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%I #13 Mar 08 2024 09:32:30
%S 1,1,5,39,411,5445,86805,1616895,34448715,826093485,22017673125,
%T 645633501975,20655688959675,715958472554325,26726481024167925,
%U 1068988088284491375,45608095005687088875,2067503007329827192125,99238033465208117605125,5027986481205385725402375
%N Expansion of e.g.f. 1 / (1 - Sum_{k>=1} (2*k - 1)!!*x^k/k!).
%F E.g.f.: 1/(2 - 1/sqrt(1 - 2*x)).
%F a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * (2*k - 1)!! * a(n-k).
%F a(n) ~ n! * 8^n / 3^(n+1). - _Vaclav Kotesovec_, Jul 01 2019
%F 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
%F 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
%p 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
%t nmax = 19; CoefficientList[Series[1/(2 - 1/Sqrt[1 - 2 x]), {x, 0, nmax}], x] Range[0, nmax]!
%t 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}]
%Y Cf. A001147, A002866, A295553.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Jul 01 2019
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