A367786 Expansion of e.g.f. exp(exp(4*x) - x - 1).

1, 3, 25, 235, 2737, 36947, 563657, 9542715, 176920417, 3555369635, 76820077945, 1772943290763, 43469116126737, 1127040956393203, 30779951676185385, 882453651485815003, 26480355971228530369, 829522636694530362691, 27064267045022876869337, 917751849133986186857003

Offset

0,2

Links

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

Formula

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

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

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

Mathematica

nmax = 19; CoefficientList[Series[Exp[Exp[4 x] - x - 1], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = -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[(-1)^(n - k) Binomial[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) - x - 1))) \\ Michel Marcus, Nov 30 2023

Crossrefs

Cf. A000110, A000296, A124311, A285064, A285065, A330605, A367785.

Sequence in context: A372458 A066221 A370280 * A332468 A134272 A335117

Adjacent sequences: A367783 A367784 A367785 * A367787 A367788 A367789

Keyword

nonn

Author

Ilya Gutkovskiy, Nov 30 2023

Status

approved