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%I #21 Jan 12 2024 22:43:58
%S 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,
%T 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,
%U 3,1,1,3,5,5,3,1,1,4,7,7,4,1,1,5,10,10,5,1,1,1,1,1,1,1,1
%N Triangle read by rows relating A074139, A074141, A078436 and A079025.
%C A proper definition is needed for this sequence.
%C 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]
%C From _M. F. Hasler_, Jan 06 2024: (Start)
%C 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.
%C 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).
%C 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)
%H M. F. Hasler, <a href="/A122172/b122172.txt">Table of n, a(n) for n = 0..9172</a> (up to partitions of 15), Jan 07 2024
%e The triangle begins
%e 1
%e 1 1
%e 1 1 1
%e 1 2 1
%e 1 1 1 1
%e 1 2 2 1
%e 1 3 3 1
%e 1 1 1 1 1
%e 1 2 2 2 1
%e 1 2 3 2 1
%e 1 3 4 3 1
%e 1 4 6 4 1
%e 1 1 1 1 1 1
%e 1 2 2 2 2 1
%e 1 2 3 3 2 1
%e 1 3 4 4 3 1
%e 1 3 5 5 3 1
%e 1 4 7 7 4 1
%e 1 5 10 10 5 1
%o From _M. F. Hasler_, Jan 06 2024: (Start)
%o (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
%o /* helper function: n-th partition as listed in A036036, A036037 or A185974 */
%o part(n)={my(c, r=0); while(n >= c = numbpart(r), n -= c; r++); partitions(r)[n+1]
%o for(n=0,5, forpart(p=n, print(A122172_row(, Vec(p))) )) \\ Illustration. (End)
%Y Cf. A036036 (partitions in A-S order), A036037 (the same, parts reversed), A185974 (corresponding Heinz numbers).
%K obsc,nonn,tabf,more
%O 0,8
%A _Alford Arnold_, Aug 23 2006
%E More terms from _M. F. Hasler_, Jan 07 2024
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