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A305840 Product_{n>=1} (1 + x^n)^a(n) = g.f. of A005169 (fountains of coins). 1
1, 1, 1, 2, 2, 4, 5, 10, 13, 23, 35, 59, 93, 154, 248, 413, 671, 1111, 1827, 3036, 5013, 8348, 13859, 23122, 38534, 64434, 107715, 180509, 302565, 508032, 853507, 1435828, 2416941, 4072943, 6868062, 11591918, 19577555, 33090308, 55964327, 94715248, 160391045 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

Inverse weigh transform of A005169.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..3000

N. J. A. Sloane, Transforms

Eric Weisstein's World of Mathematics, Rogers-Ramanujan Continued Fraction

FORMULA

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

EXAMPLE

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

MATHEMATICA

nn = 39; f[x_] := Product[(1 + x^n)^a[n], {n, 1, nn}]; sol = SolveAlways[0 == Series[f[x] - 1/(1 + ContinuedFractionK[-x^k, 1, {k, 1, nn}]), {x, 0, nn}], x]; Table[a[n], {n, 1, nn}] /. sol // Flatten

CROSSREFS

Cf. A005169, A226999.

Sequence in context: A195865 A222006 A127712 * A178113 A032090 A000014

Adjacent sequences:  A305837 A305838 A305839 * A305841 A305842 A305843

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, Jun 11 2018

STATUS

approved

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Last modified April 8 21:44 EDT 2020. Contains 333329 sequences. (Running on oeis4.)