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A295553
Expansion of 1/(1 - Sum_{k>=1} (2*k-1)!!*x^k).
1
1, 1, 4, 22, 154, 1330, 13882, 171802, 2474098, 40738594, 755322778, 15566915770, 352862768434, 8720662458754, 233285616212506, 6713983428179098, 206813607458357746, 6788092999359053410, 236481982146071359258, 8714521818620631672058, 338660320676350494328882, 13841377309645038610883266
OFFSET
0,3
COMMENTS
Invert transform of A001147.
Number of compositions (ordered partitions) of n where there are 1*3*5*...*(2*k-1) sorts of part k.
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
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).
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}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Nov 23 2017
STATUS
approved