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A269800 Convolution of A000107 and A027852. 3
0, 0, 1, 3, 10, 30, 91, 268, 790, 2308, 6737, 19609, 57044, 165796, 481823, 1400028, 4068577, 11825459, 34380152, 99981942, 290854486, 846397344, 2463892294, 7174933683, 20900764811, 60904875999, 177535250815, 517673673674, 1509950058629, 4405547856394, 12857716906991 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

This counts the arrangements of n nested circles in the plane where one pair of circles touches. a(2)=1 because the (only) pair must touch. a(3)=3 because either the third circle circumscribes the touching pair or is inside one of the touching circles or is entirely separated from the touching pair.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..500

R. J. Mathar, Topologically distinct sets of non-intersecting circles in the plane, arXiv:1603.00077 [math.CO](2016), row sums Table 8.

MATHEMATICA

b[0] = 0; b[1] = 1; b[n_] := b[n] =Sum[Sum[d b[d], {d, Divisors[j]}] b[n - j], {j, 1, n - 1}]/(n - 1);

a7[n_] := a7[n] = b[n] + Sum[ a7[n - i] b[i], {i, 1, n - 1}];

c[n_] := c[n] = If[n <= 1, n, (Sum[Sum[d c[d], {d, Divisors[j]}] c[n - j], {j, 1, n - 1}])/(n - 1)];

a52[n_] := (Sum[c[i] c[n-i], {i, 0, n}] + If[Mod[n, 2] == 0, c[n/2], 0])/2;

a[n_] := Sum[a7[k] a52[n - k + 1], {k, 0, n + 1}];

Table[a[n], {n, 0, 30}] (* Jean-Fran├žois Alcover, Dec 16 2018, after Alois P. Heinz in A000107 and A027852 *)

CROSSREFS

Cf. A000107, A027852.

Sequence in context: A026327 A014531 A062107 * A033113 A290718 A300421

Adjacent sequences:  A269797 A269798 A269799 * A269801 A269802 A269803

KEYWORD

nonn

AUTHOR

R. J. Mathar, Mar 05 2016

STATUS

approved

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Last modified August 15 12:33 EDT 2020. Contains 336499 sequences. (Running on oeis4.)