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A162507 Triangle read by rows, finite differences of an array generated by an infinite product (Cf. A162506).

%I A162507 #5 Sep 28 2016 06:38:50
%S A162507 1,2,1,2,3,1,4,3,4,1,4,9,4,5,1,6,12,12,5,6,1,6,18,24,15,6,7,1,8,24,36,
%T A162507 30,18,7,8,1,8,33,60,60,36,21,8,9,1,10,39,88,95,72,42,24,9,10,1,10,51,
%U A162507 124,160,138,84,48,27,10,11
%N A162507 Triangle read by rows, finite differences of an array generated by an infinite product (Cf. A162506).
%C A162507 Row sums = A162506 starting (1, 3, 6, 12, 23, 42, 77,...).
%F A162507 Given the infinite product and array shown in A162506, we have
%F A162507 a = [1,1,1,...], b = [1,0,2,0,2,0,2,...], c = [1,0,0,3,0,0,3,0,0,3,...];
%F A162507 The array is a, a*b, a*b*c,... Finite differences of array columns = rows of the triangle, deleting the first "1".
%e A162507 The array =
%e A162507 1,...1,...1,...1,...1,...; = a
%e A162507 1,...1,...3,...3,...5,...; = a*b
%e A162507 1,...1,...3,...6,...8,...; = a*b*c
%e A162507 1,...1,...3,...6,..12,...; = a*b*c*d
%e A162507 ...
%e A162507 taking finite differences from the top, then discarding the first "1",
%e A162507 we obtain triangle A162507:
%e A162507 1;
%e A162507 1, 2;
%e A162507 1, 2, 3;
%e A162507 1, 4, 3, 4;
%e A162507 1, 4, 9, 4, 5;
%e A162507 1, 6, 12, 12, 5, 6;
%e A162507 1, 6, 18, 24, 15, 6, 7;
%e A162507 1, 8, 24, 36, 30, 18, 7, 8;
%e A162507 1, 8, 33, 60, 60, 36, 21, 8, 9;
%e A162507 1, 10, 39, 88, 95, 72, 42, 24, 9, 10;
%e A162507 1, 10, 51, 124, 160, 138, 84, 48, 27, 10, 11;
%e A162507 ...
%Y A162507 Cf. A162506.
%K A162507 nonn,tabl
%O A162507 2,2
%A A162507 _Gary W. Adamson_, Jul 04 2009