A187770 -id:A187770 - cp's OEIS frontend

A029868 Number of connected functions on n points with a loop of length 5.

1, 1, 4, 14, 46, 145, 440, 1315, 3877, 11315, 32792, 94529, 271510, 777764, 2223865, 6350657, 18120730, 51680249, 147359335, 420163711, 1198151432, 3417475326, 9750708533, 27831153091, 79471338455, 227032777454, 648896436944, 1855571389651, 5308837191604

Offset: 5

Christian G. Bower

nonn

Andrew Howroyd, Table of n, a(n) for n = 5..500
C. G. Bower, Transforms

Column 5 of A339428.

nn = 20; f[x_] := Sum[a[n] x^n, {n, 0, nn}]; sol =
SolveAlways[
  0 == Series[
    f[x] - x Product[1/(1 - x^i)^a[i], {i, 1, nn}], {x, 0, nn}],
  x]; b = Flatten[Table[a[n], {n, 1, nn}] /. sol]; CoefficientList[
Series[CycleIndex[CyclicGroup[5], s] /.
   Table[s[i] -> Sum[b[[k]] x^(k*i), {k, 1, nn}], {i, 1, 5}], {x, 0,
nn}], x] (* Geoffrey Critzer, Aug 08 2013 *)

"CIK[ 5 ]" (necklace, indistinct, unlabeled, 5 parts) transform of A000081.
G.f.: A(x) = ( B(x)^5 +4*B(x^5) )/5 where B(x) is the o.g.f. for A000081. - Geoffrey Critzer, Aug 09 2013
a(n) ~ A187770 * A051491^n / n^(3/2). - Vaclav Kotesovec, Dec 25 2020

A029869 Number of connected functions on n points with a loop of length 6.

Original entry on oeis.org

1, 1, 5, 18, 63, 206, 671, 2087, 6434, 19472, 58375, 173316, 511452, 1500697, 4386021, 12775455, 37118209, 107621858, 311552351, 900775893, 2601887149, 7510011727, 21664873773, 62473966631, 180104101037, 519126161517, 1496177366884, 4312044894059, 12427896986697

Offset: 6

Christian G. Bower

nonn

Andrew Howroyd, Table of n, a(n) for n = 6..500
C. G. Bower, Transforms

Column 6 of A339428.

"CIK[ 6 ]" (necklace, indistinct, unlabeled, 6 parts) transform of A000081.

a(n) ~ A187770 * A051491^n / n^(3/2). - Vaclav Kotesovec, Dec 25 2020

Terms a(31) and beyond from Andrew Howroyd, Dec 08 2020

A029870 Number of connected functions on n points with a loop of length 7.

Original entry on oeis.org

1, 1, 5, 21, 80, 285, 970, 3192, 10236, 32197, 99743, 305276, 925342, 2783012, 8316994, 24726109, 73195582, 215911767, 635028054, 1863156727, 5455350409, 15946267328, 46545783253, 135702643984, 395246786050, 1150250414764, 3345193851398, 9723141517918

Offset: 7

Christian G. Bower

nonn

Andrew Howroyd, Table of n, a(n) for n = 7..500
C. G. Bower, Transforms

Column 7 of A339428.

"CIK[ 7 ]" (necklace, indistinct, unlabeled, 7 parts) transform of A000081.

a(n) ~ A187770 * A051491^n / n^(3/2). - Vaclav Kotesovec, Dec 25 2020

Terms a(32) and beyond from Andrew Howroyd, Dec 04 2020

A029871 Number of connected functions on n points with a loop of length 8.

Original entry on oeis.org

1, 1, 6, 25, 104, 384, 1380, 4729, 15806, 51478, 164788, 519296, 1617066, 4983855, 15233671, 46235252, 139506803, 418838281, 1252174861, 3730058316, 11077154790, 32808815240, 96953599162, 285945645659, 841909040785, 2475184643011, 7267678432397, 21315839832323

Offset: 8

Christian G. Bower

nonn

Andrew Howroyd, Table of n, a(n) for n = 8..500
C. G. Bower, Transforms

Column 8 of A339428.

"CIK[ 8 ]" (necklace, indistinct, unlabeled, 8 parts) transform of A000081.

a(n) ~ A187770 * A051491^n / n^(3/2). - Vaclav Kotesovec, Dec 25 2020

Terms a(32) and beyond from Andrew Howroyd, Dec 04 2020

A032205 Number of connected functions on n points with a loop of length 9.

Original entry on oeis.org

1, 1, 6, 30, 127, 506, 1902, 6823, 23673, 79936, 264036, 856772, 2739525, 8652707, 27049259, 83824636, 257847515, 788121922, 2395786374, 7248517868, 21840785100, 65574504200, 196266067194, 585822825763, 1744384025528, 5183186173513, 15372268502463, 45515472900289

Offset: 9

Christian G. Bower

nonn

Andrew Howroyd, Table of n, a(n) for n = 9..500
C. G. Bower, Transforms (2)

Column 9 of A339428.

"CIK[ 9 ]" (necklace, indistinct, unlabeled, 9 parts) transform of A000081.

a(n) ~ A187770 * A051491^n / n^(3/2). - Vaclav Kotesovec, Dec 25 2020

Terms a(33) and beyond from Andrew Howroyd, Dec 04 2020

A032206 Number of connected functions on n points with a loop of length 10.

Original entry on oeis.org

1, 1, 7, 34, 158, 655, 2578, 9619, 34659, 120966, 412214, 1375861, 4516058, 14611989, 46712942, 147798787, 463512254, 1442513910, 4459548539, 13706894100, 41915906463, 127607366453, 386953440455, 1169295277193, 3522431890950, 10581852932516, 31711045921908

Offset: 10

Christian G. Bower

nonn

Andrew Howroyd, Table of n, a(n) for n = 10..500
Christian G. Bower, Transforms (2)

Column 10 of A339428.

"CIK[ 10 ]" (necklace, indistinct, unlabeled, 10 parts) transform of A000081.

a(n) ~ A187770 * A051491^n / n^(3/2). - Vaclav Kotesovec, Dec 25 2020

Terms a(34) and beyond from Andrew Howroyd, Dec 04 2020

A032207 Number of connected functions on n points with a loop of length 11.

Original entry on oeis.org

1, 1, 7, 39, 189, 832, 3415, 13289, 49594, 178972, 628397, 2156842, 7263306, 24068685, 78667879, 254101597, 812408511, 2574398860, 8094625175, 25278311063, 78465711029, 242265995076, 744465052503, 2278029840583, 6944364119469, 21097626446747, 63901611553272

Offset: 11

Christian G. Bower

nonn

Andrew Howroyd, Table of n, a(n) for n = 11..500
Christian G. Bower, Transforms (2)

Column 11 of A339428.

"CIK[ 11 ]" (necklace, indistinct, unlabeled, 11 parts) transform of A000081.

a(n) ~ A187770 * A051491^n / n^(3/2). - Vaclav Kotesovec, Dec 25 2020

Terms a(35) and beyond from Andrew Howroyd, Dec 04 2020

A032208 Number of connected functions on n points with a loop of length 12.

Original entry on oeis.org

1, 1, 8, 45, 229, 1043, 4474, 18034, 69698, 259516, 938342, 3308793, 11427018, 38763095, 129499981, 426937452, 1391429896, 4489322347, 14356694844, 45554354359, 143546523999, 449545164857, 1400098598183, 4339078154900, 13387747332571, 41141433154653

Offset: 12

Christian G. Bower

nonn

Andrew Howroyd, Table of n, a(n) for n = 12..500
C. G. Bower, Transforms (2)

Column 12 of A339428.

"CIK[ 12 ]" (necklace, indistinct, unlabeled, 12 parts) transform of A000081.

a(n) ~ A187770 * A051491^n / n^(3/2). - Vaclav Kotesovec, Dec 25 2020

Terms a(35) and beyond from Andrew Howroyd, Dec 04 2020

A261875 Decimal expansion of the coefficient 'gamma' (see formula) appearing in Otter's result concerning the asymptotics of T_n, the number of non-isomorphic rooted trees of order n.

Original entry on oeis.org

2, 6, 8, 1, 1, 2, 8, 1, 4, 7, 2, 6, 7, 1, 1, 2, 2, 3, 8, 5, 7, 7, 3, 2, 8, 7, 8, 3, 7, 0, 3, 9, 3, 7, 0, 9, 3, 5, 4, 1, 7, 5, 3, 4, 7, 2, 0, 1, 1, 6, 1, 6, 6, 3, 5, 2, 7, 4, 9, 7, 0, 2, 5, 8, 8, 6, 4, 0, 2, 8, 4, 0, 3, 6, 5, 1, 6, 5, 3, 4, 5, 0, 6, 7, 2, 3, 9, 2, 0, 8, 5, 5, 8, 7, 7, 5, 9, 9, 1, 1

Offset: 1

Jean-François Alcover, Sep 04 2015

			2.68112814726711223857732878370393709354175347201161663527497...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.6 Otter's tree enumeration constants, p. 296.

Cf. A000055, A000081, A051491, A086308, A187770.

digits = 100; max = 250; Clear[s, a]; s[n_, k_] := s[n, k] = a[n + 1 - k] + If[n < 2*k, 0, s[n-k, k]]; a[1] = 1; a[n_] := a[n] = Sum[a[k]*s[n-1, k]*k, {k, 1, n-1}]/(n-1); A[x_] := Sum[a[k]*x^k, {k, 0, max}]; APrime[x_] := Sum[k*a[k]*x^(k-1), {k, 0, max}]; eq = Log[c] == 1 + Sum[A[c^-k]/k, {k, 2, max}]; alpha = c /. FindRoot[eq, {c, 3}, WorkingPrecision -> digits+5]; beta = (1+Sum[APrime[alpha^(-k)]/alpha^k, {k, 2, max}])^(3/2)/Sqrt[2*Pi]; gamma = 2^(2/3)*Pi^(1/6)*beta^(1/3) * Sqrt[alpha]; RealDigits[gamma, 10, digits] // First

Lim_{n->infinity} T_n*n^(3/2)/alpha^n = (beta/(2 Pi))^(1/3) = (1/(4 Pi alpha))^(1/2)*gamma, where alpha is A051491 and beta is A086308.

gamma = 2^(2/3)*Pi^(1/6)*beta^(1/3)*sqrt(alpha).

cp's OEIS Frontend

A029868 Number of connected functions on n points with a loop of length 5.

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A029869 Number of connected functions on n points with a loop of length 6.

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A029870 Number of connected functions on n points with a loop of length 7.

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A029871 Number of connected functions on n points with a loop of length 8.

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A032205 Number of connected functions on n points with a loop of length 9.

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A032206 Number of connected functions on n points with a loop of length 10.

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A032207 Number of connected functions on n points with a loop of length 11.

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A032208 Number of connected functions on n points with a loop of length 12.

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A261875 Decimal expansion of the coefficient 'gamma' (see formula) appearing in Otter's result concerning the asymptotics of T_n, the number of non-isomorphic rooted trees of order n.

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