cp's OEIS Frontend

A122172 Triangle read by rows relating A074139, A074141, A078436 and A079025.

%I A122172 #29 Jan 26 2025 03:26:57
%S A122172 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 A122172 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 A122172 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 A122172 Triangle read by rows relating A074139, A074141, A078436 and A079025.
%C A122172 A proper definition is needed for this sequence.
%C A122172 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 A122172 From _M. F. Hasler_, Jan 06 2024: (Start)
%C A122172 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 A122172 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 A122172 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 A122172 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 A122172 The triangle begins:
%e A122172   1
%e A122172   1 1
%e A122172   1 1 1
%e A122172   1 2 1
%e A122172   1 1 1 1
%e A122172   1 2 2 1
%e A122172   1 3 3 1
%e A122172   1 1 1 1 1
%e A122172   1 2 2 2 1
%e A122172   1 2 3 2 1
%e A122172   1 3 4 3 1
%e A122172   1 4 6 4 1
%e A122172   1 1 1 1 1 1
%e A122172   1 2 2 2 2 1
%e A122172   1 2 3 3 2 1
%e A122172   1 3 4 4 3 1
%e A122172   1 3 5 5 3 1
%e A122172   1 4 7 7 4 1
%e A122172   1 5 10 10 5 1
%o A122172 (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 A122172 /* helper function: n-th partition as listed in A036036, A036037 or A185974 */
%o A122172 part(n)={my(c, r=0); while(n >= c = numbpart(r), n -= c; r++); partitions(r)[n+1]}
%o A122172 for(n=0,5, forpart(p=n, print(A122172_row(, Vec(p))) )) \\ Illustration. \\ _M. F. Hasler_, Jan 06 2024
%Y A122172 Cf. A036036 (partitions in A-S order), A036037 (the same, parts reversed), A185974 (corresponding Heinz numbers).
%K A122172 obsc,nonn,tabf,more
%O A122172 0,8
%A A122172 _Alford Arnold_, Aug 23 2006
%E A122172 More terms from _M. F. Hasler_, Jan 07 2024