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proposed

Antidiagonal sums give A302020.

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proposed

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proposed

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

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

Main diagonal gives A296044 .

allocated for Ilya GutkovskiySquare 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.

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

0,8

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

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.

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 + ...

Square array begins:

1, 1, 1, 1, 1, 1, ...

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

0, 2, 5, 9, 14, 20, ...

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

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

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

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

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

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

Main diagonal gives A296044 .

Cf. A296067.

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nonn,tabl

Ilya Gutkovskiy, Dec 04 2017

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allocated for Ilya Gutkovskiy

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