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A194705 Triangle read by rows: T(k,m) = number of occurrences of k in the last section of the set of partitions of (5 + m).

%I A194705 #35 Jan 07 2024 14:14:01
%S A194705 7,4,3,2,2,3,1,1,3,2,0,1,1,2,3,1,0,1,1,2,2,0,1,0,1,1,2,2,0,0,1,0,1,1,
%T A194705 2,2,0,0,0,1,0,1,1,2,2,0,0,0,0,1,0,1,1,2,2,0,0,0,0,0,1,0,1,1,2,2,0,0,
%U A194705 0,0,0,0,1,0,1,1,2,2,0,0,0,0,0,0,0,1,0,1,1,2,2
%N A194705 Triangle read by rows: T(k,m) = number of occurrences of k in the last section of the set of partitions of (5 + m).
%C A194705 Sub-triangle of A182703 and also of A194812. Note that the sum of every row is also the number of partitions of 5. For further information see A182703 and A135010.
%H A194705 Andrew Howroyd, <a href="/A194705/b194705.txt">Table of n, a(n) for n = 1..1275</a> (rows 1..50)
%F A194705 T(k,m) = A182703(5+m,k), with T(k,m) = 0 if k > 5+m.
%F A194705 T(k,m) = A194812(5+m,k).
%e A194705 Triangle begins:
%e A194705   7,
%e A194705   4, 3,
%e A194705   2, 2, 3,
%e A194705   1, 1, 3, 2,
%e A194705   0, 1, 1, 2, 3,
%e A194705   1, 0, 1, 1, 2, 2,
%e A194705   0, 1, 0, 1, 1, 2, 2,
%e A194705   ...
%e A194705 For k = 1 and m = 1: T(1,1) = 7 because there are seven parts of size 1 in the last section of the set of partitions of 6, since 5 + m = 6, so a(1) = 7.
%e A194705 For k = 2 and m = 1: T(2,1) = 4 because there are four parts of size 2 in the last section of the set of partitions of 6, since 5 + m = 6, so a(2) = 4.
%o A194705 (PARI) P(n)={my(M=matrix(n,n), d=5); M[1,1]=numbpart(d); for(m=1, n, forpart(p=m+d, for(k=1, #p, my(t=p[k]); if(t<=n && m<=t, M[t, m]++)), [2, m+d])); M}
%o A194705 { my(T=P(10)); for(n=1, #T, print(T[n, 1..n])) } \\ _Andrew Howroyd_, Feb 19 2020
%Y A194705 Always the sum of row k = p(5) = A000041(5) = 7.
%Y A194705 The first (0-10) members of this family of triangles are A023531, A129186, A194702-A194704, this sequence, A194706-A194710.
%Y A194705 Cf. A135010, A138121, A194812.
%K A194705 nonn,tabl
%O A194705 1,1
%A A194705 _Omar E. Pol_, Feb 05 2012
%E A194705 Terms a(29) and beyond from _Andrew Howroyd_, Feb 19 2020