A211956 Coefficients of a sequence of polynomials related to the Morgan-Voyce polynomials.

1, 1, 1, 1, 1, 2, 1, 4, 2, 1, 6, 4, 1, 9, 12, 4, 1, 12, 20, 8, 1, 16, 40, 32, 8, 1, 20, 60, 56, 16, 1, 25, 100, 140, 80, 16, 1, 30, 140, 224, 144, 32, 1, 36, 210, 448, 432, 192, 32, 1, 42, 280, 672, 720, 352, 64, 1, 49, 392, 1176, 1680, 1232, 448, 64

Offset

0,6

Comments

The row generating polynomials R(n,x) of A211955 factorize in the ring Z[x] as R(n,x) = P(n,x)*P(n+1,x) for n >= 1: explicitly, P(2*n,x) = 1/2*(b(2*n,2*x) + 1)/b(n,2*x) and P(2*n+1,x) = b(n,2*x), where b(n,x) := sum {k = 0..n} binomial(n+k,2*k)*x^k are the Morgan-Voyce polynomials of A085478. This triangle lists the coefficients in ascending powers of x of the polynomials P(n,x).

The odd numbered rows of the present triangle produce triangle A123519; the even numbered row entries are recorded separately in A211957 and appear to equal the unsigned and row reversed form of A204021. The even numbered rows with a factor of 2^(k-1) removed from the k-th column entries produce triangle A208513.

Links

Table of n, a(n) for n=0..63.

Eric Weisstein's World of Mathematics, Morgan-Voyce polynomials

Formula

T(n,0) = 1; for k > 0, T(2*n,k) = 2^k*binomial(n+k,2*k) = A123519(n,k); for k > 0, T(2*n-1,k) = n/(n+k)*2^k*binomial(n+k,2*k) = 2^(k-1)*A208513(n,k).

O.g.f.: ((1+t)*(1-t^2)-t^2*x)/((1-t^2)^2-2*t^2*x) = 1 + t + (1+x)*t^2 + (1+2*x)*t^3 + (1+4*x+2*x^2)*t^4 + ....

Row generating polynomials: P(2*n,x) := 1/2*(b(2*n,2*x)+1)/b(n,2*x) and P(2*n+1,x) := b(n,2*x), where b(n,x) := sum {k = 0..n} binomial(n+k,2*k)*x^k are the Morgan-Voyce polynomials of A085478.

The product P(n,x)*P(n+1,x) is the n-th row polynomial of A211955.

In terms of T(n,x), the Chebyshev polynomials of the first kind, we have P(2*n,x) = T(2*n,u) and P(2*n+1,x) = 1/u*T(2*n+1,u), where u = sqrt((x+2)/2).

Other representations for the row polynomials include

P(2*n,x) = 1/2*(1+x+sqrt(x^2+2*x))^n + 1/2*(1+x-sqrt(x^2+2*x))^n;

P(2*n,x) = n*sum {k = 0..n}(-1)^(n-k)/(n+k)*binomial(n+k,2*k) *(2*x+4)^k for n >= 1; and P(2*n+1,x) = (2*n+1)*sum {k=0..n} (-1)^(n-k)/(n+k+1)* binomial(n+k+1,2*k+1)*(2*x+4)^k.

Recurrence equation: P(n+1,x)*P(n-2,x) - P(n,x)*P(n-1,x) = x.

Row sums A005246(n+2).

Example

Triangle begins
.n\k.|..0....1....2....3....4
= = = = = = = = = = = = = = =
..0..|..1
..1..|..1
..2..|..1....1
..3..|..1....2
..4..|..1....4....2
..5..|..1....6....4
..6..|..1....9...12....4
..7..|..1...12...20....8
..8..|..1...16...40...32....8
..9..|..1...20...60...56...16
...

Crossrefs

Cf. A005246 (row sums), A085478, A123519, A204021, A208513, A211955, A211957.

Sequence in context: A378308 A365901 A325309 * A128177 A087738 A133113

Adjacent sequences: A211953 A211954 A211955 * A211957 A211958 A211959

Keyword

nonn,easy,tabf

Author

Peter Bala, Apr 30 2012

Status

approved