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A280787
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Triangle read by rows: number of topologically distinct sets of n circles with one pair intersecting, by number of factors.
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3
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1, 3, 1, 10, 4, 1, 30, 15, 4, 1, 91, 50, 16, 4, 1, 268, 162, 55, 16, 4, 1, 790, 506, 185, 56, 16, 4, 1, 2308, 1558, 594, 190, 56, 16, 4, 1, 6737, 4727, 1878, 617, 191, 56, 16, 4, 1, 19609, 14227, 5825, 1970, 622, 191, 56, 16, 4, 1
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OFFSET
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2,2
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LINKS
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Table of n, a(n) for n=2..56.
R. J. Mathar, Topologically Distinct Sets of Non-intersecting Circles in the Plane, arXiv:1603.00077 [math.CO], 2016.
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EXAMPLE
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Triangle begins:
1;
3, 1;
10, 4, 1;
30, 15, 4, 1;
91, 50, 16, 4, 1;
268, 162, 55, 16, 4, 1;
790, 506, 185, 56, 16, 4, 1;
2308, 1558, 594, 190, 56, 16, 4, 1;
...
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MATHEMATICA
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a81[n_] := a81[n] = If[n <= 1, n, Sum[a81[n - j]*DivisorSum[j, #1*a81[#1] &], {j, n - 1}]/(n - 1)];
A027852[n_] := Module[{dh = 0, np}, For[np = 0, np <= n, np++, dh = a81[np]*a81[n - np] + dh]; If[EvenQ[n], dh = a81[n/2] + dh]; dh/2];
A280788[n_] := If[n == 0, 1, Sum[a81[np+1]*A027852[n-np+2], {np, 0, n}]];
t[n_] := t[n] = Module[{d, j}, If[n == 1, 1, Sum[Sum[d*t[d], {d, Divisors[j]}]*t[n - j], {j, 1, n - 1}]/(n - 1)]];
b[1, 1, 1] = 1;
b[n_, i_, p_] := b[n, i, p] = If[p > n, 0, If[n == 0, 1, If[Min[i, p] < 1, 0, Sum[b[n - i*j, i - 1, p - j]*Binomial[t[i] + j - 1, j], {j, 0, Min[n/i, p]}]]]]; A033185[n_, k_] := b[n, n, k];
A280786[n_] := If[n < 2, 0, Sum[A280787[n, f], {f, 1, n - 1}]];
A280787[n_, f_] := A280787[n, f] = Module[{ct}, Which[f == n, Return[0], f == n - 1, Return[1], f == 1, Return[A280786[n - 1] + A280788[n - 2]], True, ct = 0; Do[ct += A280787[np, 1]*A033185[n - np, f - 1], {np, 1, n - 1}]]; ct];
Table[A280787[n, f], {n, 2, 11}, {f, 1, n - 1}] // Flatten (* Jean-François Alcover, Nov 23 2017, after R. J. Mathar and Alois P. Heinz *)
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CROSSREFS
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Row sums give A280786.
Sequence in context: A134285 A141811 A291538 * A126954 A176992 A319375
Adjacent sequences: A280784 A280785 A280786 * A280788 A280789 A280790
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KEYWORD
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nonn,tabl
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AUTHOR
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N. J. A. Sloane, Jan 20 2017
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STATUS
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approved
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