A181506 TNP(n,k) triangle read by rows. TNP(n,k) is the number of triangular-number partitions of n with k parts.

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

Offset: 1

Author

John P. McSorley, Oct 24 2010

Keywords

• nonn
• tabl

Comments

The nonzero triangular numbers are of the form binomial(n, 2) for n = 2, 3, 4, ... namely: 1, 3, 6, 10, 15, 21, 28, ....

A triangular-number partition of n is a partition of n all of whose parts are triangular numbers. For example, (7 =) 1+6 = 1+3+3 = 1+1+1+1+3 are three different triangular-number partitions of 7; the first has 2 parts, the second has 3 parts, and the third has 5 parts.

TNP(n, k) is the number of triangular-number partitions of n with k parts.

This sequence is the TNP(n, k) triangle read by rows.

Examples

			The triangle begins
1
0 1
1 0 1
0 1 0 1
0 0 1 0 1
1 1 0 1 0 1
0 1 1 0 1 0 1
0 0 1 1 0 1 0 1
0 1 1 1 1 0 1 0 1
1 0 1 1 1 1 0 1 0 1
0 1 0 1 1 1 1 0 1 0 1
0 1 2 1 1 1 1 1 0 1 0 1
For example, row 12 is 0 1 2 1 1 1 1 1 0 1 0 1.
We have TNP(12, 1) = 0 because 12 is not a triangular number.
We have TNP(12, 2) = 1 because 6+6 is the unique triangular-number partition of 12 with 2 parts.
And we have TNP(12, 3) = 2 because 3+3+6 and 1+1+10 are the 2 triangular-number partitions of 12 with 3 parts.
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1275
• John P. McSorley and Alan H. Schoen, Rhombic tilings of (n,k)-ovals, (n, k, lambda)-cyclic difference sets, and related topics, Discrete Math., 313 (2013), 129-154.

Crossrefs

The row sums of the TNP(n, k) triangle give sequence A007294; the n-th term of A007294 is the total number of triangular-number partitions of n.

Programs

PARI

T(n)={[Vecrev(p/y) | p<-Vec(-1 + 1/prod(k=1, (sqrtint(8*n+1)-1)\2, 1 - y*x^binomial(k+1,2) + O(x*x^n)))]}
{ my(A=T(12)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Sep 26 2019

Formula

G.f.: 1 / Product_{k>=1} (1 - z*x^T(k)) where T(k) = A000217(k). - Joerg Arndt, Aug 10 2015