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A194704
Triangle read by rows: T(k,m) = number of occurrences of k in the last section of the set of partitions of (4 + m).
3
5, 1, 4, 1, 2, 2, 0, 1, 1, 3, 1, 0, 1, 1, 2, 0, 1, 0, 1, 1, 2, 0, 0, 1, 0, 1, 1, 2, 0, 0, 0, 1, 0, 1, 1, 2, 0, 0, 0, 0, 1, 0, 1, 1, 2, 0, 0, 0, 0, 0, 1, 0, 1, 1, 2, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 2
OFFSET
1,1
COMMENTS
Sub-triangle of A182703 and also of A194812. Note that the sum of every row is also the number of partitions of 4. For further information see A182703 and A135010.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)
FORMULA
T(k,m) = A182703(4+m,k), with T(k,m) = 0 if k > 4+m.
T(k,m) = A194812(4+m,k).
EXAMPLE
Triangle begins:
5,
1, 4,
1, 2, 2,
0, 1, 1, 3,
1, 0, 1, 1, 2,
...
For k = 1 and m = 1: T(1,1) = 5 because there are five parts of size 1 in the last section of the set of partitions of 5, since 4 + m = 5, so a(1) = 5.
For k = 2 and m = 1: T(2,1) = 1 because there is only one part of size 2 in the last section of the set of partitions of 5, since 4 + m = 5, so a(2) = 1.
PROG
(PARI) P(n)={my(M=matrix(n, n), d=4); 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}
{ my(T=P(10)); for(n=1, #T, print(T[n, 1..n])) } \\ Andrew Howroyd, Feb 19 2020
CROSSREFS
Always the sum of row k = p(4) = A000041(4) = 5.
The first (0-10) members of this family of triangles are A023531, A129186, A194702, A194703, this sequence, A194705-A194710.
Sequence in context: A127551 A256547 A023138 * A108170 A086988 A010132
KEYWORD
nonn,tabl
AUTHOR
Omar E. Pol, Feb 05 2012
EXTENSIONS
Terms a(16) and beyond from Andrew Howroyd, Feb 19 2020
STATUS
approved