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#45 by N. J. A. Sloane at Mon Oct 07 14:15:09 EDT 2019
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#44 by Michel Marcus at Mon Oct 07 12:29:00 EDT 2019
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Discussion
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| Mon Oct 07
| 12:30
| Petros Hadjicostas: What is the meaning of the word "frac"? Does it indicate that we have numerators or denominators of fractions?
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| 14:15
| N. J. A. Sloane: "What is the meaning of the word "frac"? Does it indicate that we have numerators or denominators of fractions?"
Yes
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#43 by Michel Marcus at Mon Oct 07 12:28:50 EDT 2019
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| KEYWORD
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nonn,frac,new
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| STATUS
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proposed
editing
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Discussion
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| Mon Oct 07
| 12:29
| Michel Marcus: ok for keyword frac ?
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#42 by Petros Hadjicostas at Mon Oct 07 12:20:50 EDT 2019
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#41 by Petros Hadjicostas at Mon Oct 07 12:20:43 EDT 2019
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| COMMENTS
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Wolfdieter Lang (see the link below) examined inverse power product expansions both for ordinary g.f.'s and for exponential g.f.'s. He connects inverse power product expansions to unital series associated to (infinite dimesionaldimensional) Witt vectors and to the so-called "Somos transformation".
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| MAPLE
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h1h := map(denom, PPE(100, 40)); # Petros Hadjicostas, Oct 06 2019 by modifying N. J. A. Sloane's program from A170912 and A170913.
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| CROSSREFS
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Numerators are in A328191.
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| STATUS
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approved
editing
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#40 by Bruno Berselli at Mon Oct 07 03:05:00 EDT 2019
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#39 by Joerg Arndt at Mon Oct 07 01:33:44 EDT 2019
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Discussion
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| Mon Oct 07
| 01:36
| Petros Hadjicostas: I will do the numerator case some other day.
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#38 by Petros Hadjicostas at Mon Oct 07 01:14:10 EDT 2019
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#37 by Petros Hadjicostas at Mon Oct 07 01:11:02 EDT 2019
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| COMMENTS
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There are more formulas for f_n and g_n in the references listed below. In all cases, we assume the g.f.'s are unital, i.e., the g.f.'s start with a constant 1.
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#36 by Petros Hadjicostas at Mon Oct 07 01:09:31 EDT 2019
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| COMMENTS
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Gingold and Knopfmacher (1995) and Alkauskas (2008, 2009) proved that f_n = -g_n for n odd, and Sum_{s|n} (-g_{n/s})^s/s = -Sum_{s|n} (-f_{n/s})^s/s. [We caution that different authors may use -g_n for g_n , or -f_n for f_n, or both.]
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