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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. 33

%I #19 Aug 16 2023 08:19:15

%S 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 25,64,114,114,56,0,1,7,33,100,219,312,282,97,0,1,8,42,146,375,676,

%U 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 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 Alois P. Heinz, <a href="/A297328/b297328.txt">Antidiagonals n = 0..200, flattened</a>

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

%F 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 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 Square array begins:

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

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

%e 0, 3, 7, 12, 18, 25, ...

%e 0, 6, 18, 37, 64, 100, ...

%e 0, 14, 49, 114, 219, 375, ...

%e 0, 25, 114, 312, 676, 1276, ...

%t 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 (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 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 Main diagonal gives A297329.

%Y Antidiagonal sums give A299162.

%Y Cf. A078308, A266941, A297321, A297323, A297325.

%K nonn,tabl

%O 0,8

%A _Ilya Gutkovskiy_, Dec 28 2017

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Last modified March 28 18:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)