Michael Burkhart - cp's OEIS frontend

A182378 G.f. satisfies A(x) = 1 + x*cycle_index(Sym(7), A(x)).

1, 1, 1, 2, 4, 9, 20, 48, 115, 285, 716, 1833, 4740, 12410, 32754, 87176, 233547, 629540, 1705809, 4644231, 12697500, 34848694, 95973026, 265142431, 734606478, 2040683413, 5682634446, 15859800889, 44355531103, 124290064228, 348904212741, 981082979409

Offset: 0

Michael Burkhart, Apr 26 2012

nonn

Number of rooted trees where each node has at most 7 children.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000081, A036717, A036718, A036719, A036720, A036721, A036722, A244372, A292553, A292554, A292555, A292556.

Column k=7 of A299038.

b:= proc(n, i, t, k) option remember; `if`(n=0, 1,
      `if`(i<1, 0, add(binomial(b((i-1)$2, k$2)+j-1, j)*
       b(n-i*j, i-1, t-j, k), j=0..min(t, n/i))))
    end:
a:= n-> `if`(n=0, 1, b(n-1$2, 7$2)):
seq(a(n), n=0..35);  # Alois P. Heinz, Sep 20 2017

b[n_, i_, t_, k_] := b[n, i, t, k] = If[n == 0, 1, If[i < 1, 0, Sum[ Binomial[ b[i-1, i-1, k, k] + j - 1, j]*b[n-i*j, i-1, t-j, k], {j, 0, Min[t, n/i]}]]];
a[n_] := If[n == 0, 1, b[n-1, n-1, 7, 7]];
Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Jan 15 2018, after Alois P. Heinz *)

a(n) = Sum_{j=1..7} A244372(n,j) for n>0, a(0) = 1. - Alois P. Heinz, Sep 19 2017

a(n) / a(n+1) ~ 0.338512011286603947719604869750539045616436718225097926729820... - Robert A. Russell, Feb 11 2023

More terms from Patrick Devlin, Apr 29 2012

A182371 Number of connected labeled graphs with n nodes and n+11 edges.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1330, 6905220, 7279892361, 3717889913655, 1255470137209650, 326123611416074340, 70993993399632155710, 13659118629343706026053, 2405832308811599670396135, 397496768417871214784702640, 62693059156926401902640364120, 9561367292987041683030275944320

Offset: 1

Michael Burkhart, Apr 26 2012

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..100
S. Janson, D. E. Knuth, T. Łuczak and B. Pittel, The Birth of the Giant Component, Random Structures and Algorithms Vol. 4 (1993), 233-358.

A diagonal of A343088.

Cf. A057500.

N:=20: [seq(coeff(op(i, [seq(coeff(taylor(log(add(x^i*(1+y)^(binomial(i, 2))/i!, i=0..N)), x=0, N+1), x, i)*i!, i=1..N)]), y, i-1+12), i=1..N)];

Offset corrected and terms a(17) and beyond from Andrew Howroyd, Apr 16 2021

A182295 Number of connected labeled graphs with n nodes and n+10 edges.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 5985, 13112470, 8535294180, 3096620034795, 800118566011380, 166591475854153740, 30012638793107746776, 4892304538906805158775, 743352352817243899253160, 107478174967432322995403280, 15008321493306766503800761840, 2046331888629918743459557040544

Offset: 1

Michael Burkhart, Apr 23 2012

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..100
Steven R. Finch, An exceptional convolutional recurrence, arXiv:2408.12440 [math.CO], 22 Aug 2024.
S. Janson, D. E. Knuth, T. Łuczak and B. Pittel, The Birth of the Giant Component, Random Structures and Algorithms Vol. 4 (1993), 233-358.

A diagonal of A343088.

Cf. A057500.

N:=20: [seq(coeff(op(i,[seq(coeff(taylor(log(add(x^i*(1+y)^(binomial(i,2))/i!,i=0..N)),x=0,N+1),x,i)*i!,i=1..N)]),y,i-1+11),i=1..N)];

Offset corrected and terms a(16) and beyond from Andrew Howroyd, Apr 16 2021

A182294 Number of connected labeled graphs with n nodes and n+9 edges.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 20349, 21426300, 8956859646, 2352103292070, 470090359867986, 79002015147719136, 11836068369346126698, 1640443794179544776604, 215598057543037336382670, 27336005392867324870778880, 3385297472808136707459580488, 413211903044379104303226531072

Offset: 1

Michael Burkhart, Apr 23 2012

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..100
Steven R. Finch, An exceptional convolutional recurrence, arXiv:2408.12440 [math.CO], 22 Aug 2024.
S. Janson, D. E. Knuth, T. Łuczak and B. Pittel, The Birth of the Giant Component, Random Structures and Algorithms Vol. 4 (1993), 233-358.

A diagonal of A343088.

Cf. A057500.

N:=20: [seq(coeff(op(i,[seq(coeff(taylor(log(add(x^i*(1+y)^(binomial(i,2))/i!,i=0..N)),x=0,N+1),x,i)*i!,i=1..N)]),y,i-1+10),i=1..N)];

Offset corrected and terms a(16) and beyond from Andrew Howroyd, Apr 16 2021

A207539 Dodecanacci numbers (12th-order Fibonacci sequence): a(n) = a(n-1) +...+ a(n-12) with a(0)=...=a(11)=1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 12, 23, 45, 89, 177, 353, 705, 1409, 2817, 5633, 11265, 22529, 45057, 90102, 180181, 360317, 720545, 1440913, 2881473, 5762241, 11523073, 23043329, 46081025, 92150785, 184279041, 368513025, 736935948, 1473691715

Offset: 0

Michael Burkhart, Feb 18 2012

G. C. Greubel, Table of n, a(n) for n = 0..1000
Kai Wang, Identities for generalized enneanacci numbers, Generalized Fibonacci Sequences (2020).
Index entries for linear recurrences with constant coefficients, signature (1,1,1,1,1,1,1,1,1,1,1,1).

Cf. A000045, A000213, A000288, A127624, A168083.

f12:=proc(n) option remember: if n<=12 then 1: else add(f12(n-i),i=1..12): fi: end:

LinearRecurrence[Table[1, {12}], Table[1, {12}], 100]

x='x+O('x^50); Vec((1-x-x^2-x^3-x^4-x^5-x^6-x^7-x^8-x^9-x^10-x^11 +10*x^12)/(1-2*x+x^13)) \\ G. C. Greubel, Jul 28 2017

G.f.: (1-x-x^2-x^3-x^4-x^5-x^6-x^7-x^8-x^9-x^10-x^11 +10*x^12)/(1 -2*x +x^13).

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User: Michael Burkhart

A182378 G.f. satisfies A(x) = 1 + x*cycle_index(Sym(7), A(x)).

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A182371 Number of connected labeled graphs with n nodes and n+11 edges.

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A182295 Number of connected labeled graphs with n nodes and n+10 edges.

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A182294 Number of connected labeled graphs with n nodes and n+9 edges.

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A207539 Dodecanacci numbers (12th-order Fibonacci sequence): a(n) = a(n-1) +...+ a(n-12) with a(0)=...=a(11)=1.

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