A162934 Shift sequence A162932 twice then subtract from the original sequence.

1, 0, 0, 1, 0, 0, 2, 0, 0, 3, 1, 0, 4, 2, 2, 5, 3, 4, 9, 5, 6, 13, 11, 10, 19, 17, 19, 28, 27, 31, 44, 41, 49, 66, 68, 74, 98, 104, 118, 145, 157, 178, 220, 234, 268, 322, 354, 397, 473, 521, 591, 686, 765, 863, 1003, 1107, 1254, 1444, 1609

Offset

6,7

Comments

From Alford Arnold, Dec 17 2009: (Start)

At n = 24, six of the partitions can be associated with the sixth row of this triangular array:

333

444 3333

555 4443 33333

666 5553 44433 333333

777 6663 55533 444333 3333333

888 7773 66633 555333 4443333 33333333

The other three partitions are new; and hence on their first row, so 6*1 + 1*3 = 9.

In a similar manner, the 44 cases at n = 36 can be computed using the array row numbers and the number of applicable partitions. Thus we have:

(10, 5, 3, 2, 1) times (1, 3, 2, 3, 7) providing 10 + 15 + 6 + 6 + 7 = 44 cases. (End)

Links

Table of n, a(n) for n=6..64.

Formula

G.f.: Sum_{n >= 0} q^(3*n+6)/Product_{k = 1..n} 1 - q^(k+2). - Peter Bala, Dec 01 2024

Example

For n = 24, the sequence counts these nine partitions of 24: 888, 7773, 66633, 55554, 555333, 4443333, 6666, 444444, 33333333.

Crossrefs

Cf. A000041, A002865, A053445, A162932.

Sequence in context: A309577 A029301 A263414 * A303908 A351592 A082660

Adjacent sequences: A162931 A162932 A162933 * A162935 A162936 A162937

Keyword

nonn,easy

Author

Alford Arnold, Aug 05 2009, Aug 06 2009

Extensions

More terms from Alford Arnold, Dec 17 2009

More terms from Joerg Arndt, Jul 16 2015

Status

approved