A178666 Irregular triangle read by rows in which row n gives expansion of the polynomial Product_{k=0..n} (1 + x^(2*k + 1)), n >= -1.

1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 2, 2, 3, 3, 3, 2, 3, 3, 3, 2, 2, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 3, 4, 4, 5, 4, 4, 5, 5, 4, 4, 5, 4, 4, 3, 4, 4, 3, 3, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 0, 1, 1

Offset

-1,27

Comments

From Álvar Ibeas, Jul 30 2020: (Start)

T(n, k) is the number of partitions of k into distinct odd parts not larger than 2*n+1. Also, the number of self-conjugate partitions of k into at most n+1 parts. Also, the number of self-conjugate partitions of 2*n+k+3 into exactly n+2 parts.

Rows are symmetric: T(n, k) = T(n, (n+1)^2 - k).

Within the range 0 <= k <= (n+1)^2, T(n, k) = 0 iff k = 2 or k = (n+1)^2 - 2.

(End)

Links

Alois P. Heinz, Rows n = -1..40, flattened

Alexander Rosa and Štefan Znám, A combinatorial problem in the theory of congruences. (Russian), Mat.-Fys. Casopis Sloven. Akad. Vied 15 1965 49-59. [Annotated scanned copy] See Table 2.

Formula

From Álvar Ibeas, Jul 30 2020: (Start)

T(n, k) = Sum_{i < n} T(i, k - 2*i - 3), if k > 0.

T(n, k) = A000700(k), if k <= 2*(n+1).

(End)

Example

Triangle begins:
[1] (the empty product)
[1, 1]
[1, 1, 0, 1, 1]
[1, 1, 0, 1, 1, 1, 1, 0, 1, 1]
[1, 1, 0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 0, 1, 1]
[1, 1, 0, 1, 1, 1, 1, 1, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 1, 1, 1, 1, 1, 0, 1, 1]
[1, 1, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 2, 2, 3, 3, 3, 2, 3, 3, 3, 2, 2, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 0, 1, 1]
[1, 1, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 3, 4, 4, 5, 4, 4, 5, 5, 4, 4, 5, 4, 4, 3, 4, 4, 3, 3, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 0, 1, 1]
...

Maple

T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(
         mul(1 + x^(2*k + 1), k=0..n)):
seq(T(n), n=-1..8);  # Alois P. Heinz, Aug 21 2015

Mathematica

Flatten[Table[CoefficientList[Product[1+x^(2k+1), {k, 0, n}], x], {n, -1, 6}]] (* Jean-François Alcover, May 23 2011 *)

Prog

(PARI)
row(n)={Vec(prod(k=0, n, 1+x^(2*k+1)))}
for(n=-1, 6, print(row(n))) \\ Andrew Howroyd, Feb 20 2018

Crossrefs

Rows give A169987-A169995. See also A002522 (row lengths). Cf. A142724, A000079(n+1) (row sums).

Sequence in context: A319797 A169987 A267611 * A206706 A302354 A000164

Adjacent sequences: A178663 A178664 A178665 * A178667 A178668 A178669

Keyword

nonn,tabf

Author

N. J. A. Sloane, Dec 25 2010

Status

approved