%I
%S 1,1,2,1,2,3,1,5,3,4,1,5,9,4,5,1,9,15,12,5,6,1,9,24,24,15,6,7,1,14,36,
%T 46,30,18,7,8,1,14,58,70,65,36,21,8,9,1,20,76,130,110,78,42,24,9,10,1,
%U 20,111,196,200,144,91,48,27,10,11,1,27,150,314,335,273,168,104,54,30
%N Plane partition triangle, row sums = A000219; derived from the Euler transform of [1, 2, 3,...].
%C Row sums = A000219, number of planar partitions of n starting with offset 1. /Q (1, 3, 6, 13, 24, 48,...).
%F Construct an array with rows = a, a*b, a*b*c,...; where a = [1, 1, 1,...], b = [1, 0, 2, 0, 3,...], c = [1, 0, 0, 3, 0, 0, 6,...], d = [1, 0, 0, 0, 4, 0, 0, 0, 10, 0, 0, 0, 20,...]...;etc, where rows converge to A000219: (1, 1, 3, 6, 13, 24,...). The triangle = finite differences of column terms starting from the top.
%e First few rows of the array =
%e 1,...1,...1,...1,...1,...1,...; = a
%e 1,...1,...3,...3,...6,...6,...; = a*b
%e 1,...1,...3,...6,...9,..15,...; = a*b*c
%e 1,...1,...3,...6,..13,..19,...; = a*b*c*d
%e 1,...1,...3,...6,..13,..24,...; = a*b*c*d*e
%e ...
%e ...then taking finite differences from the top and discarding the first "1" /Q we obtain:
%e 1;
%e 1, 2;
%e 1, 2, 3;
%e 1, 5, 3, 4;
%e 1, 5, 9, 4, 5;
%e 1, 9, 15, 12, 5, 6;
%e 1, 9, 24, 24, 15, 6, 7;
%e 1, 14, 36, 46, 30, 18, 7, 8;
%e 1, 14, 58, 70, 65, 36, 21, 8, 9;
%e 1, 20, 76, 130, 110, 78, 42, 24, 9, 10;
%e 1, 20, 111, 196, 200, 144, 91, 48, 27, 10, 11;
%e 1, 27, 150, 314, 335, 273, 168, 104, 54, 30, 11, 12;
%e ...
%Y A000219
%K nonn,tabl
%O 1,3
%A _Gary W. Adamson_, Jul 03 2009
