cp's OEIS Frontend

A290353 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the k-th Euler transform of the sequence with g.f. 1+x.

%I A290353 #25 Oct 04 2018 20:12:33
%S A290353 1,1,1,1,1,0,1,1,1,0,1,1,2,1,0,1,1,3,3,1,0,1,1,4,6,5,1,0,1,1,5,10,14,
%T A290353 7,1,0,1,1,6,15,30,27,11,1,0,1,1,7,21,55,75,58,15,1,0,1,1,8,28,91,170,
%U A290353 206,111,22,1,0,1,1,9,36,140,336,571,518,223,30,1,0
%N A290353 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the k-th Euler transform of the sequence with g.f. 1+x.
%C A290353 A(n,k) is the number of unlabeled rooted trees with exactly n leaves, all in level k.  A(3,3) = 6:
%C A290353 :    o        o       o        o        o       o
%C A290353 :    |        |       |       / \      / \     /|\
%C A290353 :    o        o       o      o   o    o   o   o o o
%C A290353 :    |       / \     /|\     |   |   ( )  |   | | |
%C A290353 :    o      o   o   o o o    o   o   o o  o   o o o
%C A290353 :   /|\    ( )  |   | | |   ( )  |   | |  |   | | |
%C A290353 :  o o o   o o  o   o o o   o o  o   o o  o   o o o
%H A290353 Alois P. Heinz, <a href="/A290353/b290353.txt">Antidiagonals n = 0..140, flattened</a>
%H A290353 B. A. Huberman and T. Hogg, <a href="https://doi.org/10.1016/0167-2789(86)90308-1">Complexity and adaptation</a>, Evolution, games and learning (Los Alamos, N.M., 1985). Phys. D 22 (1986), no. 1-3, 376-384.
%H A290353 <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F A290353 G.f. of column k=0: 1+x, of column k>0: Product_{j>0} 1/(1-x^j)^A(j,k-1).
%e A290353 Square array A(n,k) begins:
%e A290353   1, 1,  1,   1,    1,    1,     1,     1,      1, ...
%e A290353   1, 1,  1,   1,    1,    1,     1,     1,      1, ...
%e A290353   0, 1,  2,   3,    4,    5,     6,     7,      8, ...
%e A290353   0, 1,  3,   6,   10,   15,    21,    28,     36, ...
%e A290353   0, 1,  5,  14,   30,   55,    91,   140,    204, ...
%e A290353   0, 1,  7,  27,   75,  170,   336,   602,   1002, ...
%e A290353   0, 1, 11,  58,  206,  571,  1337,  2772,   5244, ...
%e A290353   0, 1, 15, 111,  518, 1789,  5026, 12166,  26328, ...
%e A290353   0, 1, 22, 223, 1344, 5727, 19193, 54046, 133476, ...
%p A290353 with(numtheory):
%p A290353 A:= proc(n, k) option remember; `if`(n<2, 1, `if`(k=0, 0, add(
%p A290353       add(A(d, k-1)*d, d=divisors(j))*A(n-j, k), j=1..n)/n))
%p A290353     end:
%p A290353 seq(seq(A(n, d-n), n=0..d), d=0..14);
%t A290353 A[n_, k_]:=b[n, k]=If[n<2, 1, If[k==0, 0, Sum[Sum[A[d, k - 1]*d, {d, Divisors[j]}] A[n - j, k], {j, n}]/n]]; Table[A[n, d - n], {d, 0, 14}, {n, 0, d}]//Flatten (* _Indranil Ghosh_, Jul 30 2017, after Maple code *)
%Y A290353 Columns k=1-10 give: A000012, A000041, A001970, A007713, A007714, A290355, A290356, A290357, A290358, A290359.
%Y A290353 Rows 0+1,2-10 give: A000012, A001477, A000217, A000330, A007715, A290360, A290361, A290362, A290363, A290364.
%Y A290353 Main diagonal gives A290354.
%Y A290353 Cf. A144150.
%K A290353 nonn,tabl
%O A290353 0,13
%A A290353 _Alois P. Heinz_, Jul 28 2017