A393272 a(0) = 0, a(n) = Sum_{k=1..n} binomial(n+1, k+1)*A000169(k) for n > 0.

0, 1, 5, 23, 139, 1199, 13901, 201711, 3497831, 70395119, 1611157129, 41304909623, 1172171636219, 36472178218839, 1234509033199469, 45156161730476639, 1774931281963830607, 74606141449005077087, 3339280622577345357329, 158560248020157678586599, 7960855389479556964189859

Offset

0,3

Comments

Partial sums of A277473 (column 1 of A203092).

Links

Table of n, a(n) for n=0..20.

Formula

E.g.f.: exp(x)*(T-1 + x*(1 + 1/T + T)) where T = -LambertW(-x).

a(n) = n + Sum_{k=0..n-2} binomial(n,k)*A258387(n-1-k) for n >= 2.

a(n) = 2*a(n-1) - a(n-2) + A038051(n) for n >= 2.

G.f.: Sum_{k>=1} k^(k-1)*x^k/(1-x)^(k+2).

Maple

T := -LambertW(-x): a := exp(x)*(T-1 + x*(1 + 1/T + T)):
ser := series(a, x = 0, 22): seq(n!*coeff(ser, x, n), n = 0 .. 20);
# Alternative:
a := proc (n) option remember; `if`(n < 2, [0, 1][n+1], n + add(binomial(n, k)*((n-k)^(n-1-k) + (n-1-k)^(n-2-k)), k=0..n-2)) end proc: seq(a(n), n = 0 .. 20);

Prog

(Python)
from math import comb
def a(n):
    return sum(comb(n+1, k+1)*k**(k-1) for k in range(1, n+1))
print([a(n) for n in range(21)])

Crossrefs

Cf. A000169, A038051, A203092, A258387, A277473, A392286.

Sequence in context: A047049 A351817 A352146 * A020034 A128884 A356436

Adjacent sequences: A393269 A393270 A393271 * A393273 A393274 A393275

Keyword

nonn,easy

Author

Mélika Tebni, Feb 08 2026

Status

approved