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A162453 Plane partition triangle, row sums = A000219; derived from the Euler transform of [1, 2, 3,...]. 1

%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

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