This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A287325 #32 Feb 16 2025 08:33:46
%S A287325 1,1,-2,1,-1,0,1,-1,-1,0,1,-1,0,0,2,1,-1,0,-1,0,0,1,-1,0,0,0,1,0,1,-1,
%T A287325 0,0,-1,0,0,0,1,-1,0,0,0,0,1,1,0,1,-1,0,0,0,-1,0,0,0,-2,1,-1,0,0,0,0,
%U A287325 0,1,0,0,0,1,-1,0,0,0,0,-1,0,0,0,0,0,1,-1,0,0,0,0,0,0,1,0,1,0,0,1,-1,0,0,0,0,0,-1,0,0,0,0,-1,0
%N A287325 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Sum_{j=-inf..inf} (-1)^j*x^(k*j*(j-1)/2 + j^2).
%H A287325 G. C. Greubel, <a href="/A287325/b287325.txt">Table of n, a(n) for the first 100 antidiagonals, flattened</a>
%H A287325 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PentagonalNumberTheorem.html">Pentagonal Number Theorem</a>
%H A287325 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%H A287325 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/JacobiThetaFunctions.html">Jacobi Theta Functions</a>
%F A287325 G.f. of column 0: Sum_{j=-inf..inf} (-1)^j*x^A000290(j) = Product_{i>=1} (1 + x^i)/(1 - x^i) (convolution inverse of A015128).
%F A287325 G.f. of column 1: Sum_{j=-inf..inf} (-1)^j*x^A000326(j) = Product_{i>=1} (1 - x^i) (convolution inverse of A000041).
%F A287325 G.f. of column 2: Sum_{j=-inf..inf} (-1)^j*x^A000384(j) = Product_{i>=1} (1 - x^(2*i))/(1 + x^(2*i-1)) (convolution inverse of A006950).
%F A287325 G.f. of column 3: Sum_{j=-inf..inf} (-1)^j*x^A000566(j) = Product_{i>=1} (1 - x^(5*i))*(1 - x^(5*i-1))*(1 - x^(5*i-4)) (convolution inverse of A036820).
%F A287325 G.f. of column 4: Sum_{j=-inf..inf} (-1)^j*x^A000567(j) = Product_{i>=1} (1 - x^(6*i))*(1 - x^(6*i-1))*(1 - x^(6*i-5)) (convolution inverse of A195848).
%F A287325 G.f. of column 5: Sum_{j=-inf..inf} (-1)^j*x^A001106(j) = Product_{i>=1} (1 - x^(7*i))*(1 - x^(7*i-1))*(1 - x^(7*i-6)) (convolution inverse of A195849).
%F A287325 G.f. of column 6: Sum_{j=-inf..inf} (-1)^j*x^A001107(j) = Product_{i>=1} (1 - x^(8*i))*(1 - x^(8*i-1))*(1 - x^(8*i-7)) (convolution inverse of A195850).
%F A287325 G.f. of column 7: Sum_{j=-inf..inf} (-1)^j*x^A051682(j) = Product_{i>=1} (1 - x^(9*i))*(1 - x^(9*i-1))*(1 - x^(9*i-8)) (convolution inverse of A195851).
%F A287325 G.f. of column 8: Sum_{j=-inf..inf} (-1)^j*x^A051624(j) = Product_{i>=1} (1 - x^(10*i))*(1 - x^(10*i-1))*(1 - x^(10*i-9)) (convolution inverse of A195852).
%F A287325 G.f. of column 9: Sum_{j=-inf..inf} (-1)^j*x^A051865(j) = Product_{i>=1} (1 - x^(11*i))*(1 - x^(11*i-1))*(1 - x^(11*i-10)) (convolution inverse of A196933).
%F A287325 G.f. of column k: Sum_{j=-inf..inf} (-1)^j*x^(k*j*(j-1)/2+j^2) = Product_{i>=1} (1 - x^((k+2)*i))*(1 - x^((k+2)*i-1))*(1 - x^((k+2)*i-k-1)).
%e A287325 Square array begins:
%e A287325 1, 1, 1, 1, 1, 1, ...
%e A287325 -2, -1, -1, -1, -1, -1, ...
%e A287325 0, -1, 0, 0, 0, 0, ...
%e A287325 0, 0, -1, 0, 0, 0, ...
%e A287325 2, 0, 0, -1, 0, 0, ...
%e A287325 0, 1, 0, 0, -1, 0, ...
%t A287325 Table[Function[k, SeriesCoefficient[Sum[(-1)^i x^(k i (i - 1)/2 + i^2), {i, -n, n}], {x, 0, n}]][j - n], {j, 0, 13}, {n, 0, j}] // Flatten
%t A287325 Table[Function[k, SeriesCoefficient[Product[(1 - x^((k + 2) i)) (1 - x^((k + 2) i - 1)) (1 - x^((k + 2) i - k - 1)), {i, 1, n}], {x, 0, n}]][j - n], {j, 0, 13}, {n, 0, j}] // Flatten
%t A287325 Table[Function[k, SeriesCoefficient[(x^(2 + k) QPochhammer[1/x, x^(2 + k)] QPochhammer[x^(-1 - k), x^(2 + k)] QPochhammer[x^(2 + k), x^(2 + k)])/((-1 + x) (-1 + x^(1 + k))), {x, 0, n}]][j - n], {j, 0, 13}, {n, 0, j}] // Flatten
%Y A287325 Columns k=0..10 give A002448, A010815, A106459, A113429, A089802, A232714, A244525, A281814 (absolute values), A205988 (absolute values), A281815 (absolute values), A247133.
%Y A287325 Cf. A000041, A000290, A000326, A000384, A000566, A000567, A001106, A001107, A006950, A015128, A036820, A051682, A051624, A051865, A195825, A195848, A195849, A195850, A195851, A195852, A196933, A211970.
%K A287325 sign,tabl
%O A287325 0,3
%A A287325 _Ilya Gutkovskiy_, Aug 13 2017