A174066 Irregular triangle, row sums = A000041, left border = A174065.

1, 1, 1, 1, 2, 1, 3, 1, 1, 4, 2, 1, 5, 3, 1, 2, 7, 4, 2, 2, 9, 5, 3, 2, 3, 12, 7, 4, 4, 3, 15, 9, 5, 6, 3, 4, 19, 12, 7, 8, 6, 4, 25, 15, 9, 10, 9, 4, 5, 31, 19, 12, 14, 12, 8, 5, 38, 25, 15, 18, 15, 12, 5, 7, 48, 31, 19, 24, 21, 16, 10, 7, 60, 38, 25, 30, 27, 20, 15, 7, 9, 73, 48, 31, 38, 36

Offset: 1

Author

Gary W. Adamson, Mar 06 2010

Keywords

• nonn
• tabf

Comments

Left border = A174065: (1, 1, 1, 2, 3, 4, 5, 7, 9, 12,...) * its aerated variant (1, 0, 1, 0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, 7,...) = A000041, the partition sequence: (1, 1, 2, 3, 5, 7, 11, 15, 22, 30,...).

Examples

			Heading and first few rows of the triangle =
.1,...1,...1,...2,...3,...4,...5,...7,...9,... = A174065.
.1;........................................... = .. 1 (A000041)
.1;........................................... = .. 1
.1,...1;...................................... = .. 2
.2,...1;...................................... = .. 3
.3,...1,...1;................................. = .. 5
.4,...2,...1;................................. = .. 7
.5,...3,...1,...2;............................ = ..11
.7,...4,...2,...2;............................ = ..15
.9....5,...3,...2,...3;....................... = ..22
.12,..7,...4,...4,...3;....................... = ..30
.15,..9,...5,...6,...3;...4;.................. = ..42
.19,.12,...7,...8,...6,...4;.................. = ..56
.25,.15,...9,..10,...9,...4,...5;............. = ..77
.31,.19,..12,..14,..12,...8,...5;............. = .101
.38,.25,..15,..18,..15,..12,...5,...7;........ = .135
.48,.31,..19,..24,..21,..16,..10,...7;........ = .176
.60,.38,..25,..30,..27,..20,..15,...7,...9;... = .231
.73,.48,..31,..38,..36,..28,..20,..14,...9;... = .297
...
Example: leftmost term in 8th row has to be 7 = (15 - (4 + 2 + 2)); so we
place a 7 as next term in the heading, then multiply * leftmost column.
Finally, shift the columns down twice.
		

Crossrefs

Cf. A000041, A174065, A174067.

Formula

The triangle is the result of three rules after beginning (1, 1, 1, 1,...):

Columns >1 are shifted down twice from previous columns; column terms = left border * (left border placed as a heading row); and row sums = A000041, the partition numbers. The rules force the next missing term in the triangle to be the leftmost term in column 1. This is found by taking p(n) for row n, then subtracting the sum of row n terms (minus leftmost term).