A325433 Triangle T read by rows: T(n,k) is the number of partitions of n in which k is the least integer that is not a part and there are more parts > k than there are < k (n >= k > 0).

0, 1, 0, 1, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 4, 1, 0, 0, 0, 0, 0, 7, 1, 0, 0, 0, 0, 0, 0, 8, 2, 0, 0, 0, 0, 0, 0, 0, 12, 3, 0, 0, 0, 0, 0, 0, 0, 0, 14, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 21, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 24, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Author

Stefano Spezia, Apr 27 2019

Keywords

• nonn
• tabl

Examples

			T(9,2) = 3 from 6 + 3 = 5 + 3 + 1 = 4 + 4 + 1 = 3 + 3 + 3.
The triangle T(n, k) begins
n\k|   1   2   3   4   5   6   7   8   9
---+------------------------------------
1  |   0
2  |   1   0
3  |   1   0   0
4  |   2   0   0   0
5  |   2   0   0   0   0
6  |   4   0   0   0   0   0
7  |   4   1   0   0   0   0   0
8  |   7   1   0   0   0   0   0   0
9  |   8   2   0   0   0   0   0   0   0
...
		

Links

• George E. Andrews and Mircea Merca, The truncated pentagonal number theorem, Journal of Combinatorial Theory, Series A, Volume 119, Issue 8, 2012, Pages 1639-1643.
• K. Banerjee and M. G. Dastidar, Inequalities for the partition function arising from truncated theta series, RISC Report Series No. 22-20, 2023. See Theorem 1.1 at p. 2.

Crossrefs

Cf. A000041, A002865, A325434 (row sums).

Programs

Mathematica

T[n_,k_]:=(-1)^(k-1)*Sum[(-1)^j*(PartitionsP[n-j*(3*j+1)/2]-PartitionsP[n-j*(3*j+5)/2-1]),{j,0,k-1}]; Flatten[Table[T[n,k],{n,1,15},{k,1,n}]]

PARI

T(n,k) = (-1)^(k-1)*sum(j=0, k-1, (-1)^j*(numbpart(n-j*(3*j+1)/2)-numbpart(n-j*(3*j+5)/2-1)));
tabl(nn) = for(n=1, nn, for(k=1, n, print1(T(n,k), ", ")); print);
tabl(10) \\ yields sequence in triangular form

Formula

T(n,k) = (-1)^(k-1)*Sum_{j=0..k-1} (-1)^j*(p(n - j*(3*j + 1)/2) - p(n - j*(3*j + 5)/2 - 1)), where p(n) = A000041(n) is the number of partitions of n (see Theorem 1.1 in Andrews and Merca).

1st column: T(n,1) = A002865(n) for n > 0.