OFFSET
0,8
COMMENTS
A proper definition is needed for this sequence.
Are the row sums A074139(n) and the row lengths A000041(n)? - R. J. Mathar, May 08 2019 [Not exactly: see below. - M. F. Hasler, Jan 07 2024]
From M. F. Hasler, Jan 06 2024: (Start)
I get this triangle as T(n,k) = # { v in S(p_n), |v| = k }, where p_n is the n-th partition as listed in A036036 or A036037 (which has a nice table of the p's), and S(p) = {0, ..., p[1]} x ... x {0, ..., p[#p]}, the set of vectors v with 0 <= v[i] <= p[i] for all indices i from 1 to #p = number of parts in p.
Then the row sums are indeed the total number of elements in S(p_n) which is equal to the product (p[1]+1)*...*(p[#p]+1) which is also the number of divisors of the Heinz number of p (cf. A185974).
The row lengths are 1 + |p| = 1 + sum of all parts of p (corresponding to the possible values of |v| ranging from 0 to |p|), repeated A000041(|p|) times: A000041(0) = 1 row of length 0+1 for the partition () of 0, A000041(1) = 1 row of length 1+1 for partition (1) of 1; A000041(2) = 2 rows of length 2+1 for the two partitions (2) and (1,1) of 2; A000041(3) = 3 rows of length 3+1 for the 3 partitions {(3), (2,1), (1,1,1)} of 3; etc. (End)
LINKS
M. F. Hasler, Table of n, a(n) for n = 0..9172 (up to partitions of 15), Jan 07 2024
EXAMPLE
The triangle begins
1
1 1
1 1 1
1 2 1
1 1 1 1
1 2 2 1
1 3 3 1
1 1 1 1 1
1 2 2 2 1
1 2 3 2 1
1 3 4 3 1
1 4 6 4 1
1 1 1 1 1 1
1 2 2 2 2 1
1 2 3 3 2 1
1 3 4 4 3 1
1 3 5 5 3 1
1 4 7 7 4 1
1 5 10 10 5 1
PROG
From M. F. Hasler, Jan 06 2024: (Start)
(PARI) A122172_row(n, p=part(n))={my(c=Vec(0, vecsum(p)+1)); forvec(v=[[0, k]| k<-p], c[vecsum(v)+1]++); c} \\ instead of n one can directly give p as 2nd arg
part(n)={my(c, r=0); while(n >= c = numbpart(r), n -= c; r++); partitions(r)[n+1]
for(n=0, 5, forpart(p=n, print(A122172_row(, Vec(p))) )) \\ Illustration. (End)
CROSSREFS
KEYWORD
obsc,nonn,tabf,more
AUTHOR
Alford Arnold, Aug 23 2006
EXTENSIONS
More terms from M. F. Hasler, Jan 07 2024
STATUS
approved