login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A305870 Product_{n>=1} (1 + x^n)^a(n) = g.f. of A001147 (double factorial of odd numbers). 3
1, 3, 12, 90, 816, 9206, 122028, 1859550, 32002076, 613891800, 12989299596, 300556868286, 7550646317520, 204687481411974, 5955892982437120, 185158929517924710, 6125200081143892800, 214837724609534836158, 7963817560398871790604, 311101285877490394183800, 12773912991134665452205048 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Inverse weigh transform of A001147.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..404

N. J. A. Sloane, Transforms

Eric Weisstein's World of Mathematics, Double Factorial

Index entries for sequences related to factorial numbers

FORMULA

Product_{n>=1} (1 + x^n)^a(n) = 1/(1 - x/(1 - 2*x/(1 - 3*x/(1 - 4*x/(1 - 5*x/(1 - ...)))))).

EXAMPLE

(1 + x) * (1 + x^2)^3 * (1 + x^3)^12 * (1 + x^4)^90 * (1 + x^5)^816 * ... * (1 + x^n)^a(n) * ... = 1 + 1*x + 1*3*x^2 + 1*3*5*x^3 + 1*3*5*7*x^4 + ... + A001147(k)*x^k + ...

MAPLE

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,

      add(binomial(a(i), j)*b(n-i*j, i-1), j=0..n/i)))

    end:

a:= proc(n) option remember; doublefactorial(2*n-1)-b(n, n-1) end:

seq(a(n), n=1..23);  # Alois P. Heinz, Jun 13 2018

MATHEMATICA

nn = 21; f[x_] := Product[(1 + x^n)^a[n], {n, 1, nn}]; sol = SolveAlways[0 == Series[f[x] - 1/(1 + ContinuedFractionK[-k x, 1, {k, 1, nn}]), {x, 0, nn}], x]; Table[a[n], {n, 1, nn}] /. sol // Flatten

CROSSREFS

Cf. A001147, A168246, A305868, A305869.

Sequence in context: A174463 A124191 A065087 * A058337 A025503 A263801

Adjacent sequences:  A305867 A305868 A305869 * A305871 A305872 A305873

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, Jun 12 2018

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified June 1 09:53 EDT 2020. Contains 334762 sequences. (Running on oeis4.)