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Revision History for A134959

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Showing entries 1-10 | older changes

A134959

Number of spanning hypertrees with n unlabeled vertices: analog of A035053 when edges of size 1 are allowed (with no two equal edges).
(history; published version)

#17 by Alois P. Heinz at Mon Feb 10 06:14:11 EST 2020

STATUS

proposed

approved

#16 by Jean-François Alcover at Mon Feb 10 02:37:41 EST 2020

STATUS

editing

proposed

#15 by Jean-François Alcover at Mon Feb 10 02:37:36 EST 2020

MATHEMATICA

etr[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d*p[d], {d, Divisors[j]}]*b[n - j], {j, 1, n}]/n]; b];

EulerT[v_List] := With[{q = etr[v[[#]] &]}, q /@ Range[Length[v]]];

ser[v_] := Sum[v[[i]] x^(i - 1), {i, 1, Length[v]}] + O[x]^Length[v];

b[n_] := Module[{v = {1}}, For[i = 2, i <= n, i++, v = Join[{1}, EulerT[EulerT[2 v]]]]; v];

seq[n_] := Module[{u = 2 b[n]}, 1 + x*ser[EulerT[u]]*(1 - x*ser[u]) + O[x]^n // CoefficientList[#, x]&];

seq[25] (* Jean-François Alcover, Feb 10 2020, after Andrew Howroyd *)

STATUS

approved

editing

#14 by Susanna Cuyler at Mon Aug 27 22:33:05 EDT 2018

STATUS

proposed

approved

#13 by Andrew Howroyd at Mon Aug 27 21:34:10 EDT 2018

STATUS

editing

proposed

#12 by Andrew Howroyd at Mon Aug 27 20:38:16 EDT 2018

DATA

1, 2, 3, 10, 35, 150, 707, 3700, 20470, 119260, 719341, 4466316, 28367118, 183620874, 1207563011, 8049914664, 54295152117, 369981325578, 2544017965638, 17633790542978, 123108792874528, 865045359778662, 6114040341515978, 43443726772579152, 310195170229429300

LINKS

Andrew Howroyd, <a href="/A134959/b134959.txt">Table of n, a(n) for n = 0..200</a>

PROG

(PARI) \\ here b(n) is A318494 as vector

EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}

b(n)={my(v=[1]); for(i=2, n, v=concat([1], EulerT(EulerT(2*v)))); v}

seq(n)={my(u=2*b(n)); Vec(1 + x*Ser(EulerT(u))*(1-x*Ser(u)))} \\ Andrew Howroyd, Aug 27 2018

CROSSREFS

Cf. A030019, A035053, A048143, A054921, A134955, A134957, A144959, A304867, A304911, A318494.

EXTENSIONS

Terms a(7) and beyond from Andrew Howroyd, Aug 27 2018

STATUS

approved

editing

#11 by Susanna Cuyler at Tue May 22 20:35:16 EDT 2018

STATUS

proposed

approved

#10 by Gus Wiseman at Tue May 22 10:17:51 EDT 2018

STATUS

editing

proposed

#9 by Gus Wiseman at Tue May 22 10:15:56 EDT 2018

CROSSREFS

Cf. A030019, A035053, A048143, A054921, A134955, A134957, A144959, A304867, A304911, A304912.

#8 by Gus Wiseman at Mon May 21 09:16:33 EDT 2018

OFFSET

1,0,2