A295553 Expansion of 1/(1 - Sum_{k>=1} (2*k-1)!!*x^k).
1, 1, 4, 22, 154, 1330, 13882, 171802, 2474098, 40738594, 755322778, 15566915770, 352862768434, 8720662458754, 233285616212506, 6713983428179098, 206813607458357746, 6788092999359053410, 236481982146071359258, 8714521818620631672058, 338660320676350494328882, 13841377309645038610883266
Offset: 0
Keywords
Links
- M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72. Erratum 320 (2000), 210. [Link to arXiv version]
- M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
- N. J. A. Sloane, Transforms
- Index entries for sequences related to factorial numbers
- Index entries for sequences related to compositions
Programs
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Mathematica
nmax = 21; CoefficientList[Series[1/(1 - Sum[(2 k - 1)!! x^k, {k, 1, nmax}]), {x, 0, nmax}], x] nmax = 21; CoefficientList[Series[1 + x/(1 - 2 x + ContinuedFractionK[-k x, 1, {k, 2, nmax}]), {x, 0, nmax}], x] a[0] = 1; a[n_] := a[n] = Sum[(2 k - 1)!! a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 21}]
Formula
G.f.: 1/(1 - Sum_{k>=1} A001147(k)*x^k).
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.
a(0) = 1; a(n) = Sum_{k=1..n} (2*k-1)!!*a(n-k).
Comments