A080242 Table of coefficients of polynomials P(n,x) defined by the relation P(n,x) = (1+x)*P(n-1,x) + (-x)^(n+1).

1, 1, 1, 1, 1, 2, 2, 1, 3, 4, 2, 1, 1, 4, 7, 6, 3, 1, 5, 11, 13, 9, 3, 1, 1, 6, 16, 24, 22, 12, 4, 1, 7, 22, 40, 46, 34, 16, 4, 1, 1, 8, 29, 62, 86, 80, 50, 20, 5, 1, 9, 37, 91, 148, 166, 130, 70, 25, 5, 1, 1, 10, 46, 128, 239, 314, 296, 200, 95

Offset: 0

Author

Paul Barry, Feb 12 2003

Keywords

• easy
• nonn
• tabf

Comments

Values generate solutions to the recurrence a(n) = a(n-1) + k(k+1)* a(n-2), a(0)=1, a(1) = k(k+1)+1. Values and sequences associated with this table are included in A072024.

Examples

			Rows are {1}, {1,1,1}, {1,2,2}, {1,3,4,2,1}, {1,4,7,6,3}, ... This is the same as table A035317 with an extra 1 at the end of every second row.
Triangle begins
  1;
  1,  1,  1;
  1,  2,  2;
  1,  3,  4,  2,  1;
  1,  4,  7,  6,  3;
  1,  5, 11, 13,  9,  3,  1;
  1,  6, 16, 24, 22, 12,  4;
  1,  7, 22, 40, 46, 34, 16,  4,  1;
  1,  8, 29, 62, 86, 80, 50, 20,  5;
		

Links

• G. C. Greubel, Rows n = 0..100 of coefficients, flattened

Crossrefs

Similar to the triangles A059259, A035317, A108561, A112555. Cf. A059260.

Cf. A001045 (row sums).

Programs

Mathematica

Table[CoefficientList[Series[((1+x)^(n+2) -(-1)^n*x^(n+2))/(1+2*x), {x, 0, n+2}], x], {n, 0, 10}]//Flatten (* G. C. Greubel, Feb 18 2019 *)

Formula

Rows are generated by P(n,x) = ((x+1)^(n+2) - (-x)^(n+2))/(2*x+1).

The polynomials P(n,-x), n > 0, satisfy a Riemann hypothesis: their zeros lie on the vertical line Re x = 1/2 in the complex plane.

O.g.f.: (1+x*t+x^2*t)/((1+x*t)(1-t-x*t)) = 1 + (1+x+x^2)*t + (1+2x+2x^2)*t^2 + ... . - Peter Bala, Oct 24 2007

T(n,k) = if(k<=2*floor((n+1)/2), Sum_{j=0..floor((n+1)/2)} binomial(n-2j,k-2j), 0). - Paul Barry, Apr 08 2011 (This formula produces the odd numbered rows correctly, but not the even. - G. C. Greubel, Feb 22 2019)