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

%I A181506 #29 Sep 27 2019 15:46:26
%S A181506 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,
%T A181506 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,
%U A181506 2,1,1,1,1,1,0,1,0,1
%N A181506 TNP(n,k) triangle read by rows. TNP(n,k) is the number of triangular-number partitions of n with k parts.
%C A181506 The nonzero triangular numbers are of the form binomial(n, 2) for n = 2, 3, 4, ... namely: 1, 3, 6, 10, 15, 21, 28, ....
%C A181506 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.
%C A181506 TNP(n, k) is the number of triangular-number partitions of n with k parts.
%C A181506 This sequence is the TNP(n, k) triangle read by rows.
%H A181506 Andrew Howroyd, <a href="/A181506/b181506.txt">Table of n, a(n) for n = 1..1275</a>
%H A181506 John P. McSorley and Alan H. Schoen, <a href="http://dx.doi.org/10.1016/j.disc.2012.08.021">Rhombic tilings of (n,k)-ovals, (n, k, lambda)-cyclic difference sets, and related topics</a>, Discrete Math., 313 (2013), 129-154.
%F A181506 G.f.: 1 / Product_{k>=1} (1 - z*x^T(k)) where T(k) = A000217(k). - _Joerg Arndt_, Aug 10 2015
%e A181506 The triangle begins
%e A181506 1
%e A181506 0 1
%e A181506 1 0 1
%e A181506 0 1 0 1
%e A181506 0 0 1 0 1
%e A181506 1 1 0 1 0 1
%e A181506 0 1 1 0 1 0 1
%e A181506 0 0 1 1 0 1 0 1
%e A181506 0 1 1 1 1 0 1 0 1
%e A181506 1 0 1 1 1 1 0 1 0 1
%e A181506 0 1 0 1 1 1 1 0 1 0 1
%e A181506 0 1 2 1 1 1 1 1 0 1 0 1
%e A181506 For example, row 12 is 0 1 2 1 1 1 1 1 0 1 0 1.
%e A181506 We have TNP(12, 1) = 0 because 12 is not a triangular number.
%e A181506 We have TNP(12, 2) = 1 because 6+6 is the unique triangular-number partition of 12 with 2 parts.
%e A181506 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.
%o A181506 (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)))]}
%o A181506 { my(A=T(12)); for(n=1, #A, print(A[n])) } \\ _Andrew Howroyd_, Sep 26 2019
%Y A181506 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.
%K A181506 nonn,tabl
%O A181506 1,69
%A A181506 _John P. McSorley_, Oct 24 2010