A367940 Expansion of e.g.f. exp(exp(4*x) - 1 - 3*x).

1, 1, 17, 113, 1377, 17185, 252401, 4104721, 73500609, 1430779713, 30026750161, 674586467505, 16130795165473, 408560492670049, 10915540174130353, 306531211899158609, 9019774516570506113, 277345675943850865281, 8889954225208868308369, 296408283056785166556401

Offset

0,3

Links

Table of n, a(n) for n=0..19.

Formula

G.f. A(x) satisfies: A(x) = 1 - x * ( 3 * A(x) - 4 * A(x/(1 - 4*x)) / (1 - 4*x) ).

a(n) = exp(-1) * Sum_{k>=0} (4*k-3)^n / k!.

a(0) = 1; a(n) = -3 * a(n-1) + Sum_{k=1..n} binomial(n-1,k-1) * 4^k * a(n-k).

a(n) = Sum_{k=0..n} binomial(n,k) * (-3)^(n-k) * 4^k * Bell(k).

Mathematica

nmax = 19; CoefficientList[Series[Exp[Exp[4 x] - 1 - 3 x], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = -3 a[n - 1] + Sum[Binomial[n - 1, k - 1] 4^k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 19}]
Table[Sum[Binomial[n, k] (-3)^(n - k) 4^k BellB[k], {k, 0, n}], {n, 0, 19}]

Prog

(PARI) my(x='x+O('x^30)); Vec(serlaplace(exp(exp(4*x) - 1 - 3*x))) \\ Michel Marcus, Dec 07 2023

Crossrefs

Cf. A000110, A124311, A355163, A346738, A367786, A367939.

Sequence in context: A298332 A299223 A197360 * A289472 A172521 A349879

Adjacent sequences: A367937 A367938 A367939 * A367941 A367942 A367943

Keyword

nonn

Author

Ilya Gutkovskiy, Dec 05 2023

Status

approved