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

Author

Stefano Spezia, Aug 12 2023

Keywords

• sign
• tabl

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.

Crossrefs

Cf. A000041, A002865, A035457, A325433, A364892 (row sums), A364893.

Programs

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}]]

Formula

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

abs(T(n,n)) = A035457(n).