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A194707 Triangle read by rows: T(k,m) = number of occurrences of k in the last section of the set of partitions of (7 + m).

%I A194707 #31 Jan 07 2024 14:14:31
%S A194707 15,8,7,3,6,6,3,2,5,5,1,2,3,4,5,1,1,2,2,5,4,0,1,1,2,2,4,5,1,0,1,1,2,2,
%T A194707 4,4,0,1,0,1,1,2,2,4,4,0,0,1,0,1,1,2,2,4,4,0,0,0,1,0,1,1,2,2,4,4,0,0,
%U A194707 0,0,1,0,1,1,2,2,4,4,0,0,0,0,0,1,0,1,1,2,2,4,4
%N A194707 Triangle read by rows: T(k,m) = number of occurrences of k in the last section of the set of partitions of (7 + m).
%C A194707 Sub-triangle of A182703 and also of A194812. Note that the sum of every row is also the number of partitions of 7. For further information see A182703 and A135010.
%H A194707 Andrew Howroyd, <a href="/A194707/b194707.txt">Table of n, a(n) for n = 1..1275</a> (rows 1..50)
%F A194707 T(k,m) = A182703(7+m,k), with T(k,m) = 0 if k > 7+m.
%F A194707 T(k,m) = A194812(7+m,k).
%e A194707 Triangle begins:
%e A194707   15,
%e A194707    8, 7,
%e A194707    3, 6, 6,
%e A194707    3, 2, 5, 5,
%e A194707   ...
%e A194707 For k = 1 and m = 1: T(1,1) = 15 because there are 15 parts of size 1 in the last section of the set of partitions of 8, since 7 + m = 8, so a(1) = 15.
%e A194707 For k = 2 and m = 1: T(2,1) = 8 because there are eight parts of size 2 in the last section of the set of partitions of 8, since 7 + m = 8, so a(2) = 8.
%o A194707 (PARI) P(n)={my(M=matrix(n,n), d=7); 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 A194707 { my(T=P(10)); for(n=1, #T, print(T[n, 1..n])) } \\ _Andrew Howroyd_, Feb 19 2020
%Y A194707 Always the sum of row k = p(7) = A000041(7) = 15.
%Y A194707 The first (0-10) members of this family of triangles are A023531, A129186, A194702-A194706, this sequence, A194708-A194710.
%Y A194707 Cf. A135010, A138121, A194812.
%K A194707 nonn,tabl
%O A194707 1,1
%A A194707 _Omar E. Pol_, Feb 05 2012
%E A194707 Terms a(11) and beyond from _Andrew Howroyd_, Feb 19 2020