A380155 - OEIS

%I #10 Jan 23 2025 05:22:18

%S 1,1,7,63,785,12545,244407,5619775,148977313,4473497601,150078670055,

%T 5563415292479,225832882678449,9962766560986369,474619650950131351,

%U 24283168467229957695,1327993894505461755713,77305844496338607597569,4772660185400974888323015

%N Expansion of e.g.f. 1/sqrt(1 - 2*x*exp(2*x)).

%F a(n) = 2^n * n! * Sum_{k=0..n} (-1)^k * k^(n-k) * binomial(-1/2,k)/(n-k)!.

%F a(n) == 1 (mod 2).

%F a(n) ~ 2^(n + 1/2) * n^n / (sqrt(1 + LambertW(1)) * exp(n) * LambertW(1)^n). - _Vaclav Kotesovec_, Jan 23 2025

%o (PARI) a(n) = 2^n*n!*sum(k=0, n, (-1)^k*k^(n-k)*binomial(-1/2, k)/(n-k)!);

%Y Cf. A006153, A380156.

%Y Cf. A380014, A380015.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Jan 13 2025