A394045 - OEIS

%I #4 Mar 17 2026 16:39:52

%S 0,0,0,1,17,237,3497,58237,1104261,23676237,568261317,15119327517,

%T 442065987217,14096919853013,487072495995537,18130790565917117,

%U 723480290047622797,30811812079135831197,1395085912631686052301,66922971714793668780477,3390739311328091468712537

%N a(0) = 0, a(n) = Sum_{k=1..n} binomial(n+1, k+1)*A053507(k) for n > 0.

%C Partial sums of column k=3 of A203092.

%C a(n)+3 is divisible by 2*A007531(n) if n is of the form 6*k+5 (A016969).

%F E.g.f.: exp(x)*(-3 + (1+x)*T^3/6 + x*T^2/2 + 3*x*T/2 + 3*x*(1+1/T)) where T = -LambertW(-x).

%F G.f.: Sum_{k>=1} (k-1)*(k-2)/2*k^(k-3)*x^k/(1-x)^(k+2).

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

%p T := -LambertW(-x): a := exp(x)*(-3 + (1+x)*T^3/6 + x*T^2/2 + 3*x*T/2 + 3*x*(1+1/T)):

%p ser := series(a, x = 0, 22): seq(n!*coeff(ser, x, n), n = 0 .. 20);

%o (Python)

%o from math import comb

%o def a(n):

%o     return sum(comb(n+1, k+1)*comb(k-1, 2)*k**(k-3) for k in range(3, n+1))

%o print([a(n) for n in range(21)])

%Y Cf. A000169, A007531, A016969, A053507, A073229, A203092, A393272.

%K nonn,easy

%O 0,5

%A _Mélika Tebni_, Mar 08 2026