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A007719 Number of independent polynomial invariants of symmetric matrix of order n. 11
1, 2, 4, 11, 30, 95, 328, 1211, 4779, 19902, 86682, 393072, 1847264, 8965027, 44814034, 230232789, 1213534723, 6552995689, 36207886517, 204499421849, 1179555353219, 6942908667578, 41673453738272, 254918441681030, 1588256152307002, 10073760672179505 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Also, number of connected multigraphs with n edges (allowing loops) and any number of nodes.

Also the number of non-isomorphic connected multiset partitions of {1, 1, 2, 2, 3, 3, ..., n, n}. - Gus Wiseman, Jul 18 2018

LINKS

Table of n, a(n) for n=0..25.

R. J. Mathar, Statistics on Small Graphs, arXiv:1709.09000 [math.CO] (2017) Table 63.

FORMULA

Inverse Euler transform of A007717.

EXAMPLE

From Gus Wiseman, Jul 18 2018: (Start)

Non-isomorphic representatives of the a(3) = 11 connected multiset partitions of {1, 1, 2, 2, 3, 3}:

  (112233),

  (1)(12233), (12)(1233), (112)(233), (123)(123),

  (1)(2)(1233), (1)(12)(233), (1)(23)(123), (12)(13)(23),

  (1)(2)(3)(123), (1)(2)(13)(23).

(End)

MATHEMATICA

mob[m_, n_] := If[Mod[m, n] == 0, MoebiusMu[m/n], 0];

EULERi[b_] := Module[{a, c, i, d}, c = {}; For[i = 1, i <= Length[b], i++,

  c = Append[c, i*b[[i]] - Sum[c[[d]]*b[[i - d]], {d, 1, i - 1}]]]; a = {};

  For[i = 1, i <= Length[b], i++, a = Append[a, (1/i)*Sum[mob[i, d]*c[[d]], {d, 1, i}]]]; Return[a]];

permcount[v_] := Module[{m = 1, s = 0, k = 0, t}, For[i = 1, i <= Length[v], i++, t = v[[i]]; k = If[i > 1 && t == v[[i - 1]], k + 1, 1]; m *= t k; s += t]; s!/m];

Kq[q_, t_, k_] := SeriesCoefficient[1/Product[g = GCD[t, q[[j]]]; (1 - x^(q[[j]]/g))^g, {j, 1, Length[q]}], {x, 0, k}];

RowSumMats[n_, m_, k_] := Module[{s = 0}, Do[s += permcount[q]* SeriesCoefficient[ Exp[Sum[Kq[q, t, k]/t x^t, {t, 1, n}]], {x, 0, n}], {q, IntegerPartitions[m]}]; s/m!];

A007717 = Table[Print[n]; RowSumMats[n, 2 n, 2], {n, 0, 20}];

Join[{1}, EULERi[Rest[A007717]]] (* Jean-Fran├žois Alcover, Oct 29 2018, using Andrew Howroyd's code for A007717 *)

CROSSREFS

Cf. A002905, A007716, A007717, A007719, A020555, A050535, A053419, A076864, A191970, A316972, A316974.

Sequence in context: A148153 A148154 A148155 * A148156 A148157 A141268

Adjacent sequences:  A007716 A007717 A007718 * A007720 A007721 A007722

KEYWORD

nonn,nice

AUTHOR

Colin Mallows

EXTENSIONS

a(0)=1 added by Alberto Tacchella, Jun 20 2011

a(7)-a(25) from Franklin T. Adams-Watters, Jun 21 2011

STATUS

approved

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Last modified May 25 22:17 EDT 2019. Contains 323576 sequences. (Running on oeis4.)