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Triangle read by rows: T(n,k) = coefficient of x^n in expansion of (-1 + Product_{j>=1} 1 / (1 + (-x)^j))^k, n >= 0, 0 <= k <= n.
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%I #10 Feb 09 2021 06:37:47

%S 1,0,1,0,0,1,0,1,0,1,0,1,2,0,1,0,1,2,3,0,1,0,1,3,3,4,0,1,0,1,4,6,4,5,

%T 0,1,0,2,5,9,10,5,6,0,1,0,2,8,13,16,15,6,7,0,1,0,2,9,21,26,25,21,7,8,

%U 0,1,0,2,12,27,44,45,36,28,8,9,0,1,0,3,15,40,63,80,71,49,36,9,10,0,1

%N Triangle read by rows: T(n,k) = coefficient of x^n in expansion of (-1 + Product_{j>=1} 1 / (1 + (-x)^j))^k, n >= 0, 0 <= k <= n.

%H Alois P. Heinz, <a href="/A341279/b341279.txt">Rows n = 0..200, flattened</a>

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

%F Sum_{k=0..n} (-1)^(n-k) * T(n,k) = A000009(n).

%e Triangle T(n,k) begins:

%e 1;

%e 0, 1;

%e 0, 0, 1;

%e 0, 1, 0, 1;

%e 0, 1, 2, 0, 1;

%e 0, 1, 2, 3, 0, 1;

%e 0, 1, 3, 3, 4, 0, 1;

%e 0, 1, 4, 6, 4, 5, 0, 1;

%e 0, 2, 5, 9, 10, 5, 6, 0, 1;

%e 0, 2, 8, 13, 16, 15, 6, 7, 0, 1;

%e 0, 2, 9, 21, 26, 25, 21, 7, 8, 0, 1;

%e 0, 2, 12, 27, 44, 45, 36, 28, 8, 9, 0, 1;

%e ...

%p g:= proc(n) option remember; `if`(n=0, 1, add(add([0, d, -d, d]

%p [1+irem(d, 4)], d=numtheory[divisors](j))*g(n-j), j=1..n)/n)

%p end:

%p T:= proc(n, k) option remember;

%p `if`(k=0, `if`(n=0, 1, 0), `if`(k=1, `if`(n=0, 0, g(n)),

%p (q-> add(T(j, q)*T(n-j, k-q), j=0..n))(iquo(k, 2))))

%p end:

%p seq(seq(T(n, k), k=0..n), n=0..12); # _Alois P. Heinz_, Feb 09 2021

%t T[n_, k_] := SeriesCoefficient[(-1 + 2/QPochhammer[-1, -x])^k, {x, 0, n}]; Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten

%Y Columns k=0-10 give A000007, A000700, A338463, A341241, A341243, A341244, A341245, A341246, A341247, A341251, A341253.

%Y Main diagonal and lower diagonals give A000012, A000004, A001477, A000217, A000290.

%Y Row sums give A307058.

%Y T(2n,n) gives A341265.

%Y Cf. A000009, A047265, A060642, A286352, A308680.

%K nonn,tabl

%O 0,13

%A _Ilya Gutkovskiy_, Feb 08 2021