A141664 An irregular triangular sequence formed by partition-like complex polynomials.

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

Offset

0,20

Comments

Row sums are: {1, 1, 0, -2, -4, -4, 0, 8, 16, 16, 0, ...}.

Links

G. C. Greubel, Rows n=0..30 of triangle, flattened

Formula

Let p(x,n) = Product_{j=1..n} (1 + i*x^j), p(x,0)=1, with i being the imaginary unit, then the n-th row is the real part of the coefficients of p(x,n).

Example

Irregular triangle begins as:
1.
1, 0.
1, 0, 0, -1.
1, 0, 0, -1, -1, -1,  0.
1, 0, 0, -1, -1, -2, -1, -1,  0,  0, 1.
1, 0, 0, -1, -1, -2, -2, -2, -1, -1, 1, 1, 1, 1, 1, 0. ...

Mathematica

p[x_, n_]:= If[n == 0, 1, Product[(1 + I*x^i), {i, 1, n}]];
Table[Expand[p[x, n]], {n, 0, 10}];
Table[Re[CoefficientList[p[x, n], x]], {n, 0, 10}]//Flatten

Prog

(PARI) row(n) = if (n==0, 1, apply(x->real(x), Vecrev(prod(j=1, n, (1 + I*x^j))))); \\ Michel Marcus, Apr 02 2019

Crossrefs

Cf. A053632.

Sequence in context: A068934 A035200 A198066 * A056979 A087812 A288384

Adjacent sequences: A141661 A141662 A141663 * A141665 A141666 A141667

Keyword

sign,look,tabf

Author

Roger L. Bagula, Sep 05 2008

Extensions

Edited by G. C. Greubel, Apr 01 2019

Status

approved