A392286 a(n) = Sum_{k=1..n} binomial(n+1, k+1)*A000272(k).

0, 1, 4, 13, 51, 301, 2668, 31865, 473077, 8347321, 170297766, 3940771165, 101950138447, 2915344627469, 91302800186536, 3107800435564081, 114235243085526249, 4509551519090855857, 190275672302226201034, 8545566392958010663541, 407015614186583759406211, 20491642557407081350307557

Offset

0,3

Comments

Partial sums of A270593.

Links

Table of n, a(n) for n=0..21.

Formula

E.g.f.: exp(x)*(T - exp(-T) + (1-x) - (1+x)/2*T^2) where T = T(x) is the e.g.f. of A000169.

Limit_{n->oo} a(n) / n^(n-2) = exp(exp(-1)) = A073229.

a(n) = 2*a(n-1) - a(n-2) + A088957(n-1) for n > 1.

a(n) == 1 (mod A000217(n)) for n > 0.

G.f.: Sum_{k>=1} k^(k-2)*x^k/(1-x)^(k+2). - Mélika Tebni, Feb 20 2026

Maple

T := -LambertW(-x):
a := exp(x)*(T-exp(-T)+1-x-(1+x)/2*T^2):
ser := series(a, x = 0, 22): seq(n!*coeff(ser, x, n), n = 0 .. 21);
# Recurrence:
a := proc (n) option remember; `if`(n < 2, n, 2*a(n-1)-a(n-2)+add(binomial(n-1, k)*(n-k)^(n-k-2), k=0..n-1)) end:
seq(a(n), n = 0 .. 21);

Prog

(Python)
from math import comb
def a(n):
    return sum(comb(n+1, k+1) * (1 if k==1 else k**(k-2)) for k in range(1, n+1))
print([a(n) for n in range(22)])
(PARI) a(n) = sum(k=1, n, sum(i=1, k, binomial(k, i)*(i^(i-2)))); \\ Michel Marcus, Jan 12 2026

Crossrefs

Cf. A000169, A000217, A000272, A073229, A088957, A270593.

Sequence in context: A245156 A223899 A357962 * A097169 A149463 A323791

Adjacent sequences: A392283 A392284 A392285 * A392287 A392288 A392289

Keyword

nonn,easy

Author

Mélika Tebni, Jan 06 2026

Status

approved