Revision History for A125702
(Underlined text is an addition;
strikethrough text is a deletion.)
Showing entries 1-10
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#12 by Michel Marcus at Sun Nov 03 01:43:35 EST 2019
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#11 by Joerg Arndt at Sun Nov 03 01:10:19 EST 2019
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#10 by Andrew Howroyd at Sat Nov 02 20:52:06 EDT 2019
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#9 by Andrew Howroyd at Sat Nov 02 18:19:01 EDT 2019
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| LINKS
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Andrew Howroyd, <a href="/A125702/b125702.txt">Table of n, a(n) for n = 1..500</a>
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| FORMULA
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a(n) = A122086(n) for n > 1.
G.f.: 2*f(x) - f(x)^2 - x where f(x) is the g.f. of A000081. - Andrew Howroyd, Nov 02 2019
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| PROG
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(PARI) \\ TreeGf gives gf of A000081.
TreeGf(N)={my(A=vector(N, j, 1)); for (n=1, N-1, A[n+1] = 1/n * sum(k=1, n, sumdiv(k, d, d*A[d]) * A[n-k+1] ) ); x*Ser(A)}
seq(n)={Vec(2*TreeGf(n) - TreeGf(n)^2 - x)} \\ Andrew Howroyd, Nov 02 2019
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| CROSSREFS
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Cf. A000081, A000272, A007716, A007717, A030019, A052888, A134954, A317631, A317632, A318697, A320921, A321155.
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| STATUS
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approved
editing
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#8 by Susanna Cuyler at Wed Oct 31 21:53:19 EDT 2018
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#7 by Gus Wiseman at Tue Oct 30 23:19:19 EDT 2018
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#6 by Gus Wiseman at Tue Oct 30 23:18:41 EDT 2018
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| CROSSREFS
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Cf. A000272, A007716, A007717, A030019, A052888, A134954, A317631, A317632, A317679A318697, A320921, A321155.
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#5 by Gus Wiseman at Tue Oct 30 21:08:42 EDT 2018
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| EXAMPLE
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Non-isomorphic representatives of the a(1) = 1 through a(6) = 10 multi-hypertrees of weight n - 1 with singletons allowed:
{} {{1}} {{1,2}} {{1,2,3}} {{1,2,3,4}} {{1,2,3,4,5}}
{{1},{1}} {{2},{1,2}} {{1,3},{2,3}} {{1,4},{2,3,4}}
{{1},{1},{1}} {{3},{1,2,3}} {{4},{1,2,3,4}}
{{1},{2},{1,2}} {{2},{1,3},{2,3}}
{{2},{2},{1,2}} {{2},{3},{1,2,3}}
{} {{1}} {{12}} {{123}} {{1234}} {{12345}}
{{1}{1}} {{2}{12}} {{13}{23}} {{14}{234}}
{{1}{1}{1}} {{3}{123}} {{4}{1234}}
{{1}{2}{12}} {{2}{13}{23}}
{{2}{2}{12}} {{2}{3}{123}}
{{ {{1},{}{1},{}{1},{}{1}} {{3},{1,3},{2,3}{13}{23}}
{{3},{3},{1,2,3}}
{{3}{3}{123}}
{{ {{1},{2},{}{2},{1,}{2}{12}}
{{2},{ {{2},{}{2},{1,}{2}{12}}
{{ {{1},{}{1},{}{1},{}{1},{}{1}}
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#4 by Gus Wiseman at Tue Oct 30 21:05:22 EDT 2018
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| COMMENTS
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Also the number of non-isomorphic multi-hypertrees of weight n - 1 with singletons allowed. A multi-hypertree with singletons allowed is a connected set multipartition (multiset of sets) with density -1, where the density of a set multipartition is the weight (sum of sizes of the parts) minus the number of parts minus the number of vertices. - Gus Wiseman, Oct 30 2018
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| EXAMPLE
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From Gus Wiseman, Oct 30 2018: (Start)
Non-isomorphic representatives of the a(1) = 1 through a(6) = 10 multi-hypertrees with singletons allowed:
{} {{1}} {{1,2}} {{1,2,3}} {{1,2,3,4}} {{1,2,3,4,5}}
{{1},{1}} {{2},{1,2}} {{1,3},{2,3}} {{1,4},{2,3,4}}
{{1},{1},{1}} {{3},{1,2,3}} {{4},{1,2,3,4}}
{{1},{2},{1,2}} {{2},{1,3},{2,3}}
{{2},{2},{1,2}} {{2},{3},{1,2,3}}
{{1},{1},{1},{1}} {{3},{1,3},{2,3}}
{{3},{3},{1,2,3}}
{{1},{2},{2},{1,2}}
{{2},{2},{2},{1,2}}
{{1},{1},{1},{1},{1}}
(End)
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| CROSSREFS
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Cf. A000272, A007716, A007717, A030019, A052888, A134954, A317631, A317632, A317679, A320921, A321155.
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| STATUS
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approved
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#3 by Russ Cox at Sat Mar 31 20:01:59 EDT 2012
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Discussion
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| Sat Mar 31
| 20:01
| OEIS Server: https://oeis.org/edit/global/999
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