A033995 -id:A033995 - cp's OEIS frontend

1, 1, 1, 1, 1, 3, 3, 1, 6, 15, 16, 3, 1, 10, 45, 110, 140, 60, 10, 1, 15, 105, 435, 1125, 1701, 1200, 480, 105, 10, 1, 21, 210, 1295, 5355, 14952, 26572, 26670, 17535, 7840, 2331, 420, 35, 1, 28, 378, 3220, 19075, 81228, 246414, 507424, 666015, 620900, 431368

Offset: 0

Vladeta Jovovic, Jun 23 2007

			Triangle begins:
  1;
  1;
  1,  1;
  1,  3,  3;
  1,  6, 15,  16,   3;
  1, 10, 45, 110, 140, 60, 10;
  ...

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.5.

Andrew Howroyd, Table of n, a(n) for n = 0..1403 (rows 0..25)

Row sums give A047864,
Columns k=1..5 are A000217(n-1), A050534, A053526, A053527, A053528.
The unlabeled version is A297877.
Cf. A052296, A033995.

nn=10;f[x_,y_]:=Sum[Sum[Binomial[n,k](1+y)^(k(n-k)),{k,0,n}]x^n/n!,{n,0,nn}];Map[Select[#,#>0&]&,Range[0,nn]!CoefficientList[Series[Exp[Log[f[x,y]]/2],{x,0,nn}],{x,y}]]//Grid (* Geoffrey Critzer, Sep 05 2013 *)

T(n)={[Vecrev(p) | p<-Vec(serlaplace(sqrt(sum(k=0, n, exp(x*(1+y)^k + O(x*x^n))*x^k/k! ))))]}
{ my(A=T(6)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Jan 10 2022

E.g.f.: sqrt(Sum_{n>=0} exp(x*(1+q)^n)*x^n/n!).

A008324 Number of simple regular bipartite graphs with 2n nodes.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 18, 40, 230, 4296, 431206, 162267272, 201636689771, 777816803942186, 9865957936943931964, 395886667549681689591841, 53716176608076643470621234239, 23524515269630339982914646821899537, 35682168849414944013547274452501768506834

Offset: 0

Brendan McKay

First differs from A333732 at n = 12. - Andrew Howroyd, Apr 03 2020

Brendan McKay and E. Rogoyski, Latin squares of order ten, Electron. J. Combinatorics, 2 (1995) #N3.

Row sums of A008327.

Cf. A033995, A008323, A333732.

a(0)=1 prepended and terms a(11) and beyond from Andrew Howroyd, Apr 03 2020

A165659 Numerators of A007504 divided by A033955, starting from the second term of A033955.

Original entry on oeis.org

2, 5, 5, 17, 28, 41, 58, 77, 50, 129, 16, 197, 119, 281, 164, 127, 440, 501, 568, 71, 356, 791, 46, 321, 530, 1161, 1264, 457, 1480, 1593, 344, 1851, 284, 2127, 2276, 809, 2584, 2747, 1457, 441, 1633, 1149, 3638, 3831, 1007, 4227, 4438

Offset: 1

Creighton Dement, Sep 24 2009

Conjecture: with the exception of the second term, 2 <= a(n)/A165660(n) < 3.

Cf. A007504, A033995, A002110, A102647, A165657, A165658, A165660.

a1(n)=sum(i=1, n, prime(i));
b1(n)=sum(i=1, n, prime(n+1)%prime(i));
a(n)=if(n<0, 0, numerator(a1(n)/b1(n)));
for(n=1, 50, print1(a(n) ", "))

Typo in definition corrected by Creighton Dement, Oct 09 2009

A165660 Denominators of A007504 divided by A033955, starting from the second term of A033955.

Original entry on oeis.org

1, 3, 2, 8, 13, 18, 27, 29, 23, 56, 7, 74, 44, 98, 67, 49, 171, 200, 217, 28, 137, 309, 17, 116, 209, 448, 471, 174, 571, 629, 137, 739, 111, 793, 853, 318, 997, 1002, 560, 164, 610, 446, 1419, 1466, 385, 1615, 1573, 1633, 1707, 1825, 946, 662, 2221, 781, 1198

Offset: 1

Creighton Dement, Sep 24 2009

Conjecture: with the exception of the second term, 2 <= A165659(n)/a(n) < 3.

Cf. A007504, A033995, A002110, A102647, A165657, A165658, A165659

a1(n)=sum(i=1, n, prime(i)); b1(n)=sum(i=1, n, prime(n+1)%prime(i)); a(n)=if(n<0, 0, denominator(a1(n)/b1(n))); for(n=1, 50, print1(a(n) ", "))

Terms corrected by Creighton Dement, Oct 03 2009
Removed a conjecture - R. J. Mathar, Oct 09 2009
Typo in definition corrected by Creighton Dement, Oct 09 2009

A297877 Triangle T(n,k) read by rows, giving number of bipartite graphs with n nodes (n >= 0) and k edges (0 <= k <= floor(n/2*n/2)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 3, 4, 1, 1, 1, 1, 2, 4, 7, 8, 6, 3, 2, 1, 1, 1, 2, 4, 8, 13, 19, 14, 13, 7, 4, 1, 1, 1, 1, 2, 4, 9, 16, 32, 45, 52, 48, 40, 24, 16, 7, 3, 2, 1, 1, 1, 2, 4, 9, 17, 38, 70, 120, 150, 179, 164, 143, 94, 63, 32, 19, 7, 4, 1, 1, 1, 1, 2, 4, 9, 18, 41, 85, 181, 324, 500, 659

Offset: 0

Juergen Will, Jan 07 2018

The sum of the m-th row is the (m-1)-st member of A033995, number of bipartite graphs with n nodes.

			Triangle begins:
0:  1;
1:  1;
2:  1,  1;
3:  1,  1,  1;
4:  1,  1,  2,  2,  1;
5:  1,  1,  2,  3,  4,  1,  1;
6:  1,  1,  2,  4,  7,  8,  6,  3,  2,  1;
7:  1,  1,  2,  4,  8, 13, 19, 14, 13,  7,  4,  1,  1;

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, New York / London, 1973.

P. Hanlon, The enumeration of bipartite graphs, Discrete Math. 28 (1979), 49-57.
Juergen Will, Rows 0 to 10 of triangle.

Cf. A033995 (row sums).

A079571 Number of unlabeled, connected graphs on n vertices whose complements are bipartite.

Original entry on oeis.org

1, 1, 1, 2, 5, 11, 32, 85, 299, 1115, 5474, 32298, 251129, 2527706, 33985846, 611846933, 14864650916, 488222721984, 21712049275189, 1308300679611460, 106897965189674281, 11852113048215107812, 1784730721403509209204, 365323537513403184463262

Offset: 0

Jim Nastos, Jan 24 2003

nonn

Equivalently, number of bipartite graphs whose complement is connected. The only bipartite graphs with disconnected complement are complete bipartite graphs. - Falk Hüffner, Jan 22 2016

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

Cf. A005142, A033995.

A005142 = Import["https://oeis.org/A005142/b005142.txt", "Table"][[All, 2]];
etr[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d*p[d], {d, Divisors[j]}]*b[n - j], {j, 1, n}]/n]; b];
b = etr[A005142[[# + 1]]&];
a[n_] := b[n] - Floor[n/2];
a /@ Range[0, 50] (* Jean-François Alcover, Sep 17 2019 *)

a(n) = A033995(n) - floor(n/2).

Corrected and extended using formula by Falk Hüffner, Jan 22 2016

a(0)=1 prepended and terms a(21) and beyond from Andrew Howroyd, Sep 05 2018

cp's OEIS Frontend

A076318 Number of 6-colorable (i.e., chromatic number <= 6) simple graphs on n nodes.

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A076319 Number of 7-colorable (i.e., chromatic number <= 7) simple graphs on n nodes.

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A076320 Number of 8-colorable (i.e., chromatic number <= 8) simple graphs on n nodes.

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A076321 Number of 9-colorable (i.e., chromatic number <= 9) simple graphs on n nodes.

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A117279 Triangle read by rows: T(n,k) is number of labeled bipartite graphs with n nodes and k edges.

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A008324 Number of simple regular bipartite graphs with 2n nodes.

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A165659 Numerators of A007504 divided by A033955, starting from the second term of A033955.

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A165660 Denominators of A007504 divided by A033955, starting from the second term of A033955.

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PARI

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A297877 Triangle T(n,k) read by rows, giving number of bipartite graphs with n nodes (n >= 0) and k edges (0 <= k <= floor(n/2*n/2)).

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