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A341279 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.

%I A341279 #10 Feb 09 2021 06:37:47
%S A341279 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 A341279 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 A341279 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 A341279 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 A341279 Alois P. Heinz, <a href="/A341279/b341279.txt">Rows n = 0..200, flattened</a>
%F A341279 G.f. of column k: (-1 + Product_{j>=1} (1 + x^(2*j-1)))^k.
%F A341279 Sum_{k=0..n} (-1)^(n-k) * T(n,k) = A000009(n).
%e A341279 Triangle T(n,k) begins:
%e A341279   1;
%e A341279   0,  1;
%e A341279   0,  0,  1;
%e A341279   0,  1,  0,  1;
%e A341279   0,  1,  2,  0,  1;
%e A341279   0,  1,  2,  3,  0,  1;
%e A341279   0,  1,  3,  3,  4,  0,  1;
%e A341279   0,  1,  4,  6,  4,  5,  0,  1;
%e A341279   0,  2,  5,  9, 10,  5,  6,  0,  1;
%e A341279   0,  2,  8, 13, 16, 15,  6,  7,  0,  1;
%e A341279   0,  2,  9, 21, 26, 25, 21,  7,  8,  0,  1;
%e A341279   0,  2, 12, 27, 44, 45, 36, 28,  8,  9,  0,  1;
%e A341279   ...
%p A341279 g:= proc(n) option remember; `if`(n=0, 1, add(add([0, d, -d, d]
%p A341279       [1+irem(d, 4)], d=numtheory[divisors](j))*g(n-j), j=1..n)/n)
%p A341279     end:
%p A341279 T:= proc(n, k) option remember;
%p A341279       `if`(k=0, `if`(n=0, 1, 0), `if`(k=1, `if`(n=0, 0, g(n)),
%p A341279       (q-> add(T(j, q)*T(n-j, k-q), j=0..n))(iquo(k, 2))))
%p A341279     end:
%p A341279 seq(seq(T(n, k), k=0..n), n=0..12);  # _Alois P. Heinz_, Feb 09 2021
%t A341279 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 A341279 Columns k=0-10 give A000007, A000700, A338463, A341241, A341243, A341244, A341245, A341246, A341247, A341251, A341253.
%Y A341279 Main diagonal and lower diagonals give A000012, A000004, A001477, A000217, A000290.
%Y A341279 Row sums give A307058.
%Y A341279 T(2n,n) gives A341265.
%Y A341279 Cf. A000009, A047265, A060642, A286352, A308680.
%K A341279 nonn,tabl
%O A341279 0,13
%A A341279 _Ilya Gutkovskiy_, Feb 08 2021