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

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

Offset: 1

Author

Gary W. Adamson, Jul 03 2009

Keywords

• nonn
• tabl

Comments

Row sums = A000219, number of planar partitions of n starting with offset 1.

Examples

			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" 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

Cf. A000219.

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.