OFFSET
1,1
COMMENTS
See eq. (27) of the reference for a recurrence.
LINKS
Alois P. Heinz, Rows n = 1..141, flattened
R. J. Mathar, Topologically distinct sets of non-intersecting circles in the plane, arXiv:1603:00077 [math.CO] (2016), Table 4.
EXAMPLE
3 ;
9 6 ;
45 27 10;
246 180 54 15;
1485 1143 405 90 21;
9432 7704 2856 720 135 28;
62625 52731 20682 5385 1125 189 36;
428319 369969 150282 40914 8730 1620 252 45;
3000393 2638332 1104702 309510 68400 12891 2205 324 55;
21410436 19097802 8183943 2353989 531702 103140 17868 2880 405 66;
155106693 139921470 61122222 17954262 4140105 816858 145134 23661 3645 495 78;
1137703869 1035882315 459695791 137490273 32241834 6466053 1164978 194382 30270 4500 594 91 ;
8432624850 7737370857 3479520051 1056731244 251493255 51104574 9331833 1576062 250884 37695 5445 702 105 ;
MAPLE
g:= proc(n) option remember; `if`(n<2, 3*n, (add(add(d*g(d),
d=numtheory[divisors](j))*g(n-j), j=1..n-1))/(n-1))
end:
b:= proc(n, i, p) option remember; `if`(p>n, 0, `if`(n=0, 1,
`if`(min(i, p)<1, 0, add(b(n-i*j, i-1, p-j)*
binomial(g(i)+j-1, j), j=0..min(n/i, p)))))
end:
T:= (n, k)-> b(n$2, k):
seq(seq(T(n, k), k=1..n), n=1..14); # Alois P. Heinz, Apr 13 2017
MATHEMATICA
g[n_] := g[n] = If[n < 2, 3*n, (Sum[Sum[d*g[d], {d, Divisors[j]}]*g[n - j], {j, 1, n - 1}])/(n - 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[g[i] + j - 1, j], {j, 0, Min[n/i, p]}]]]];
T[n_, k_] := b[n, n, k];
Table[Table[T[n, k], {k, 1, n}], {n, 1, 14}] // Flatten (* Jean-François Alcover, Nov 10 2017, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
R. J. Mathar, Apr 16 2016
STATUS
approved