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A308939
Expansion of e.g.f. 1 / (1 - Sum_{k>=1} (2*k - 1)!!*x^k/k!).
3
1, 1, 5, 39, 411, 5445, 86805, 1616895, 34448715, 826093485, 22017673125, 645633501975, 20655688959675, 715958472554325, 26726481024167925, 1068988088284491375, 45608095005687088875, 2067503007329827192125, 99238033465208117605125, 5027986481205385725402375
OFFSET
0,3
FORMULA
E.g.f.: 1/(2 - 1/sqrt(1 - 2*x)).
a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * (2*k - 1)!! * a(n-k).
a(n) ~ n! * 8^n / 3^(n+1). - Vaclav Kotesovec, Jul 01 2019
D-finite with recurrence: +3*a(n) +(-14*n+9)*a(n-1) +8*(n-1)*(2*n-3)*a(n-2)=0. - R. J. Mathar, Jan 27 2020
a(n) = 2^n*Sum_{k=0..n} Sum_{j=0..k} (-1)^(k-j)*binomial(k, j)*Pochhammer(j/2, n). - Peter Luschny, Mar 08 2024
MAPLE
a := n -> local j, k; 2^n*add(add((-1)^(k-j)*binomial(k, j)*pochhammer(j/2, n), j = 0..k), k = 0..n): seq(a(n), n = 0..19); # Peter Luschny, Mar 08 2024
MATHEMATICA
nmax = 19; CoefficientList[Series[1/(2 - 1/Sqrt[1 - 2 x]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] (2 k - 1)!! a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 19}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Jul 01 2019
STATUS
approved