737, 701091001, 20221662241, 649032795951, 22945630163017, 886151307346501, 37121193546044609, 1676607954371120611, 81222976991097364321, 4201418329450141471473, 231127287514383805458625
OFFSET
0,3
FORMULA
a(n) = n! * Sum_{k=0..n} (n-k+1)^(k-1) * binomial(2*k,n-k)/k!.
a(n) ~ sqrt((1 + r*s)*(1 + 3*r*s) / (2*(1 + 2*r + 4*r^2*s + 2*r^3*s^2))) * n^(n-1) / (exp(n) * r^(n+1)), where r = 0.302307732979052080722256232095444259577495... and s = 2.910394288602135748195482733301939282588478379746... are real roots of the system of equations exp(r*(1 + r*s)^2) = s, 2*s*r^2*(1 + r*s) = 1. - Vaclav Kotesovec, Nov 18 2023
MATHEMATICA
Join[{1}, Table[n! * Sum[(n-k+1)^(k-1) * Binomial[2*k, n-k]/k!, {k, 0, n}], {n, 1, 20}]] (* Vaclav Kotesovec, Nov 18 2023 *)
PROG
(PARI) a(n) = n!*sum(k=0, n, (n-k+1)^(k-1)*binomial(2*k, n-k)/k!);
CROSSREFS
Cf. A161635.
Sequence in context: A176302 A129586 A135744 * A372162 A307615 A020514
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Aug 17 2023
STATUS
approved