A211955 Triangle of coefficients of a polynomial sequence related to the Morgan-Voyce polynomials A085478.

1, 1, 1, 1, 3, 2, 1, 6, 10, 4, 1, 10, 30, 28, 8, 1, 15, 70, 112, 72, 16, 1, 21, 140, 336, 360, 176, 32, 1, 28, 252, 840, 1320, 1056, 416, 64, 1, 36, 420, 1848, 3960, 4576, 2912, 960, 128, 1, 45, 660, 3696, 10296, 16016, 14560, 7680, 2176, 256

Offset

0,5

Comments

Let b(n,x) = sum {k = 0..n} binomial(n+k,2*k)*x^k denote the Morgan-Voyce polynomials of A085478. This triangle lists the coefficients (in ascending powers of x) of the related polynomial sequence R(n,x) := 1/2*b(n,2*x) + 1/2. Several sequences already in the database are of the form (R(n,x))n>=0 for a fixed value of x. These include A101265 (x = 1), A011900 (x = 2), A182432 (x = 3), A054318 (x = 4) as well as signed versions of A133872 (x = -1), A109613(x = -2), A146983 (x = -3) and A084159(x = -4).

The polynomials R(n,x) 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). The coefficients of P(n,x) occur in several tables in the database, although without the connection to the Morgan-Voyce polynomials being noted - see A211956 for more details. 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). Hence R(n,x) = 1/u*T(n,u)*T(n+1,u).

Links

Table of n, a(n) for n=0..54.

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

Formula

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

O.g.f. for column k (except column 0): 2^(k-1)*x^k/(1-x)^(2*k+1). O.g.f.: (1-t*(x+2)+t^2)/((1-t)*(1-2*t(x+1)+t^2)) = 1 + (1+x)*t + (1+3*x+2*x^2)*t^2 + ....

Removing the first column from the triangle produces the Riordan array [x/(1-x)^3, 2*x/(1-x)^2].

The row polynomials R(n,x) := 1/2*b(n,2*x) + 1/2 = 1 + x*sum {k = 1..n} binomial(n+k,2*k)*(2*x)^(k-1).

Recurrence equation: R(n,x) = 2*(1+x)*R(n-1,x) - R(n-2,x) - x with initial conditions R(0,x) = 1, R(1,x) = 1+x. Another recurrence is R(n,x)*R(n-2,x) = R(n-1,x)*(R(n-1,x) + x).

With P(n,x) as defined in the Comments section we have (x+2)/x - {sum {k = 0..2n} 1/R(k,x)}^2 = 2/(x*P(2*n+1,x)^2); (x+2)/x - {sum {k = 0..2n+1} 1/R(k,x)}^2 = (x+2)/(x*P(2*n+2,x)^2); consequently sum {k = 0..inf} 1/R(k,x) = sqrt((x+2)/x) for either x > 0 or x <= -2.

Row sums R(n,1) = A101265(n+1); Alt. row sums R(n,-1) = A133872(n+1);

R(n,2) = A011900(n); R(n,-2) = (-1)^n*A109613(n); R(n,3) = A182432;

R(n,-3) = (-1)^n*A146983(n); R(n,4) = A054318(n+1); R(n,-4) = (-1)^n*A084159(n).

Example

Triangle begins
.n\k.|..0....1....2....3....4....5....6
= = = = = = = = = = = = = = = = = = = =
..0..|..1
..1..|..1....1
..2..|..1....3....2
..3..|..1....6...10....4
..4..|..1...10...30...28....8
..5..|..1...15...70..112...72...16
..6..|..1...21..140..336..360..176...32

Crossrefs

Cf. A011900, A084159, A085478, A101265 (row sums), A109613, A112373, A123519, A133872 (alt row sums), A146983, A182432, A204021, A208513, A211956, A211957.

Sequence in context: A337977 A212207 A111049 * A088617 A190909 A144250

Adjacent sequences: A211952 A211953 A211954 * A211956 A211957 A211958

Keyword

nonn,easy,tabl

Author

Peter Bala, Apr 30 2012

Status

approved