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

2,13

References

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

Links

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.

Formula

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

Example

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 ...
  ...

Prog

(PARI)
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

Crossrefs

Central coefficients are A005334.

Cf. A004100, A123474, A262307.

Sequence in context: A228426 A023928 A022070 * A328913 A269482 A037933

Adjacent sequences: A123298 A123299 A123300 * A123302 A123303 A123304

Keyword

nonn,tabl

Author

N. J. A. Sloane, Nov 12 2006

Extensions

Offset corrected by Andrew Howroyd, Jan 04 2021

Status

approved