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.

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

Alois P. Heinz, Rows n = 0..200, flattened

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

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

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

Row sums give A307058.

T(2n,n) gives A341265.

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

Sequence in context: A279628 A241914 A324393 * A071482 A071483 A340499

Adjacent sequences: A341276 A341277 A341278 * A341280 A341281 A341282

Keyword

nonn,tabl

Author

Ilya Gutkovskiy, Feb 08 2021

Status

approved