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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).

%I A123301 #24 May 22 2021 21:01:52
%S A123301 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,
%T A123301 0,1,3889,117071,117071,3889,1,0,0,1,15982,1535601,6317926,1535601,
%U A123301 15982,1,0,0,1,64747,18271947,228842801,228842801,18271947
%N 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).
%D A123301 R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1977.
%H A123301 Andrew Howroyd, <a href="/A123301/b123301.txt">Table of n, a(n) for n = 2..1276</a> (first 50 rows; first 24 rows from R. W. Robinson)
%H A123301 F. Harary and R. W. Robinson, <a href="http://dx.doi.org/10.4153/CJM-1979-007-3">Labeled bipartite blocks</a>, Canad. J. Math., 31 (1979), 60-68.
%F A123301 A004100(n) = (1/2) * Sum_{k=1..n-1} binomial(n,k)*T(n,k). - _Andrew Howroyd_, Jan 03 2021
%e A123301 Triangle begins:
%e A123301   1;
%e A123301   0, 0;
%e A123301   0, 1,    0;
%e A123301   0, 1,    1,      0;
%e A123301   0, 1,   34,      1,      0;
%e A123301   0, 1,  199,    199,      1,    0;
%e A123301   0, 1,  916,   7037,    916,    1, 0;
%e A123301   0, 1, 3889, 117071, 117071, 3889, 1, 0;
%e A123301   ...
%e A123301 Formatted as an array:
%e A123301 =================================================
%e A123301 k/j | 1 2    3       4         5           6
%e A123301 --- +-------------------------------------------
%e A123301   1 | 1 0    0       0         0           0 ...
%e A123301   2 | 0 1    1       1         1           1 ...
%e A123301   3 | 0 1   34     199       916        3889 ...
%e A123301   4 | 0 1  199    7037    117071     1535601 ...
%e A123301   5 | 0 1  916  117071   6317926   228842801 ...
%e A123301   6 | 0 1 3889 1535601 228842801 21073662977 ...
%e A123301   ...
%o A123301 (PARI)
%o A123301 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)}
%o A123301 \\ this switches x/y halfway through because PARI only does serreverse in x.
%o A123301 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)))}
%o A123301 M(n)={my(p=B(n)); matrix(n,n,i,j,polcoef(polcoef(p,j),i)*i!*j!)}
%o A123301 { my(A=M(6)); for(n=1, #A~, print(A[n,])) } \\ _Andrew Howroyd_, Jan 04 2021
%Y A123301 Central coefficients are A005334.
%Y A123301 Cf. A004100, A123474, A262307.
%K A123301 nonn,tabl
%O A123301 2,13
%A A123301 _N. J. A. Sloane_, Nov 12 2006
%E A123301 Offset corrected by _Andrew Howroyd_, Jan 04 2021