A361142 E.g.f. satisfies A(x) = exp( x*A(x)^2/(1 - x*A(x)) ).
1, 1, 7, 91, 1773, 46401, 1529593, 60911103, 2845757449, 152663425633, 9250206248781, 624880915165959, 46569571425664477, 3795729136868379777, 335902071304953561073, 32074779600414913885231, 3287242849289861637185937, 359917016243351870997841473
Offset: 0
Keywords
Links
- Winston de Greef, Table of n, a(n) for n = 0..338
Programs
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Mathematica
Table[n! * Sum[(n+k+1)^(k-1) * Binomial[n-1,n-k]/k!, {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Mar 03 2023 *) -
PARI
a(n) = n!*sum(k=0, n, (n+k+1)^(k-1)*binomial(n-1, n-k)/k!);
Formula
a(n) = n! * Sum_{k=0..n} (n+k+1)^(k-1) * binomial(n-1,n-k)/k!.
a(n) ~ s^2 * sqrt((2 - r*s)/(2 + r*s*(-2 + s*(2 - r*s)^2))) * n^(n-1) / (exp(n) * r^(n - 1/2)), where r = 0.14220768719194290600038416000340972911571484385125... and s = 1.549730657609106944767484487465870359529391502493... are roots of the system of equations exp(r*s^2/(1 - r*s)) = s, r*s^2*(2 - r*s) = (1 - r*s)^2. - Vaclav Kotesovec, Mar 03 2023