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A210764 Square array T(n,k), n>=0, k>=0, read by antidiagonals in which column k gives the partial sums of column k of A144064.

%I A210764 #26 Apr 19 2018 16:24:56
%S A210764 1,1,1,1,2,1,1,4,3,1,1,7,8,4,1,1,12,18,13,5,1,1,19,38,35,19,6,1,1,30,
%T A210764 74,86,59,26,7,1,1,45,139,194,164,91,34,8,1,1,67,249,415,416,281,132,
%U A210764 43,9,1,1,97,434,844,990,787,447,183,53,10,1
%N A210764 Square array T(n,k), n>=0, k>=0, read by antidiagonals in which column k gives the partial sums of column k of A144064.
%C A210764 It appears that row 2 is A034856.
%C A210764 Observation:
%C A210764 Column 1 is the EULER transform of 2,1,1,1,1,1,1,1...
%C A210764 Column 2 is the EULER transform of 3,2,2,2,2,2,2,2...
%H A210764 Alois P. Heinz, <a href="/A210764/b210764.txt">Rows n = 0..140, flattened</a>
%e A210764 Array begins:
%e A210764 1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,
%e A210764 1,   2,   3,   4,   5,   6,   7,   8,   9,  10,
%e A210764 1,   4,   8,  13,  19,  26,  34,  43,  53,
%e A210764 1,   7,  18,  35,  59,  91, 132, 183,
%e A210764 1,  12,  38,  86, 164, 281, 447,
%e A210764 1,  19,  74, 194, 416, 787,
%e A210764 1,  30, 139, 415, 990,
%e A210764 1,  45, 249, 844,
%e A210764 1,  67, 434,
%e A210764 1,  97,
%e A210764 1,
%p A210764 with(numtheory):
%p A210764 etr:= proc(p) local b;
%p A210764         b:= proc(n) option remember; `if`(n=0, 1,
%p A210764               add(add(d*p(d), d=divisors(j))*b(n-j), j=1..n)/n)
%p A210764             end
%p A210764       end:
%p A210764 A:= (n, k)-> etr(j-> k +`if`(j=1, 1, 0))(n):
%p A210764 seq(seq(A(d-k, k), k=0..d), d=0..14); # _Alois P. Heinz_, May 20 2013
%t A210764 etr[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d*p[d], {d, Divisors[ j]}]*b[n-j], {j, 1, n}]/n]; b]; A[n_, k_] := etr[Function[{j}, k + If[j == 1, 1, 0]]][n]; Table[Table[A[d-k, k], {k, 0, d}], {d, 0, 14}] // Flatten (* _Jean-François Alcover_, Mar 05 2015, after _Alois P. Heinz_ *)
%Y A210764 Columns (0-3): A000012, A000070, A000713, A210843.
%Y A210764 Rows (0-1): A000012, A000027.
%Y A210764 Main diagonal gives A303070.
%Y A210764 Cf. A000007, A000041, A005758, A006922, A000712, A000716, A023003-A023021, A144064, A195825, A211970.
%K A210764 nonn,tabl
%O A210764 0,5
%A A210764 _Omar E. Pol_, Jun 27 2012