A293136 Irregular triangle T(n,k) read by rows: T(n,k) is the number of strongly unimodal compositions of n (A059618) into k parts.

1, 0, 1, 0, 1, 0, 1, 2, 0, 1, 2, 1, 0, 1, 4, 1, 0, 1, 4, 5, 0, 1, 6, 6, 2, 0, 1, 6, 10, 4, 0, 1, 8, 14, 6, 1, 0, 1, 8, 19, 14, 1, 0, 1, 10, 23, 20, 5, 0, 1, 10, 31, 30, 10, 0, 1, 12, 36, 42, 18, 2, 0, 1, 12, 44, 60, 27, 4, 0, 1, 14, 52, 76, 48, 8, 0, 1, 14, 61, 102, 68, 16, 1, 0, 1, 16, 69, 126, 101, 30, 1, 0, 1, 16, 81, 160, 138, 50, 5, 0

Offset

0,8

Comments

Conjecture: index k of last nonzero entry in row n of is A293137(n).

Links

Joerg Arndt, Table of n, a(n) for n = 0..1793 (rows 0...125)

Formula

G.f.: 1 + Sum_{n>=1} t*x^n * ( Product_{k=1..n-1} 1 + t*x^k )^2.

Example

Triangle starts:
00:  [1]
01:  [0, 1]
02:  [0, 1]
03:  [0, 1, 2]
04:  [0, 1, 2, 1]
05:  [0, 1, 4, 1]
06:  [0, 1, 4, 5]
07:  [0, 1, 6, 6, 2]
08:  [0, 1, 6, 10, 4]
09:  [0, 1, 8, 14, 6, 1]
10:  [0, 1, 8, 19, 14, 1]
11:  [0, 1, 10, 23, 20, 5]
12:  [0, 1, 10, 31, 30, 10]
13:  [0, 1, 12, 36, 42, 18, 2]
14:  [0, 1, 12, 44, 60, 27, 4]
15:  [0, 1, 14, 52, 76, 48, 8]
16:  [0, 1, 14, 61, 102, 68, 16, 1]
17:  [0, 1, 16, 69, 126, 101, 30, 1]
18:  [0, 1, 16, 81, 160, 138, 50, 5]
19:  [0, 1, 18, 90, 194, 191, 80, 10]
20:  [0, 1, 18, 102, 238, 252, 118, 22]
...
Row n=7 is [0, 1, 6, 6, 2] because in the 15 partitions of 7 there is 0 into zero parts, 1 into one part, 6 into two parts, 6 into three parts, and 2 into four parts:
[ 1]   [ 1 2 3 1 ]
[ 2]   [ 1 2 4 ]
[ 3]   [ 1 3 2 1 ]
[ 4]   [ 1 4 2 ]
[ 5]   [ 1 5 1 ]
[ 6]   [ 1 6 ]
[ 7]   [ 2 3 2 ]
[ 8]   [ 2 4 1 ]
[ 9]   [ 2 5 ]
[10]   [ 3 4 ]
[11]   [ 4 2 1 ]
[12]   [ 4 3 ]
[13]   [ 5 2 ]
[14]   [ 6 1 ]
[15]   [ 7 ]

Prog

(PARI) N=25;  x='x+O('x^N);
T=Vec(1 + sum(n=1, N, t*x^(n) * prod(k=1, n-1, 1+t*x^k)^2));
for(r=1, #T, print(Vecrev(T[r])) );  \\ as triangle

Crossrefs

Cf. A059618 (row sums), A293137.

Cf. A072704 (same for weakly unimodal compositions).

Sequence in context: A320341 A054523 A161363 * A106351 A360764 A096800

Adjacent sequences: A293133 A293134 A293135 * A293137 A293138 A293139

Keyword

nonn,tabf

Author

Joerg Arndt, Oct 01 2017

Status

approved