A194709 Triangle read by rows: T(k,m) = number of occurrences of k in the last section of the set of partitions of (9 + m).

30, 15, 15, 6, 10, 14, 5, 5, 10, 10, 3, 4, 5, 8, 10, 2, 2, 5, 4, 8, 9, 1, 2, 2, 4, 5, 7, 9, 1, 1, 2, 2, 4, 4, 8, 8, 0, 1, 1, 2, 2, 4, 4, 7, 9, 1, 0, 1, 1, 2, 2, 4, 4, 7, 8, 0, 1, 0, 1, 1, 2, 2, 4, 4, 7, 8, 0, 0, 1, 0, 1, 1, 2, 2, 4, 4, 7, 8, 0, 0, 0, 1, 0, 1, 1, 2, 2, 4, 4, 7, 8

Offset: 1

Author

Omar E. Pol, Feb 05 2012

Keywords

• nonn
• tabl

Comments

Sub-triangle of A182703 and also of A194812. Note that the sum of every row is also the number of partitions of 9. For further information see A182703 and A135010.

Examples

			Triangle begins:
  30;
  15, 15;
   6, 10, 14;
   5,  5, 10, 10;
   3,  4,  5,  8, 10;
   2,  2,  5,  4,  8, 9;
  ...
For k = 1 and  m = 1; T(1,1) = 30 because there are 30 parts of size 1 in the last section of the set of partitions of 10, since 9 + m = 10, so a(1) = 30. For k = 2 and m = 1; T(2,1) = 15 because there are 15 parts of size 2 in the last section of the set of partitions of 10, since 9 + m = 10, so a(2) = 15.
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

Crossrefs

Always the sum of row k = p(9) = A000041(n) = 30.

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

Cf. A135010, A138121, A194812.

Programs

PARI

P(n)={my(M=matrix(n,n), d=9); 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

Formula

T(k,m) = A182703(9+m,k), with T(k,m) = 0 if k > 9+m.

T(k,m) = A194812(9+m,k).

Extensions

Terms a(7) and beyond from Andrew Howroyd, Feb 19 2020