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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.
3

%I #26 Apr 19 2018 16:24:56

%S 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 74,86,59,26,7,1,1,45,139,194,164,91,34,8,1,1,67,249,415,416,281,132,

%U 43,9,1,1,97,434,844,990,787,447,183,53,10,1

%N 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 It appears that row 2 is A034856.

%C Observation:

%C Column 1 is the EULER transform of 2,1,1,1,1,1,1,1...

%C Column 2 is the EULER transform of 3,2,2,2,2,2,2,2...

%H Alois P. Heinz, <a href="/A210764/b210764.txt">Rows n = 0..140, flattened</a>

%e Array begins:

%e 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,

%e 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,

%e 1, 4, 8, 13, 19, 26, 34, 43, 53,

%e 1, 7, 18, 35, 59, 91, 132, 183,

%e 1, 12, 38, 86, 164, 281, 447,

%e 1, 19, 74, 194, 416, 787,

%e 1, 30, 139, 415, 990,

%e 1, 45, 249, 844,

%e 1, 67, 434,

%e 1, 97,

%e 1,

%p with(numtheory):

%p etr:= proc(p) local b;

%p b:= proc(n) option remember; `if`(n=0, 1,

%p add(add(d*p(d), d=divisors(j))*b(n-j), j=1..n)/n)

%p end

%p end:

%p A:= (n, k)-> etr(j-> k +`if`(j=1, 1, 0))(n):

%p seq(seq(A(d-k, k), k=0..d), d=0..14); # _Alois P. Heinz_, May 20 2013

%t 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 Columns (0-3): A000012, A000070, A000713, A210843.

%Y Rows (0-1): A000012, A000027.

%Y Main diagonal gives A303070.

%Y Cf. A000007, A000041, A005758, A006922, A000712, A000716, A023003-A023021, A144064, A195825, A211970.

%K nonn,tabl

%O 0,5

%A _Omar E. Pol_, Jun 27 2012