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A297328 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1 - j*x^j)^k.

%I A297328 #19 Aug 16 2023 08:19:15
%S A297328 1,1,0,1,1,0,1,2,3,0,1,3,7,6,0,1,4,12,18,14,0,1,5,18,37,49,25,0,1,6,
%T A297328 25,64,114,114,56,0,1,7,33,100,219,312,282,97,0,1,8,42,146,375,676,
%U A297328 855,624,198,0,1,9,52,203,594,1276,2030,2178,1422,354,0,1,10,63,272,889,2196,4155,5736,5496,3058,672,0
%N A297328 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1 - j*x^j)^k.
%H A297328 Alois P. Heinz, <a href="/A297328/b297328.txt">Antidiagonals n = 0..200, flattened</a>
%F A297328 G.f. of column k: Product_{j>=1} 1/(1 - j*x^j)^k.
%F A297328 A(0,k) = 1; A(n,k) = (k/n) * Sum_{j=1..n} A078308(j) * A(n-j,k). - _Seiichi Manyama_, Aug 16 2023
%e A297328 G.f. of column k: A_k(x) = 1 + k*x + (1/2)*k*(k + 5)*x^2 + (1/6)*k*(k^2 + 15*k + 20)*x^3 + (1/24)*k*(k^3 + 30*k^2 + 155*k + 150)*x^4 + (1/120)*k*(k^4 + 50*k^3 + 575*k^2 + 1750*k + 624)*x^5 + ...
%e A297328 Square array begins:
%e A297328   1,   1,    1,    1,    1,     1,  ...
%e A297328   0,   1,    2,    3,    4,     5,  ...
%e A297328   0,   3,    7,   12,   18,    25,  ...
%e A297328   0,   6,   18,   37,   64,   100,  ...
%e A297328   0,  14,   49,  114,  219,   375,  ...
%e A297328   0,  25,  114,  312,  676,  1276,  ...
%t A297328 Table[Function[k, SeriesCoefficient[Product[1/(1 - i x^i)^k, {i, 1, n}], {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten
%o A297328 (PARI) first(n, k) = my(res = matrix(n, k)); for(u=1, k, my(col = Vec(prod(j=1, n, 1/(1 - j*x^j)^(u-1)) + O(x^n))); for(v=1, n, res[v, u] = col[v])); res \\ _Iain Fox_, Dec 28 2017
%Y A297328 Columns k=0..32 give A000007, A006906, A022726, A022727, A022728, A022729, A022730, A022731, A022732, A022733, A022734, A022735, A022736, A022737, A022738, A022739, A022740, A022741, A022742, A022743, A022744, A022745, A022746, A022747, A022748, A022749, A022750, A022751, A022752, A022753, A022754, A022755, A022756.
%Y A297328 Main diagonal gives A297329.
%Y A297328 Antidiagonal sums give A299162.
%Y A297328 Cf. A078308, A266941, A297321, A297323, A297325.
%K A297328 nonn,tabl
%O A297328 0,8
%A A297328 _Ilya Gutkovskiy_, Dec 28 2017