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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.
4
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, 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, 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
OFFSET
0,13
LINKS
FORMULA
G.f. of column k: (-1 + Product_{j>=1} (1 + x^(2*j-1)))^k.
Sum_{k=0..n} (-1)^(n-k) * T(n,k) = A000009(n).
EXAMPLE
Triangle T(n,k) begins:
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, 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, 0, 1;
0, 2, 12, 27, 44, 45, 36, 28, 8, 9, 0, 1;
...
MAPLE
g:= proc(n) option remember; `if`(n=0, 1, add(add([0, d, -d, d]
[1+irem(d, 4)], d=numtheory[divisors](j))*g(n-j), j=1..n)/n)
end:
T:= proc(n, k) option remember;
`if`(k=0, `if`(n=0, 1, 0), `if`(k=1, `if`(n=0, 0, g(n)),
(q-> add(T(j, q)*T(n-j, k-q), j=0..n))(iquo(k, 2))))
end:
seq(seq(T(n, k), k=0..n), n=0..12); # Alois P. Heinz, Feb 09 2021
MATHEMATICA
T[n_, k_] := SeriesCoefficient[(-1 + 2/QPochhammer[-1, -x])^k, {x, 0, n}]; Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten
CROSSREFS
Main diagonal and lower diagonals give A000012, A000004, A001477, A000217, A000290.
Row sums give A307058.
T(2n,n) gives A341265.
Sequence in context: A279628 A241914 A324393 * A071482 A071483 A340499
KEYWORD
nonn,tabl
AUTHOR
Ilya Gutkovskiy, Feb 08 2021
STATUS
approved