A362163 -id:A362163 - cp's OEIS frontend

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

Seiichi Manyama, Apr 11 2023

Cf. A025168, A362163.

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 *)

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

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)

A362165 Expansion of e.g.f. exp(-x * sqrt(1-2*x)).

Original entry on oeis.org

1, -1, 3, -4, 25, 24, 721, 5942, 82209, 1186280, 19956241, 373942194, 7768988833, 177018731876, 4389959146665, 117700102748654, 3392361669670081, 104592876707106672, 3434908281762030049, 119702402508549928490, 4411764405039931048641

Offset: 0

Seiichi Manyama, Apr 10 2023

sign

Cf. A115329, A362161, A362163.

A362165 := proc(n)
    (-1)^n*n!*add(2^k * binomial((n-k)/2,k)/(n-k)!,k=0..n) ;
end proc:
seq(A362165(n),n=0..70) ; # R. J. Mathar, Dec 04 2023

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-x*sqrt(1-2*x))))

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

D-finite with recurrence a(n) +2*(-n+3)*a(n-1) +2*(-3*n+10)*a(n-2) +6*(n-2)*a(n-3) -9*(n-3)^2*a(n-4) -27*(n-3)*(n-4)*a(n-5)=0. - R. J. Mathar, Dec 04 2023

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

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