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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.
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%I #14 Aug 27 2023 10:28:54

%S 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,

%T -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,

%U 0,0,0,0,0,0,21,-2,4,-4,4,-4,4,-4,4,-4,4,-4

%N 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.

%H K. Banerjee and M. G. Dastidar, <a href="https://doi.org/10.35011/risc.22-20">Inequalities for the partition function arising from truncated theta series</a>, RISC Report Series No. 22-20, 2023. See Corollary 1.4 at p. 2.

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

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

%e The triangle begins:

%e 0;

%e 1, -1;

%e 1, 0, 0;

%e 2, -1, 1, -1;

%e 2, 0, 0, 0, 0;

%e 4, -2, 2, -2, 2, -2;

%e 4, 0, 0, 0, 0, 0, 0;

%e 7, -2, 2, -2, 2, -2, 2, -2;

%e 8, 0, 0, 0, 0, 0, 0, 0, 0;

%e 12, -2, 3, -3, 3, -3, 3, -3, 3, -3;

%e 14, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0;

%e ...

%t 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}]]

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

%K sign,tabl

%O 1,7

%A _Stefano Spezia_, Aug 12 2023