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
KEYWORD
nonn,tabf
AUTHOR
Joerg Arndt, Oct 01 2017
STATUS
approved