A343482 Expansion of the e.g.f. sqrt(-1 + 2 / (1 - x) / exp(x)).
1, 0, 1, 2, 6, 24, 135, 930, 7105, 59192, 549360, 5746080, 66713361, 839528052, 11308954657, 163038260294, 2520332282910, 41640324943968, 730119174449151, 13507292654421390, 263004450921933817, 5385277610047242620, 115775314245285797256, 2606072891349667903152, 61248210450060537498321
Offset: 0
Keywords
Examples
sqrt(-1+2/(1-x)/exp(x)) = 1 + x^2/2! + 2*x^3/3! + 6*x^4/4! + 24*x^5/5! + 135*x^6/6! + 930*x^7/7! + 7105*x^8/8! + 59192*x^9/9! + ...
a(23) = Sum_{k=1..11} (-1)^(k-1)*A014304(k-1)*A008306(23,k) = 2606072891349667903152.
For k=1, (-1)^(1-1)*A014304(1-1)*A008306(23,1) == -1 (mod 23), because A014304(0) = 1 and A008306(23,1) = (23-1)!
For k>=2, (-1)^(k-1)*A014304(k-1)*A008306(23,k) == 0 (mod 23), because A008306(23,k) == 0 (mod 23), result a(23) == -1 (mod 23).
a(18) = Sum_{k=1..9} (-1)^(k-1)*A014304(k-1)*A008306(18,k) = 730119174449151.
a(18) == 0 (mod (18-1)), because for k >= 1, A008306(18,k) == 0 (mod 17).
Programs
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Maple
A014304:= proc(n) option remember; `if`(n=0, 1, (-1)^n + add(binomial(n,k)*A014304(k)* A014304(n-k-1), k=0..n-1)) end: A008306 := proc(n, k): if k=1 then (n-1)! ; elif n<=2*k-1 then 0; else (n-1)*procname(n-1, k)+(n-1)*procname(n-2, k-1) ; end if; end proc: a:= n-> add(((-1)^(k-1)*A014304(k-1)*A008306(n,k)), k=1..iquo(n,2)):a(0):=1 ; seq(a(n), n=0..24); # second program: a := series(sqrt(-1+2/(1-x)/exp(x)), x=0, 25):seq(n!*coeff(a, x, n), n=0..24);
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Mathematica
CoefficientList[Series[Sqrt[-1+2/(1-x)/E^x], {x, 0, 24}], x] * Range[0, 24]! -
PARI
my(x='x+O('x^30)); Vec(serlaplace(sqrt(-1 + 2 / (1 - x) / exp(x)))) \\ Michel Marcus, Jul 06 2021
Formula
E.g.f. y(x) satisfies y*y' = exp(-x)*x/(1-x)^2.
For all p prime, a(p) == -1 (mod p).
For n > 1, a(n) == 0 (mod (n-1)).
a(n) ~ 2 * n^n / exp(n + 1/2). - Vaclav Kotesovec, Jul 06 2021