A156039 Number of compositions (ordered partitions) of n into 4 parts, where the first is at least as great as each of the others.

1, 1, 4, 7, 11, 17, 26, 35, 48, 63, 81, 102, 127, 154, 187, 223, 263, 308, 359, 413, 474, 540, 612, 690, 775, 865, 964, 1069, 1181, 1301, 1430, 1565, 1710, 1863, 2025, 2196, 2377, 2566, 2767, 2977, 3197, 3428, 3671, 3923, 4188, 4464, 4752, 5052, 5365, 5689

Offset

0,3

Comments

For n=1,2 these are just the tetrahedral numbers. a(n) is always at least 1/4 of the corresponding tetrahedral number, since each partition of this type gives up to four ordered partitions with the same cyclical order.

Diagonal sums of the irregular triangle A109439, for example a(0)=1, a(1)=1, a(2)=1+3, a(3)=1+3+3, a(4)=1+3+6+1. - Bob Selcoe, Feb 09 2014

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (2, -1, 1, -1, -1, 1, -1, 2, -1).

Formula

G.f.: ( 1-x+3*x^2-x^3+x^4 ) / ( (1+x)*(1+x^2)*(1+x+x^2)*(x-1)^4 ). - Alois P. Heinz, Jun 14 2009

Example

For n = 3 the a(3) = 7 compositions are: (3 0 0 0) (2 1 0 0) (2 0 1 0) (2 0 0 1) (1 1 1 0) (1 1 0 1) (1 0 1 1).

Maple

a:= proc(n) local m, r; m:= iquo(n, 12, 'r'); r:= r+1; (9 +(27 +72*m +18*r)*m +((9 +3*r) *r-12) /2)*m +[1, 1, 4, 7, 11, 17, 26, 35, 48, 63, 81, 102][r] end: seq(a(n), n=0..60); # Alois P. Heinz, Jun 14 2009

Mathematica

LinearRecurrence[{2, -1, 1, -1, -1, 1, -1, 2, -1}, {1, 1, 4, 7, 11, 17, 26, 35, 48}, 60] (* Jean-François Alcover, May 17 2018 *)

Crossrefs

For partitions into 3 summands see A156040; also see A156041 and A156042.

Sequence in context: A049648 A211647 A133257 * A350477 A310767 A207871

Adjacent sequences: A156036 A156037 A156038 * A156040 A156041 A156042

Keyword

nonn

Author

Jack W Grahl, Feb 02 2009, Feb 11 2009

Extensions

More terms from Alois P. Heinz, Jun 14 2009

Status

approved