A361361 -id:A361361 - cp's OEIS frontend

A321304 Triangle T(n,f): the number of bicolored connected cubic graphs on 2n vertices with f vertices of the first color.

1, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 5, 5, 5, 2, 2, 5, 10, 31, 46, 63, 46, 31, 10, 5, 19, 64, 248, 542, 931, 1052, 931, 542, 248, 64, 19, 85, 490, 2382, 7011, 15199, 23405, 27336, 23405, 15199, 7011, 2382, 490, 85, 509, 4595, 27233, 101002, 268675, 523246, 776657, 882321, 776657, 523246, 268675, 101002, 27233, 4595, 509

Offset: 0

R. J. Mathar, Nov 03 2018

These are connected, undirected, simple cubic graphs where each vertex has either the first or the second color. Row n has 2n+1 entries, 0<=f<=2n. The column f=0 (1, 0, 2, 5,...) counts the cubic graphs (A002851). The column f=1 (0, 1, 2, 10, 64, 490...) counts the rooted cubic graphs.

			The triangle starts:
0 vertices:   1;
2 vertices:   0,  0,   0;
4 vertices:   1,  1,   1,   1,   1;
6 vertices:   2,  2,   5,   5,   5,    2,   2;
8 vertices:   5, 10,  31,  46,  63,   46,  31,  10,   5;
10 vertices: 19, 64, 248, 542, 931, 1052, 931, 542, 248, 64, 19;

Andrew Howroyd, Table of n, a(n) for n = 0..440 (rows 0..20)

Columns f=0, 1, 2 are A002851, A361407, A361408.
Row sums are A361403.
Central coefficients are A361406.
Cf. A294783 (bicolored trees), A321305 (signed edges), A361361 (not necessarily connected).

T(n,f) = T(n,2n-f).

Terms a(49) and beyond from Andrew Howroyd, Mar 11 2023

A361410 Number of cubic graphs on 2n unlabeled vertices rooted at a vertex.

Original entry on oeis.org

0, 1, 2, 11, 68, 510, 4712, 51877, 664520, 9662968, 156490473, 2783955994, 53863486240, 1124886942314, 25206326633070, 603048386506505, 15339533779133582, 413338072569232815, 11760801736217845686, 352342902996056683824

Offset: 1

Andrew Howroyd, Mar 11 2023

nonn

Column k=1 of A361361.

Cf. A005638, A361407, A361411.

A361411 Number of cubic graphs on 2n unlabeled vertices rooted at a pair of indistinguishable vertices.

Original entry on oeis.org

0, 1, 5, 33, 257, 2443, 27682, 363759, 5405697, 89134360, 1609418390, 31525697245, 665263778962, 15039817276939, 362579178545598, 9284375250749758, 251643492565059981, 7197256536139662143, 216621907269166632361, 6844093745422473471562

Offset: 1

Andrew Howroyd, Mar 11 2023

nonn

Column k=2 of A361361.

Cf. A005638, A361408, A361410.

A361404 Triangle read by rows: T(n,k) is the number of graphs with loops on n unlabeled vertices with k loops.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 4, 6, 6, 4, 11, 20, 28, 20, 11, 34, 90, 148, 148, 90, 34, 156, 544, 1144, 1408, 1144, 544, 156, 1044, 5096, 13128, 20364, 20364, 13128, 5096, 1044, 12346, 79264, 250240, 472128, 580656, 472128, 250240, 79264, 12346

Offset: 0

Andrew Howroyd, Mar 11 2023

T(n,k) is the number of bicolored graphs on n nodes with k vertices having the first color. Adjacent vertices may have the same color.

			Triangle begins:
     1;
     1,    1;
     2,    2,     2;
     4,    6,     6,     4;
    11,   20,    28,    20,    11;
    34,   90,   148,   148,    90,    34;
   156,  544,  1144,  1408,  1144,   544,  156;
  1044, 5096, 13128, 20364, 20364, 13128, 5096, 1044;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Columns k=0..2 are A000088, A000666(n-1), A303829.
Row sums are A000666.
Central coefficients are A361405.
Cf. A361361 (cubic).

permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
edges(v) = {sum(i=2, #v, sum(j=1, i-1, gcd(v[i], v[j]))) + sum(i=1, #v, v[i]\2)}
row(n) = {my(s=0); forpart(p=n, s+=permcount(p)*2^edges(p)*prod(i=1, #p, 1 + x^p[i])); Vecrev(s/n!)}

T(n,k) = T(n, n-k).

A361362 Number of bicolored cubic graphs on 2n unlabeled vertices.

Original entry on oeis.org

1, 0, 5, 23, 262, 4775, 126026, 4315481, 177939133, 8486268015, 457398466292, 27442206452816, 1812456359735759, 130630783430897459, 10200930403740584232, 857888417749736680977, 77299388952584465682198, 7429004444540543143978901, 758559920648248499878180973, 82006219796827162656265186759, 9357477001574426557631620060473

Offset: 0

Andrew Howroyd, Mar 10 2023

nonn

Adjacent vertices may have the same color.

Row sums of A361361.

Cf. A361403 (connected).

Euler transform of A361403.

A361409 Number of bicolored cubic graphs on 2n unlabeled vertices with n vertices of each color.

Original entry on oeis.org

1, 0, 1, 5, 66, 1071, 27606, 887305, 34583357, 1562797351, 80177945542, 4597212665432, 291214532031215, 20193430937073303, 1521240318892230748, 123711268485285686123, 10801367759750192440520, 1007762402877770768660697, 100058924666668698411972015, 10533938778032068908299390227, 1172080056205294525370971027435

Offset: 0

Andrew Howroyd, Mar 11 2023

nonn

Adjacent vertices may have the same color.

Central coefficients of A361361.

Cf. A005638, A361406, A361410, A361411.

cp's OEIS Frontend

A321304 Triangle T(n,f): the number of bicolored connected cubic graphs on 2n vertices with f vertices of the first color.

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A361410 Number of cubic graphs on 2n unlabeled vertices rooted at a vertex.

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A361411 Number of cubic graphs on 2n unlabeled vertices rooted at a pair of indistinguishable vertices.

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A361404 Triangle read by rows: T(n,k) is the number of graphs with loops on n unlabeled vertices with k loops.

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A361362 Number of bicolored cubic graphs on 2n unlabeled vertices.

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A361409 Number of bicolored cubic graphs on 2n unlabeled vertices with n vertices of each color.

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