A364891 Triangle read by rows: T(n,k) = (-1)^(k-1)*Sum_{j=0..k-1} (-1)^j*(p(n - j*(2*j + 1)) - p(n - (j + 1)*(2*j + 1))), where p(n) = A000041(n) is the number of partitions of n, and 1 <= k <= n.
0, 1, -1, 1, 0, 0, 2, -1, 1, -1, 2, 0, 0, 0, 0, 4, -2, 2, -2, 2, -2, 4, 0, 0, 0, 0, 0, 0, 7, -2, 2, -2, 2, -2, 2, -2, 8, 0, 0, 0, 0, 0, 0, 0, 0, 12, -2, 3, -3, 3, -3, 3, -3, 3, -3, 14, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 21, -2, 4, -4, 4, -4, 4, -4, 4, -4, 4, -4
Offset: 1
Examples
The triangle begins: 0; 1, -1; 1, 0, 0; 2, -1, 1, -1; 2, 0, 0, 0, 0; 4, -2, 2, -2, 2, -2; 4, 0, 0, 0, 0, 0, 0; 7, -2, 2, -2, 2, -2, 2, -2; 8, 0, 0, 0, 0, 0, 0, 0, 0; 12, -2, 3, -3, 3, -3, 3, -3, 3, -3; 14, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0; ...
Links
- K. Banerjee and M. G. Dastidar, Inequalities for the partition function arising from truncated theta series, RISC Report Series No. 22-20, 2023. See Corollary 1.4 at p. 2.
Programs
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Mathematica
T[n_, k_]:=(-1)^(k-1)*Sum[(-1)^j*(PartitionsP[n-j(2j+1)]-PartitionsP[n-(j+1)(2j+1)]), {j, 0, k-1}]; Flatten[Table[T[n, k], {n, 1, 12}, {k, 1, n}]]