A367920 Expansion of e.g.f. exp(4*(exp(x) - 1) - 2*x).

1, 2, 8, 36, 196, 1196, 8116, 60108, 481140, 4126540, 37671540, 364068172, 3707910772, 39645022540, 443540780660, 5177560304972, 62903920321140, 793654042136908, 10378403752717940, 140413475790402892, 1962339063781284468, 28287778534523140428, 420059992540347885172

Offset

0,2

Links

Table of n, a(n) for n=0..22.

Formula

G.f. A(x) satisfies: A(x) = 1 - 2 * x * ( A(x) - 2 * A(x/(1 - x)) / (1 - x) ).

a(n) = exp(-4) * Sum_{k>=0} 4^k * (k-2)^n / k!.

a(0) = 1; a(n) = -2 * a(n-1) + 4 * Sum_{k=1..n} binomial(n-1,k-1) * a(n-k).

Mathematica

nmax = 22; CoefficientList[Series[Exp[4 (Exp[x] - 1) - 2 x], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = -2 a[n - 1] + 4 Sum[Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 22}]

Crossrefs

Cf. A000296, A078944, A126617, A194689, A367888, A367891, A367919, A367921.

Sequence in context: A001540 A129044 A052582 * A222468 A020021 A213076

Adjacent sequences: A367917 A367918 A367919 * A367921 A367922 A367923

Keyword

nonn

Author

Ilya Gutkovskiy, Dec 04 2023

Status

approved