A158985 Coefficients of polynomials (in descending powers of x) P(n,x) := 1 + P(n-1,x)^2, where P(1,x) = x + 1.
1, 1, 1, 2, 2, 1, 4, 8, 8, 5, 1, 8, 32, 80, 138, 168, 144, 80, 26, 1, 16, 128, 672, 2580, 7664, 18208, 35296, 56472, 74944, 82432, 74624, 54792, 31776, 13888, 4160, 677, 1, 32, 512, 5440, 43048, 269920, 1393728, 6082752, 22860480, 75010560, 217147904
Offset: 1
Examples
Row 1: 1 1 (from x + 1) Row 2: 1 2 2 (from x^2 + 2*x + 2) Row 3: 1 4 8 8 5 Row 4: 1 8 32 80 138 168 144 80 26
Links
- Clark Kimberling, Polynomials defined by a second-order recurrence, interlacing zeros, and Gray codes, The Fibonacci Quarterly 48 (2010) 209-218.
Programs
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PARI
tabf(nn) = {my(P = x+1); print(Vec(P)); for (n=1, nn, P = 1 + P^2; print(Vec(P)););} \\ Michel Marcus, Jul 01 2015
Formula
From Peter Bala, Jul 01 2015: (Start)
P(n,x) = P(n,-2 - x) for n >= 2.
P(n+1,x)= P(n,(1 + x)^2). Thus if alpha is a zero of P(n,x) then sqrt(alpha) - 1 is a zero of P(n+1,x).
Define a sequence of polynomials Q(n,x) by setting Q(1,x) = 1 + x^2 and Q(n,x) = Q(n-1, 1 + 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) = A003095(n+1);
P(n,1) = P(n+1,0) = P(n+1,-2); P(n,1) = P(n,-3) for n >= 2.
P(n,2) = A062013(n). (End)
Comments