A296068 - OEIS

%I #9 Apr 01 2018 05:53:05

%S 1,1,0,1,1,0,1,2,2,0,1,3,5,3,0,1,4,9,10,4,0,1,5,14,22,18,6,0,1,6,20,

%T 40,48,32,9,0,1,7,27,65,101,99,55,12,0,1,8,35,98,185,236,194,90,16,0,

%U 1,9,44,140,309,481,518,363,144,22,0,1,10,54,192,483,882,1165,1080,657,226,29,0,1,11,65,255,718,1498,2330,2665,2162,1155,346,38,0

%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 - x^(2*j-1)))^k.

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

%F G.f. of column k: Product_{j>=1} ((1 - x^(4*j))/(1 - x^j))^k.

%F G.f. of column k: 2^(-k/2)*(theta_2(0,x)/(x^(1/8)*theta_2(Pi/4,sqrt(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,  2,   5,   9,   14,   20,  ...

%e 0,  3,  10,  22,   40,   65,  ...

%e 0,  4,  18,  48,  101,  185,  ...

%e 0,  6,  32,  99,  236,  481,  ...

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

%t Table[Function[k, SeriesCoefficient[Product[((1 - x^(4 i))/(1 - x^i))^k, {i, 1, n}], {x, 0, n}]][j - n], {j, 0, 12}, {n, 0, j}] // Flatten

%t Table[Function[k, SeriesCoefficient[2^(-k/2) (EllipticTheta[2, 0, x]/(x^(1/8) EllipticTheta[2, Pi/4, Sqrt[x]]))^k, {x, 0, n}]][j - n], {j, 0, 12}, {n, 0, j}] // Flatten

%Y Columns k=0..8 give A000007, A001935, A001936, A001937, A093160, A001939, A001940, A001941, A092877.

%Y Main diagonal gives A296044.

%Y Antidiagonal sums give A302020.

%Y Cf. A296067.

%K nonn,tabl

%O 0,8

%A _Ilya Gutkovskiy_, Dec 04 2017