A397423 a(n) = Sum_{j=0..n} abs(Stirling1(n, j)) * binomial(n + j, n) * n^j.

1, 2, 30, 834, 34120, 1846970, 124495056, 10047807546, 944801560704, 101438766492282, 12244506725818400, 1641472208008246548, 241972783517997222912, 38901934135879029948980, 6774014909661031663285760, 1270083656346147360159806250, 255106807553373977988264132608

Offset

0,2

Links

Table of n, a(n) for n=0..16.

Formula

a(n) ~ c * (exp(r)*(1+r)*(1+2*r)/r)^n * n! / n, where r = 0.82506987245889649414011656390500836009994111212734792413605... is the root of the equation -(1+r)*(LambertW(-1, -r*exp(-r)) + r) = r^2 and c = 0.18810075704917016248053554671907469507232504783794203152534...

Maple

A397421 := (n, k) -> (-1)^(n-k)*Stirling1(n, k)*binomial(n + k, n)*n^k:
A397423 := n -> local j; add(A397421(n, j), j = 0..n):
seq(A397423(n), n = 0..16);  # Peter Luschny, Jun 24 2026

Mathematica

Join[{1}, Table[Sum[Abs[StirlingS1[n, j]]*Binomial[n + j, n]*n^j, {j, 0, n}], {n, 1, 20}]]

Crossrefs

Cf. A132393, A397324, A397421.

Sequence in context: A270531 A229781 A397325 * A114938 A082653 A382958

Adjacent sequences: A397420 A397421 A397422 * A397426 A397427 A397430

Keyword

nonn,new

Author

Vaclav Kotesovec, Jun 24 2026

Status

approved