A206438 Triangle read by rows which lists the squares of the parts of A135010.

1, 1, 4, 1, 1, 9, 1, 1, 1, 4, 4, 16, 1, 1, 1, 1, 1, 4, 9, 25, 1, 1, 1, 1, 1, 1, 1, 4, 4, 4, 4, 16, 9, 9, 36, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 4, 9, 4, 25, 9, 16, 49, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 4, 4, 4, 4, 4, 16, 4, 9, 9, 4, 36, 9, 25

Offset: 1

Author

Omar E. Pol, Feb 08 2012

Keywords

• nonn
• tabf

Comments

Volumes of the parts in the section model of partitions version "boxes" in which each part of size k has a volume = k^2. Row sums of this triangle give A206440 and partial sums of A206440 give A066183.

Examples

			Written as a triangle:
1;
1,4;
1,1,9;
1,1,1,4,4,16;
1,1,1,1,1,4,9,25;
1,1,1,1,1,1,1,4,4,4,4,16,9,9,36;
1,1,1,1,1,1,1,1,1,1,1,4,4,9,4,25,9,16,49;
		

Links

• Alois P. Heinz, Rows n = 1..23, flattened

Crossrefs

Row n has length A138137(n).

Row sums give A206440.

Right border gives positives A000290.

Cf. A066183, A135010, A138121, A141285.

Programs

Mathematica

Table[Reverse@ConstantArray[{1}, PartitionsP[n - 1]] ~Join~ DeleteCases[Sort@PadRight[Reverse/@Cases[IntegerPartitions[n], x_ /; Last[x] != 1]], x_ /; x == 0, 2], {n, 1, 8}] ^2  // Flatten (* Robert Price, May 28 2020 *)

Formula

a(n) = A135010(n)^2.