login
Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} ((1 + x^(2*j-1))/(1 + x^(2*j)))^k.
3

%I #8 May 10 2018 23:07:18

%S 1,1,0,1,1,0,1,2,-1,0,1,3,-1,0,0,1,4,0,-2,1,0,1,5,2,-5,3,0,0,1,6,5,-8,

%T 3,2,-1,0,1,7,9,-10,-1,9,-4,-1,0,1,8,14,-10,-10,20,-7,-4,2,0,1,9,20,

%U -7,-24,31,-2,-15,5,1,0,1,10,27,0,-42,36,20,-40,9,8,-2,0,1,11,35,12,-62,28,65,-75,3,27,-8,-1,0

%N Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} ((1 + x^(2*j-1))/(1 + x^(2*j)))^k.

%H G. C. Greubel, <a href="/A296067/b296067.txt">Rows n=0..100 of antidiagonals, flattened</a>

%F G.f. of column k: Product_{j>=1} ((1 + x^(2*j-1))/(1 + x^(2*j)))^k.

%F G.f. of column k: (x^(1/8)*theta_2(sqrt(x))/theta_2(x))^k, where theta_() is the Jacobi theta function.

%e G.f. of column k: A_k(x) = 1 + k*x + (1/2)*k*(k - 3)*x^2 + (1/6)*k*(k^2 - 9*k + 8)*x^3 + (1/24)*k*(k^3 - 18*k^2 + 59*k - 18)*x^4 + (1/120)*k*(k^4 - 30*k^3 + 215*k^2 - 330*k + 144)*x^5 + ...

%e Square array begins:

%e 1, 1, 1, 1, 1, 1, ...

%e 0, 1, 2, 3, 4, 5, ...

%e 0, -1, -1, 0, 2, 5, ...

%e 0, 0, -2, -5, -8, -10, ...

%e 0, 1, 3, 3, -1, -10, ...

%e 0, 0, 2, 9, 20, 31, ...

%t Table[Function[k, SeriesCoefficient[Product[((1 + x^(2 i - 1))/(1 + x^(2 i)))^k, {i, 1, n}], {x, 0, n}]][j - n], {j, 0, 12}, {n, 0, j}] // Flatten

%t Table[Function[k, SeriesCoefficient[(x^(1/8) EllipticTheta[2, 0, x^(1/2)]/EllipticTheta[2, 0, x])^k, {x, 0, n}]][j - n], {j, 0, 12}, {n, 0, j}] // Flatten

%Y Columns k=0..8 give A000007, A029838, A029839, A029840, A029841, A029842, A029843, A029844, A029845 (with offset 0).

%Y Main diagonal gives A296043.

%Y Cf. A296068.

%K sign,tabl

%O 0,8

%A _Ilya Gutkovskiy_, Dec 04 2017