

A162453


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


1



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
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OFFSET

1,3


COMMENTS

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


LINKS

Table of n, a(n) for n=1..76.


FORMULA

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.


EXAMPLE

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;
...


CROSSREFS

A000219
Sequence in context: A327632 A117704 A078032 * A008313 A334550 A232177
Adjacent sequences: A162450 A162451 A162452 * A162454 A162455 A162456


KEYWORD

nonn,tabl


AUTHOR

Gary W. Adamson, Jul 03 2009


STATUS

approved



