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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.

%I A364891 #14 Aug 27 2023 10:28:54
%S A364891 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 A364891 -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 A364891 0,0,0,0,0,0,21,-2,4,-4,4,-4,4,-4,4,-4,4,-4
%N 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.
%H A364891 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 A364891 1st column: T(n,1) = A002865(n) for n > 0.
%F A364891 abs(T(n,n)) = A035457(n).
%e A364891 The triangle begins:
%e A364891    0;
%e A364891    1, -1;
%e A364891    1,  0, 0;
%e A364891    2, -1, 1, -1;
%e A364891    2,  0, 0,  0, 0;
%e A364891    4, -2, 2, -2, 2, -2;
%e A364891    4,  0, 0,  0, 0,  0, 0;
%e A364891    7, -2, 2, -2, 2, -2, 2, -2;
%e A364891    8,  0, 0,  0, 0,  0, 0,  0, 0;
%e A364891   12, -2, 3, -3, 3, -3, 3, -3, 3, -3;
%e A364891   14,  1, 0,  0, 0,  0, 0,  0, 0,  0, 0;
%e A364891   ...
%t A364891 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 A364891 Cf. A000041, A002865, A035457, A325433, A364892 (row sums), A364893.
%K A364891 sign,tabl
%O A364891 1,7
%A A364891 _Stefano Spezia_, Aug 12 2023