cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A215772 Number of undirected labeled graphs on n nodes with exactly 2 cycle graphs as connected components.

Original entry on oeis.org

1, 3, 7, 25, 127, 777, 5547, 45216, 414144, 4209480, 47009880, 572101920, 7535302560, 106791531840, 1620314539200, 26205248563200, 450022496716800, 8178211565798400, 156798308067609600, 3162998405887488000, 66967168288624128000, 1484773164338365440000
Offset: 2

Views

Author

Alois P. Heinz, Aug 23 2012

Keywords

Examples

			a(4) = 7:  .1.2.  .1-2.  .1.2.  o1.2.  .1.2o  .1-2.  .1.2.
.          .|.|.  .....  ..X..  ../|.  .|\..  ..\|.  .|\..
.          .3.4.  .3-4.  .3.4.  .3-4.  .3-4.  o3.4.  .3-4o

Links

Crossrefs

Column k=2 of A215771.

Programs

  • Magma
    I:=[1, 3, 7, 25, 127, 777,5547]; [n le 7 select I[n] else ((2*n^3 - 21*n^2 + 65*n - 58)*Self(n-1) - (n^4 - 13*n^3 + 60*n^2 - 116*n + 80)*Self(n-2))/((n-3)*(n- 6)): n in [1..30]]; // G. C. Greubel, Aug 30 2018
  • Maple
    a:= proc(n) option remember; `if`(n<7, [0, 0, 1, 3, 7, 25, 127][n+1],
            ((2*n^3-21*n^2+65*n-58)*a(n-1)
      -(n^4-13*n^3+60*n^2-116*n+80)*a(n-2))/((n-3)*(n-6)))
    end:
seq(a(n), n=2..30);
  • Mathematica
    Join[{1, 3, 7, 25, 127}, RecurrenceTable[{a[n] == ((2*n^3 - 21*n^2 + 65*n - 58)*a[n-1] - (n^4 - 13*n^3 + 60*n^2 - 116*n + 80)*a[n-2])/((n-3)*(n- 6)), a[7] == 777, a[8] == 5547}, a, {n, 7, 20}]] (* G. C. Greubel, Aug 30 2018 *)
  • PARI
    m=30; v=concat([1,3,7,25,127, 777, 5547], vector(m-6)); for(n=7, m, v[n] = ((2*n^3-21*n^2+65*n-58)*v[n-1]-(n^4-13*n^3+60*n^2-116*n +80)*v[n-2] )/((n-3)*(n-6))); v \\ G. C. Greubel, Aug 30 2018

Formula

See Maple program.
a(n) ~ (n-1)! * (log(n) + 3/2 + gamma)/4, where gamma is the Euler-Mascheroni constant (A001620). - Vaclav Kotesovec, Apr 27 2015

A215773 Number of undirected labeled graphs on n+3 nodes with exactly n cycle graphs as connected components.

Original entry on oeis.org

0, 3, 25, 120, 420, 1190, 2898, 6300, 12540, 23265, 40755, 68068, 109200, 169260, 254660, 373320, 534888, 750975, 1035405, 1404480, 1877260, 2475858, 3225750, 4156100, 5300100, 6695325, 8384103, 10413900, 12837720, 15714520, 19109640, 23095248, 27750800
Offset: 0

Views

Author

Alois P. Heinz, Aug 23 2012

Keywords

Examples

			a(1) = 3:  .1-2.  .1.2.  .1-2.
.          .|.|.  .|X|.  ..X..
.          .3-4.  .3.4.  .3-4.

Links

Crossrefs

A diagonal of A215771.

Programs

  • Maple
    a:= n-> (24+(50+(41+(21+(7+n)*n)*n)*n)*n)*n/48:
seq (a(n), n=0..40);

Formula

G.f.: (8*x^2+4*x+3)*x/(1-x)^7.
a(n) = n*(n+1)*(n+2)*(n+3)*(n^2+n+4)/48.

A215774 Number of undirected labeled graphs on n+4 nodes with exactly n cycle graphs as connected components.

Original entry on oeis.org

0, 12, 127, 742, 3157, 10857, 31899, 82929, 195459, 425139, 864864, 1662661, 3045406, 5349526, 9059946, 14858646, 23684298, 36804558, 55902693, 83180328, 121478203, 174416935, 246559885, 343600335, 472575285, 642108285, 862683822, 1146955887, 1510093452
Offset: 0

Views

Author

Alois P. Heinz, Aug 23 2012

Keywords

Examples

			a(1) = 12 = 4!/2: (1-2-3-4-5-1), (1-2-3-5-4-1), (1-2-4-3-5-1), (1-2-4-5-3-1), (1-2-5-3-4-1), (1-2-5-4-3-1), (1-3-2-4-5-1), (1-3-2-5-4-1), (1-3-4-2-5-1), (1-3-5-2-4-1), (1-4-2-3-5-1), (1-4-3-2-5-1).

Links

Crossrefs

A diagonal of A215771.

Programs

  • Maple
    a:= n-> (6864+(20180+(22980+(13295+(4536+(1070+(180+15*n)*
               n)*n)*n)*n)*n)*n)*n/5760:
seq(a(n), n=0..40);

Formula

G.f.: (43*x^3+31*x^2+19*x+12)*x/(1-x)^9.
a(n) = n*(n+1)*(n+2)*(n+3)*(n+4)*(15*n^3+30*n^2+245*n+286)/5760.

A253276 Number of undirected labeled graphs on 2n nodes with exactly n cycle graphs as connected components.

Original entry on oeis.org

1, 1, 7, 120, 3157, 109935, 4754200, 245722477, 14779601837, 1014260971581, 78214593177825, 6696084566881710, 630196627700087272, 64671387743952373150, 7186999700934499032405, 859879811676654352591875, 110201017079975901129209565, 15061748014412378814910531365
Offset: 0

Views

Author

Alois P. Heinz, May 01 2015

Keywords

Links

Crossrefs

Cf. A215771.

Programs

  • Maple
    b:= proc(n, k) option remember; `if`(k<0 or k>n, 0, `if`(n=0, 1,
      add(binomial(n-1, i)*b(n-1-i, k-1)*ceil(i!/2), i=0..n-k)))
    end:
a:= n-> b(2*n, n):
seq(a(n), n=0..20);
  • Mathematica
    b[n_, k_] := b[n, k] = If[k<0 || k>n, 0, If[n==0, 1, Sum[Binomial[n-1, i]*b[n-1-i, k-1]*Ceiling[i!/2], {i, 0, n-k}]]]; a[n_] := b[2 n, n]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Feb 26 2017, translated from Maple *)

Formula

a(n) = A215771(2n,n).
a(n) ~ c * d^n * (n-1)!, where d = 8.52944416851968239902405793921886268..., c = 0.1101477123991489575407024889... . - Vaclav Kotesovec, May 01 2015
Previous Showing 11-14 of 14 results.

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