A355501 Expansion of e.g.f. exp(3 * x * exp(x)).

1, 3, 15, 90, 633, 5028, 44217, 424434, 4399953, 48858984, 577372809, 7221983838, 95192539641, 1317190650636, 19071213218745, 288112248054882, 4530217559806497, 73976635012027344, 1252091246140278153, 21926952634345281030, 396671314081806278601

Offset

0,2

Links

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

Formula

G.f.: Sum_{k>=0} (3 * x)^k/(1 - k*x)^(k+1).

a(0) = 1; a(n) = 3 * Sum_{k=1..n} k * binomial(n-1,k-1) * a(n-k).

a(n) = Sum_{k=0..n} 3^k * k^(n-k) * binomial(n,k).

a(n) ~ n^(n + 1/2) * exp(3*r*exp(r) - r/2 - n) / (sqrt(3*(1 + 3*r + r^2)) * r^(n + 1/2)), where r = 2*w - 1/(2*w) + 5/(8*w^2) - 19/(24*w^3) + 209/(192*w^4) - 763/(480*w^5) + 4657/(1920*w^6) - 6855/(1792*w^7) + 199613/(32256*w^8) + ... and w = LambertW(sqrt(n/3)/2). - Vaclav Kotesovec, Jul 06 2022

Prog

(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(3*x*exp(x))))
(PARI) my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (3*x)^k/(1-k*x)^(k+1)))
(PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=3*sum(j=1, i, j*binomial(i-1, j-1)*v[i-j+1])); v;
(PARI) a(n) = sum(k=0, n, 3^k*k^(n-k)*binomial(n, k));

Crossrefs

Column 3 of A187105.

Cf. A000248, A275707, A295623.

Cf. A351763.

Sequence in context: A366085 A271930 A201953 * A185369 A024339 A319950

Adjacent sequences: A355498 A355499 A355500 * A355502 A355503 A355504

Keyword

nonn

Author

Seiichi Manyama, Jul 04 2022

Status

approved