1,3

Row sums = A000219, number of planar partitions of n starting with offset 1. /Q (1, 3, 6, 13, 24, 48,...).

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.

First few rows of the array =

1,...1,...1,...1,...1,...1,...; = a

1,...1,...3,...3,...6,...6,...; = a*b

1,...1,...3,...6,...9,..15,...; = a*b*c

1,...1,...3,...6,..13,..19,...; = a*b*c*d

1,...1,...3,...6,..13,..24,...; = a*b*c*d*e

...

...then taking finite differences from the top and discarding the first "1" /Q we obtain:

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, 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, 20, 111, 196, 200, 144, 91, 48, 27, 10, 11;

1, 27, 150, 314, 335, 273, 168, 104, 54, 30, 11, 12;

...

A000219

Sequence in context: A101391 A117704 A078032 this_sequence A008313 A111377 A014046

Adjacent sequences: A162450 A162451 A162452 this_sequence A162454 A162455 A162456

nonn,tabl

Gary W. Adamson (qntmpkt(AT)yahoo.com), Jul 03 2009