A026797 -id:A026797 - cp's OEIS frontend

A185644 Triangular array E(n,k) counting, not necessarily connected, k-regular simple graphs on n vertices with girth exactly 4.

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, 1, 0, 0, 0, 0, 0, 5, 2, 1, 0, 0, 1, 0, 2, 0, 0, 0, 2, 21, 12, 1, 1, 0, 0, 2, 0, 31, 0, 0, 0, 0, 3, 103, 220, 7, 1, 1, 0, 0, 3, 0, 1606, 0, 1, 0, 0, 0, 5, 752, 16829, 388, 9, 1, 1, 0, 0, 5, 0, 193900, 0, 6, 0, 0, 0

Offset: 1

Jason Kimberley, Feb 22 2013

In the n-th row 0 <= 2k <= n.

			01: 0;
02: 0, 0;
03: 0, 0;
04: 0, 0, 1;
05: 0, 0, 0;
06: 0, 0, 0, 1;
07: 0, 0, 0, 0;
08: 0, 0, 1, 2, 1;
09: 0, 0, 1, 0, 0;
10: 0, 0, 0, 5, 2, 1;
11: 0, 0, 1, 0, 2, 0;
12: 0, 0, 2, 21, 12, 1, 1;
13: 0, 0, 2, 0, 31, 0, 0;
14: 0, 0, 3, 103, 220, 7, 1, 1;
15: 0, 0, 3, 0, 1606, 0, 1, 0;
16: 0, 0, 5, 752, 16829, 388, 9, 1, 1;
17: 0, 0, 5, 0, 193900, 0, 6, 0, 0;
18: 0, 0, 7, 7385, 2452820, 406824, 267, 8, 1, 1;
19: 0, 0, 8, 0, 32670331, 0, 3727, 0, 0, 0;
20: 0, 0, 11, 91939, 456028487, 1125022326, 483012, 741, 13, 1, 1;
21: 0, 0, 12, 0, 6636066126, 0, 69823723, 0, 1, 0, 0;
22: 0, 0, 16, 1345933, 100135577863, 3813549359275, 14836130862, 2887493, ?, 14, 1;

Jason Kimberley, Table of n, a(i)=E(n,k) for i = 1..139 (n = 1..22)
Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth exactly g

The sum of the n-th row of this sequence is A198314(n).

Not necessarily connected k-regular simple graphs girth exactly 4: A198314 (any k), this sequence (triangle); fixed k: A026797 (k=2), A185134 (k=3), A185144 (k=4).

E(n,k) = A186734(n,k) + A210704(n,k), noting the differing row lengths.

E(n,k) = A185304(n,k) - A185305(n,k), noting the differing row lengths.

A027196 Number of partitions of n into an even number of parts, the least being 4; also, a(n+4) = number of partitions of n into an odd number of parts, each >=4.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 9, 12, 14, 17, 20, 25, 29, 35, 41, 50, 58, 70, 81, 97, 113, 134, 156, 185, 214, 252, 292, 343, 396, 463, 534, 623, 718, 833, 958, 1110, 1274, 1471, 1686, 1943, 2223, 2555, 2919, 3347, 3818, 4368

Offset: 1

Clark Kimberling

nonn

Cf. A026797, A027190.

b:= proc(n, i, t) option remember; `if`(n=0, t,
     `if`(i>n, 0, b(n, i+1, t)+b(n-i, i, 1-t)))
    end:
a:= n-> `if`(n<4, 0, b(n-4, 4, 0)):
seq(a(n), n=1..100);  # Alois P. Heinz, Oct 18 2019

b[n_, i_, t_] := b[n, i, t] = If[n == 0, t, If[i > n, 0, b[n, i + 1, t] + b[n - i, i, 1 - t]]];
a[n_] := If[n < 4, 0, b[n - 4, 4, 0]];
Array[a, 100] (* Jean-François Alcover, May 17 2020, after Alois P. Heinz *)

a(n) + A027190(n) = A026797(n). - R. J. Mathar, Oct 18 2019

G.f.: x^8 * Sum_{k>=0} x^(8*k)/Product_{j=1..2*k+1} (1-x^j). - Seiichi Manyama, May 15 2023

cp's OEIS Frontend

A185644 Triangular array E(n,k) counting, not necessarily connected, k-regular simple graphs on n vertices with girth exactly 4.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A027196 Number of partitions of n into an even number of parts, the least being 4; also, a(n+4) = number of partitions of n into an odd number of parts, each >=4.

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

Formula