A362204 Expansion of e.g.f. exp(x/sqrt(1-2*x)).

1, 1, 3, 16, 121, 1176, 13921, 193978, 3106881, 56201176, 1132709041, 25162197006, 610668537073, 16073212005436, 455980333073721, 13868451147012946, 450140785396634881, 15529495879187075088, 567427732311438658081, 21889446540911251445206

Offset

0,3

Links

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

Formula

a(n) = n! * Sum_{k=0..n} (-2)^k * binomial(-(n-k)/2,k)/(n-k)! = n! * Sum_{k=0..n} 2^(n-k) * binomial(n-k/2-1,n-k)/k!.

From Vaclav Kotesovec, Feb 20 2024: (Start)

a(n) ~ 3^(-1/2) * 2^(n - 1/6) * exp(3*2^(-4/3)*n^(1/3) - n) * n^(n - 1/3) * (1 - 3/(16*(n/2)^(1/3))).

Recurrence (for n>5): (n-5)*a(n) = 3*(2*n^2 - 13*n + 16)*a(n-1) - (12*n^3 - 108*n^2 + 299*n - 259)*a(n-2) + (n-2)*(8*n^3 - 84*n^2 + 290*n - 327)*a(n-3) + (n-4)*(n-3)*(n-2)*a(n-4). (End)

Mathematica

Table[n! * Sum[2^(n-k) * Binomial[n-k/2-1, n-k]/k!, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Feb 20 2024 *)
Join[{1, 1}, RecurrenceTable[{(-4 + n) (-3 + n) (-2 + n) a[-4 + n] + (-2 + n) (-327 + 290 n - 84 n^2 + 8 n^3) a[-3 + n] + (259 - 299 n + 108 n^2 - 12 n^3) a[-2 + n] + 3 (16 - 13 n + 2 n^2) a[-1 + n] + (5 - n) a[n] == 0, a[2] == 3, a[3] == 16, a[4] == 121, a[5] == 1176}, a, {n, 2, 20}]] (* Vaclav Kotesovec, Feb 20 2024 *)

Prog

(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x/sqrt(1-2*x))))

Crossrefs

Cf. A025168, A362163.

Sequence in context: A166883 A145158 A132070 * A121629 A351218 A200793

Adjacent sequences: A362201 A362202 A362203 * A362205 A362206 A362207

Keyword

nonn,easy

Author

Seiichi Manyama, Apr 11 2023

Status

approved