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A123301
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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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3
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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
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OFFSET
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2,13
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REFERENCES
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R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1977.
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LINKS
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FORMULA
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EXAMPLE
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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 ...
...
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PROG
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(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
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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