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%I #4 Nov 23 2017 20:22:52
%S 1,1,4,22,154,1330,13882,171802,2474098,40738594,755322778,
%T 15566915770,352862768434,8720662458754,233285616212506,
%U 6713983428179098,206813607458357746,6788092999359053410,236481982146071359258,8714521818620631672058,338660320676350494328882,13841377309645038610883266
%N Expansion of 1/(1 - Sum_{k>=1} (2*k-1)!!*x^k).
%C Invert transform of A001147.
%C Number of compositions (ordered partitions) of n where there are 1*3*5*...*(2*k-1) sorts of part k.
%H M. Bernstein and N. J. A. Sloane, <a href="http://arXiv.org/abs/math.CO/0205301">Some canonical sequences of integers</a>, Linear Alg. Applications, 226-228 (1995), 57-72. Erratum 320 (2000), 210. [Link to arXiv version]
%H M. Bernstein and N. J. A. Sloane, <a href="/A003633/a003633_1.pdf">Some canonical sequences of integers</a>, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>
%H <a href="/index/Com#comp">Index entries for sequences related to compositions</a>
%F G.f.: 1/(1 - Sum_{k>=1} A001147(k)*x^k).
%F G.f.: 1 + x/(1 - 2*x - 2*x/(1 - 3*x/(1 - 4*x/(1 - 5*x/(1 - 6*x/(1 - ...)))))), a continued fraction.
%F a(0) = 1; a(n) = Sum_{k=1..n} (2*k-1)!!*a(n-k).
%t nmax = 21; CoefficientList[Series[1/(1 - Sum[(2 k - 1)!! x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
%t nmax = 21; CoefficientList[Series[1 + x/(1 - 2 x + ContinuedFractionK[-k x, 1, {k, 2, nmax}]), {x, 0, nmax}], x]
%t a[0] = 1; a[n_] := a[n] = Sum[(2 k - 1)!! a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 21}]
%Y Cf. A001147, A051296, A185971, A292778.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Nov 23 2017
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