A158982 -id:A158982 - cp's OEIS frontend

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

Clark Kimberling, Apr 02 2009

			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

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

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

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)

A158985 Coefficients of polynomials (in descending powers of x) P(n,x) := 1 + P(n-1,x)^2, where P(1,x) = x + 1.

Original entry on oeis.org

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

Clark Kimberling, Apr 02 2009

			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

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

Cf. A158982, A158983, A158984, A158986, A003095 (row sums), A062013.

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

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)

A158984 Coefficients of polynomials (in descending powers of x) P(n,x) := -1 + P(n-1,x)^2, where P(1,x) = x - 1.

Original entry on oeis.org

1, -1, 1, -2, 0, 1, -4, 4, -1, 1, -8, 24, -32, 14, 8, -8, 0, 0, 1, -16, 112, -448, 1116, -1744, 1552, -384, -700, 736, -160, -128, 64, 0, 0, 0, -1, 1, -32, 480, -4480, 29112, -139552, 509600, -1441024, 3166616, -5345344, 6668992, -5473536, 1494624

Offset: 1

Clark Kimberling, Apr 02 2009

			Row 1: 1 -1 (from x-1)
Row 2: 1 -2 0 (from x^2-2x)
Row 3: 1 -4 4 -1
Row 4: 1 -8 24 -32 14 8 -8 0 0

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

Cf. A158982, A158983, A158985, A158986, A003096.

tabf(nn) = {p = x-1; print(Vec(p)); for (n=2, nn, p = -1 + p^2; print(Vec(p)););} \\ Michel Marcus, Mar 01 2016

From Peter Bala, Jul 01 2015: (Start)
P(n,x) = P(n,2 - x) for n >= 2.
P(n+1,x)= P(n,(x - 1)^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.
P(n,x)^(2^k) divides P(n + 2*k,x) in Z[x] for k = 1,2,....
P(n,4) = A003096(n). (End)

A158986 Coefficients of polynomials Q(n,x):=-2+(1+Q(n-1,x))^2, where Q(1,x)=x-2.

Original entry on oeis.org

1, -2, 1, -2, -1, 1, -4, 4, 0, -2, 1, -8, 24, -32, 14, 8, -8, 0, -1, 1, -16, 112, -448, 1116, -1744, 1552, -384, -700, 736, -160, -128, 64, 0, 0, 0, -2, 1, -32, 480, -4480, 29112, -139552, 509600, -1441024, 3166616, -5345344, 6668992, -5473536, 1494624, 3005056, -4820608

Offset: 1

Clark Kimberling, Apr 02 2009

Let P(n,x) be the n-th polynomial at A158984. Then Q(n,x)=P(n-1,x)-1 is a factor of P(n,x).

			Row 1: 1 -2 (from x-2)
Row 2: 1 -2 -1 (from x^2-2x-1)
Row 3: 1 -4 4 0 -2
Row 4: 1 -8 24 -32 14 8 -8 0 -1

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

Cf. A158982, A158983, A158984, A158985.

tabf(nn) = {p = x-2; print(Vec(p)); for (n=2, nn, p = -2 + (p+1)^2; print(Vec(p)););} \\ Michel Marcus, Mar 01 2016

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

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A158985 Coefficients of polynomials (in descending powers of x) P(n,x) := 1 + P(n-1,x)^2, where P(1,x) = x + 1.

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A158984 Coefficients of polynomials (in descending powers of x) P(n,x) := -1 + P(n-1,x)^2, where P(1,x) = x - 1.

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A158986 Coefficients of polynomials Q(n,x):=-2+(1+Q(n-1,x))^2, where Q(1,x)=x-2.

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