Revision History for A286310
(Underlined text is an addition;
strikethrough text is a deletion.)
Showing entries 1-10
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#11 by Vaclav Kotesovec at Thu Sep 16 14:04:53 EDT 2021
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#10 by Vaclav Kotesovec at Thu Sep 16 14:04:46 EDT 2021
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| FORMULA
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a(n) ~ 2^(n + 1/2) * n^n / exp(n). - Vaclav Kotesovec, Sep 16 2021
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| STATUS
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approved
editing
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#9 by Bruno Berselli at Wed May 10 11:07:16 EDT 2017
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#8 by Ilya Gutkovskiy at Wed May 10 07:51:41 EDT 2017
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#7 by Ilya Gutkovskiy at Wed May 10 07:51:31 EDT 2017
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| MATHEMATICA
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nn = 20; f[x_] := 1 + Sum[a[n] x^n/(1 - x^n), {n, 1, nn}]; sol = SolveAlways[0 == Series[f[x] - 1/(1 + ContinuedFractionK[-n x, 1, {n, 1, nn}]), {x, 0, nn}], x]; ]; Table[a[n], {n, 1, nn}] /. sol // Flatten
Table[a[n], {n, 1, nn}] /. sol // Flatten
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| STATUS
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approved
editing
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#6 by N. J. A. Sloane at Sat May 06 19:54:12 EDT 2017
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#5 by Ilya Gutkovskiy at Sat May 06 03:50:04 EDT 2017
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#4 by Ilya Gutkovskiy at Sat May 06 03:49:31 EDT 2017
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| EXAMPLE
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G.f.: 1 + x/(1 - x) + 2*x^2/(1 - x^32) + 14*x^3/(1 - x^43) + 102*x^4/(1 - x^54) + ... = 1/(1 - x/(1 - 2*x/(1 - 3*x/(1 - 4*x/(1 - ...))))).
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#3 by Ilya Gutkovskiy at Sat May 06 03:44:45 EDT 2017
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| EXAMPLE
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G.f.: 1 + x/(1 - x) + 2*x^2/(1 - x^3) + 14*x^3/(1 - x^4) + 102*x^4/(1 - x^5) + ... = 1/(1 - x/(1 - 2*x/(1 - 3*x^/(/(1 - 4*x/(1 - ...))))).
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#2 by Ilya Gutkovskiy at Sat May 06 03:40:06 EDT 2017
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| NAME
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allocated for Ilya Gutkovskiy
G.f.: 1 + Sum_{n>=1} a(n)*x^n/(1 - x^n) = 1/(1 - x/(1 - 2*x/(1 - 3*x/(1 - 4*x/(1 - ...))))).
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| DATA
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1, 2, 14, 102, 944, 10378, 135134, 2026920, 34459410, 654728128, 13749310574, 316234132728, 7905853580624, 213458046541738, 6190283353628416, 191898783960483600, 6332659870762850624, 221643095476665302070, 8200794532637891559374, 319830986772877116086448
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| OFFSET
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1,2
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| LINKS
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N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
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| FORMULA
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Sum_{d|n) a(d) = A001147(n) for n > 0.
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| EXAMPLE
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G.f.: 1 + x/(1 - x) + 2*x^2/(1 - x^3) + 14*x^3/(1 - x^4) + 102*x^4/(1 - x^5) + ... = 1/(1 - x/(1 - 2*x/(1 - 3*x^/(1 - 4*x/(1 - ...))))).
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| MATHEMATICA
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nn = 20; f[x_] := 1 + Sum[a[n] x^n/(1 - x^n), {n, 1, nn}]; sol = SolveAlways[0 == Series[f[x] - 1/(1 + ContinuedFractionK[-n x, 1, {n, 1, nn}]), {x, 0, nn}], x];
Table[a[n], {n, 1, nn}] /. sol // Flatten
a[n_] := Sum[MoebiusMu[n/d] (2 d - 1)!!, {d, Divisors[n]}]; Array[a, 20]
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| CROSSREFS
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Cf. A001147, A002996, A062794.
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| KEYWORD
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allocated
nonn
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| AUTHOR
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Ilya Gutkovskiy, May 06 2017
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| STATUS
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approved
editing
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