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A288515 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} ((1 + x^j)/(1 - x^j))^k.

%I A288515 #16 Mar 29 2019 15:51:20
%S A288515 1,1,0,1,2,0,1,4,4,0,1,6,12,8,0,1,8,24,32,14,0,1,10,40,80,76,24,0,1,
%T A288515 12,60,160,234,168,40,0,1,14,84,280,552,624,352,64,0,1,16,112,448,
%U A288515 1110,1712,1552,704,100,0,1,18,144,672,2004,3912,4896,3648,1356,154,0,1,20,180,960,3346,7896,12600,13120,8184,2532,232,0
%N A288515 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} ((1 + x^j)/(1 - x^j))^k.
%H A288515 Seiichi Manyama, <a href="/A288515/b288515.txt">Antidiagonals n = 0..139, flattened</a>
%H A288515 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F A288515 G.f. of column k: Product_{j>=1} ((1 + x^j)/(1 - x^j))^k.
%F A288515 G.f. of column k: 1/theta_4(x)^k, where theta_4() is the Jacobi theta function.
%F A288515 For asymptotics of column k see comment from _Vaclav Kotesovec_ in A001934.
%e A288515 Square array begins:
%e A288515 1,   1,    1,    1,     1,     1,  ...
%e A288515 0,   2,    4,    6,     8,    10,  ...
%e A288515 0,   4,   12,   24,    40,    60,  ...
%e A288515 0,   8,   32,   80,   160,   280,  ...
%e A288515 0,  14,   76,  234,   552,  1110,  ...
%e A288515 0,  24,  168,  624,  1712,  3913,  ...
%t A288515 Table[Function[k, SeriesCoefficient[Product[((1 + x^i)/(1 - x^i))^k, {i, 1, n}], {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten
%t A288515 Table[Function[k, SeriesCoefficient[1/EllipticTheta[4, 0, x]^k, {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten
%o A288515 (Julia) # JacobiTheta4 is defined in A002448.
%o A288515 A288515Column(k, len) = JacobiTheta4(len, -k)
%o A288515 for k in 0:8 A288515Column(k, 8) |> println end # _Peter Luschny_, Mar 12 2018
%Y A288515 Columns k=0-24 give: A000007, A015128, A001934, A004404 (alternating values), A284286, A004406-A004425 (alternating values).
%Y A288515 Rows n=0-2 give: A000012, A005843, A046092.
%Y A288515 Main diagonal gives A270919.
%Y A288515 Antidiagonal sums give A299108.
%Y A288515 Cf. A122141, A286815.
%K A288515 nonn,tabl
%O A288515 0,5
%A A288515 _Ilya Gutkovskiy_, Jun 10 2017