A367819 Expansion of e.g.f. exp(1 - 4*x - exp(x)).

1, -5, 24, -111, 497, -2166, 9239, -38765, 160658, -659773, 2691205, -10922544, 44166173, -178098121, 716703848, -2879774019, 11558005677, -46348854134, 185746261419, -744036460097, 2979305960426, -11926715433881, 47735079979633, -191026723545976, 764362047956073, -3058170811731677

Offset

0,2

Links

Table of n, a(n) for n=0..25.

Formula

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

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

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

a(n) = Sum_{k=0..n} binomial(n,k) * (-4)^(n-k) * A000587(k).

Mathematica

nmax = 25; CoefficientList[Series[Exp[1 - 4 x - Exp[x]], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = -4 a[n - 1] - Sum[Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 25}]
Table[Sum[Binomial[n, k] (-4)^(n - k) BellB[k, -1], {k, 0, n}], {n, 0, 25}]

Prog

(PARI) my(x='x+O('x^30)); Vec(serlaplace(exp(1 - 4*x - exp(x)))) \\ Michel Marcus, Dec 02 2023

Crossrefs

Cf. A000587, A045379, A109747, A153732, A193684, A346739, A367818.

Sequence in context: A183934 A171310 A225116 * A296770 A370035 A081104

Adjacent sequences: A367816 A367817 A367818 * A367820 A367821 A367822

Keyword

sign

Author

Ilya Gutkovskiy, Dec 01 2023

Status

approved