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).

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, 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, 0, 0, 1, 0, 1, 1, 2, 2, 4, 4, 0, 0, 0, 0, 0, 1, 0, 1, 1, 2, 2, 4, 4

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 7. 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(7+m,k), with T(k,m) = 0 if k > 7+m.

T(k,m) = A194812(7+m,k).

Example

Triangle begins:
  15,
   8, 7,
   3, 6, 6,
   3, 2, 5, 5,
  ...
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.
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.

Prog

(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}
{ 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(7) = A000041(7) = 15.

The first (0-10) members of this family of triangles are A023531, A129186, A194702-A194706, this sequence, A194708-A194710.

Cf. A135010, A138121, A194812.

Sequence in context: A126070 A305998 A103241 * A094501 A320572 A090636

Adjacent sequences: A194704 A194705 A194706 * A194708 A194709 A194710

Keyword

nonn,tabl

Author

Omar E. Pol, Feb 05 2012

Extensions

Terms a(11) and beyond from Andrew Howroyd, Feb 19 2020

Status

approved