A158983 Coefficients of polynomials (in descending powers of x) P(n,x) := 2 + P(n-1,x)^2, where P(1,x) = x + 2.

1, 2, 1, 4, 6, 1, 8, 28, 48, 38, 1, 16, 120, 544, 1628, 3296, 4432, 3648, 1446, 1, 32, 496, 4928, 35064, 189248, 800992, 2711424, 7419740, 16475584, 29610272, 42666880, 48398416, 41867904, 26125248, 10550016, 2090918, 1, 64, 2016, 41600, 631536

Offset: 1

Author

Clark Kimberling, Apr 02 2009

Keywords

• nonn
• tabf

Examples

			Row 1: 1 2 (from x+2)
Row 2: 1 4 6 (from x^2+4x+6)
Row 3: 1 8 28 48 38
Row 4: 1 16 120 544 1628 3296 4432 3648 1446
		

Links

• Clark Kimberling, Polynomials defined by a second-order recurrence, interlacing zeros, and Gray codes, The Fibonacci Quarterly 48 (2010) 209-218.

Crossrefs

Cf. A158982, A158984, A158985, A158986. A102847 (row sums).

Formula

From Peter Bala, Jul 01 2015: (Start)

P(n,x) = P(n,-4 - x) for n >= 2.

P(n+1,x)= P(n,(2 + x)^2). Thus if alpha is a zero of P(n,x) then sqrt(alpha) - 2 is a zero of P(n+1,x).

Define a sequence of polynomials Q(n,x) by setting Q(1,x) = 2 + x^2 and Q(n,x) = Q(n-1, 2 + x^2) for n >= 2. Then P(n,x) = Q(n,sqrt(x)).

Q(n,x) = Q(k,Q(n-k,x)) for 1 <= k <= n-1; P(n,x) = P(k,P(n-k,x)^2) for 1 <= k <= n - 1.

n-th row sum = P(n,1) = A102847(n);

P(n,1) = P(n+1,-1) = P(n+1,-3); P(n,1) = P(n,-5) for n >= 2.

(End)