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A211955 Triangle of coefficients of a polynomial sequence related to the Morgan-Voyce polynomials A085478. 9
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 (list; table; graph; refs; listen; history; text; internal format)
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: A181897 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

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Last modified November 20 02:34 EST 2019. Contains 329323 sequences. (Running on oeis4.)