A005335
Number of labeled nonseparable (or 2-connected) bipartite graphs with 2n nodes and n nodes in each part.
Original entry on oeis.org
1, 3, 340, 246295, 796058676, 9736032295374, 432386386904461704, 70004505120317453723895, 41988978212639552393332333300, 95055430627597798399511262461524570, 826275345303020411581696428212189429357784, 27965998400207183955394390590886658323558240477654
Offset: 1
- 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.
- F. Harary and R. W. Robinson, Labeled bipartite blocks, Canad. J. Math., 31 (1979), 60-68. (Annotated scanned copy)
Main diagonal of
A123474 as an array.
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).
Original entry on oeis.org
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
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.
-
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
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 and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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