A004100 -id:A004100 - cp's OEIS frontend

A123301 Triangle read by rows: T(n,k) is the number of specially labeled bicolored nonseparable graphs with k points in one color class and n-k points in the other class. "Special" means there are separate labels 1,2,...,k and 1,2,...,n-k for the two color classes (n >= 2, k = 1,...,n-1).

1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 34, 1, 0, 0, 1, 199, 199, 1, 0, 0, 1, 916, 7037, 916, 1, 0, 0, 1, 3889, 117071, 117071, 3889, 1, 0, 0, 1, 15982, 1535601, 6317926, 1535601, 15982, 1, 0, 0, 1, 64747, 18271947, 228842801, 228842801, 18271947

Offset: 2

N. J. A. Sloane, Nov 12 2006

			Triangle begins:
  1;
  0, 0;
  0, 1,    0;
  0, 1,    1,      0;
  0, 1,   34,      1,      0;
  0, 1,  199,    199,      1,    0;
  0, 1,  916,   7037,    916,    1, 0;
  0, 1, 3889, 117071, 117071, 3889, 1, 0;
  ...
Formatted as an array:
=================================================
k/j | 1 2    3       4         5           6
--- +-------------------------------------------
  1 | 1 0    0       0         0           0 ...
  2 | 0 1    1       1         1           1 ...
  3 | 0 1   34     199       916        3889 ...
  4 | 0 1  199    7037    117071     1535601 ...
  5 | 0 1  916  117071   6317926   228842801 ...
  6 | 0 1 3889 1535601 228842801 21073662977 ...
  ...

R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1977.

Andrew Howroyd, Table of n, a(n) for n = 2..1276 (first 50 rows; first 24 rows from R. W. Robinson)
F. Harary and R. W. Robinson, Labeled bipartite blocks, Canad. J. Math., 31 (1979), 60-68.

Central coefficients are A005334.

Cf. A004100, A123474, A262307.

G(n)={sum(i=0, n, x^i*(sum(j=0, n, y^j*2^(i*j)/(i!*j!)) + O(y*y^n))) + O(x*x^n)}
\\ this switches x/y halfway through because PARI only does serreverse in x.
B(n)={my(p=log(G(n))); p=subst(deriv(p,y), x, serreverse(x*deriv(p,x))); p=substvec(p, [x,y], [y,x]); intformal(log(x/serreverse(x*p)))}
M(n)={my(p=B(n)); matrix(n,n,i,j,polcoef(polcoef(p,j),i)*i!*j!)}
{ my(A=M(6)); for(n=1, #A~, print(A[n,])) } \\ Andrew Howroyd, Jan 04 2021

A004100(n) = (1/2) * Sum_{k=1..n-1} binomial(n,k)*T(n,k). - Andrew Howroyd, Jan 03 2021

Offset corrected by Andrew Howroyd, Jan 04 2021

A123474 Triangle read by rows: T(n,k) = number of labeled bicolored nonseparable graphs with k points in one color class and n-k points in the other class. The classes are interchangeable if k = n-k. Here n >= 2, k=1..n-1.

Original entry on oeis.org

1, 0, 0, 0, 3, 0, 0, 10, 10, 0, 0, 15, 340, 15, 0, 0, 21, 6965, 6965, 21, 0, 0, 28, 51296, 246295, 51296, 28, 0, 0, 36, 326676, 14750946, 14750946, 326676, 36, 0, 0, 45, 1917840, 322476210, 796058676, 322476210, 1917840, 45, 0, 0, 55, 10683255

Offset: 2

N. J. A. Sloane, Nov 12 2006

			Triangle begins:
  1;
  0,  0;
  0,  3,     0;
  0, 10,    10,      0;
  0, 15,   340,     15,     0;
  0, 21,  6965,   6965,    21,  0;
  0, 28, 51296, 246295, 51296, 28, 0;
  ...
Formatted as an array:
==========================================================
m/n | 1  2       3        4            5             6
----+-----------------------------------------------------
  1 | 1  0      0         0            0             0 ...
  2 | 0  3     10        15           21            28 ...
  3 | 0 10    340      6965        51296        326676 ...
  4 | 0 15   6965    246295     14750946     322476210 ...
  5 | 0 21  51296  14750946    796058676  105725374062 ...
  6 | 0 28 326676 322476210 105725374062 9736032295374 ...
  ...

R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1977.

Andrew Howroyd, Table of n, a(n) for n = 2..1276 (first 50 rows; first 24 rows from R. W. Robinson)
F. Harary and R. W. Robinson, Labeled bipartite blocks, Canad. J. Math., 31 (1979), 60-68.

Central coefficients are A005335.

Cf. A004100, A123301, A262307.

From Andrew Howroyd, Jan 03 2021: (Start)
T(n,k) = f(n-2*k) * binomial(n,k) * A123301(n, k) where f(0) = 1/2 and 1 otherwise.
A004100(n) = Sum_{k=0..floor(n/2)} T(n,k). (End)

A005336 Number of labeled nonseparable (or 2-connected) bipartite graphs with 2n nodes.

Original entry on oeis.org

1, 3, 355, 297619, 1120452771, 15350524923547, 738416821509929731, 126430202628042630866787, 78847417416749666369637926851, 183373380693566591129149674727445419, 1623847327688450079238401833083018045926051, 55669578575421273854874611540671620662810228887603

Offset: 1

N. J. A. Sloane

nonn

Nonseparable graphs are also called blocks.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Andrew Howroyd, Table of n, a(n) for n = 1..50
F. Harary and R. W. Robinson, Labeled bipartite blocks, Canad. J. Math., 31 (1979), 60-68.

Bisection of A004100.

Cf. A005334, A005335 (each part with n nodes).

Name clarified and more terms added by Andrew Howroyd, Jan 03 2021

cp's OEIS Frontend

A123301 Triangle read by rows: T(n,k) is the number of specially labeled bicolored nonseparable graphs with k points in one color class and n-k points in the other class. "Special" means there are separate labels 1,2,...,k and 1,2,...,n-k for the two color classes (n >= 2, k = 1,...,n-1).

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A123474 Triangle read by rows: T(n,k) = number of labeled bicolored nonseparable graphs with k points in one color class and n-k points in the other class. The classes are interchangeable if k = n-k. Here n >= 2, k=1..n-1.

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Formula

A005336 Number of labeled nonseparable (or 2-connected) bipartite graphs with 2n nodes.

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Extensions