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

0,3

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

Table of n, a(n) for n=0..21.

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

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

Cf. A001147, A051296, A185971, A292778.

Sequence in context: A294346 A049376 A083410 * A302548 A052772 A365124

Adjacent sequences: A295550 A295551 A295552 * A295554 A295555 A295556

Keyword

nonn

Author

Ilya Gutkovskiy, Nov 23 2017

Status

approved