cp's OEIS Frontend

A174067 Triangle, row sums = A000041 starting (1, 2, 3, 5, 7, ...); derived from finite differences of p(x) = A(x)*A(x^2) = B(x)*B(x^3) = C(x)*C(x^4) = ...

%I A174067 #8 Feb 08 2022 22:44:16
%S A174067 1,1,1,2,0,1,3,1,0,1,4,1,1,0,1,5,2,2,1,0,1,7,2,3,1,1,0,1,9,4,3,3,1,1,
%T A174067 0,1,12,5,5,3,3,0,1,0,1,15,8,6,5,3,2,1,1,0,1,19,10,9,6,5,2,2,1,1,0,1,
%U A174067 25,13,12,10,5,5,2,2,1,1,0,1,31,17,16,12,9,5,4,2,2,1,1,0,1
%N A174067 Triangle, row sums = A000041 starting (1, 2, 3, 5, 7, ...); derived from finite differences of p(x) = A(x)*A(x^2) = B(x)*B(x^3) = C(x)*C(x^4) = ...
%C A174067 Row sums = A000041 starting with offset 1: (1, 2, 3, 5, 7, 11, ...).
%F A174067 Given an array of rows satisfying p(x) = A(x)*A(x^2) = row 1 = A174065; row = 2 A174068 satisfying p(x) = B(x)*B(x^3); row 3 satisfies p(x) = C(x)*C(x^4), ... and so on; take finite differences from the top, becoming rows of triangle A174067.
%e A174067 First few rows of the array:
%e A174067   1, 1, 1, 2, 3, 4,  5,  7,  9, 12, 15, 19, ... = A174065
%e A174067   1, 1, 2, 2, 4, 5,  7,  9, 13, 17, 23, 29, ... = A174068
%e A174067   1, 1, 2, 3, 4, 6,  9, 12, 16, 22, 29, 38, ... satisfies p(x) = C(x)*C(x^4)
%e A174067   1, 1, 2, 3, 5, 6, 10, 13, 19, 25, 34, 44, ... analogous for k=5
%e A174067   1, 1, 2, 3, 5, 7, 10, 14, 20, 28, 37, 49, ..................k=6
%e A174067   1, 1, 2, 3, 5, 7, 11, 14, 21, 28, 39, 51, ..................k=7
%e A174067   1, 1, 2, 3, 5, 7, 11, 15, 21, 29, 40, 53, ..................k=8
%e A174067   1, 1, 2, 3, 5, 7, 11, 15, 22, 29, 41, 54, ..................k=9
%e A174067   1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 41, 55, ..................k=10
%e A174067   ...
%e A174067 Finally, take finite differences from the top, deleting the first 1, to obtain triangle A174067:
%e A174067    1;
%e A174067    1,  1;
%e A174067    2,  0,  1;
%e A174067    3,  1,  0,  1;
%e A174067    4,  1,  1,  0,  1;
%e A174067    5,  2,  2,  1,  0,  1;
%e A174067    7,  2,  3,  1,  1,  0,  1;
%e A174067    9,  4,  3,  3,  1,  1,  0,  1;
%e A174067   12,  5,  5,  3,  3,  0,  1,  0,  1;
%e A174067   15,  8,  6,  5,  3,  2,  1,  1,  0,  1;
%e A174067   19, 10,  9,  6,  5,  2,  2,  1,  1,  0,  1;
%e A174067   25, 13, 12, 10,  5,  5,  2,  2,  1,  1,  0,  1;
%e A174067   31, 17, 16, 12,  9,  5,  4,  2,  2,  1,  1,  0,  1;
%e A174067   38, 24, 20, 18, 11,  8,  5,  4,  2,  2,  1,  1,  0,  1;
%e A174067   ...
%Y A174067 Cf. A000041, A174065, A174066, A174068.
%K A174067 nonn,tabl
%O A174067 1,4
%A A174067 _Gary W. Adamson_, Mar 06 2010