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A211957 Triangle of coefficients of a polynomial sequence related to the Morgan-Voyce polynomials A085478. 4
1, 1, 1, 1, 4, 2, 1, 9, 12, 4, 1, 16, 40, 32, 8, 1, 25, 100, 140, 80, 16, 1, 36, 210, 448, 432, 192, 32, 1, 49, 392, 1176, 1680, 1232, 448, 64, 1, 64, 672, 2688, 5280, 5632, 3328, 1024, 128, 1, 81, 1080, 5544, 14256, 20592, 17472, 8640, 2304, 256, 1, 100, 1650, 10560, 34320, 64064, 72800, 51200, 21760, 5120, 512 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Triangle formed from the even numbered rows of A211956.

The coefficients of the  Morgan-Voyce polynomials b(n,x) := sum {k = 0..n} binomial(n+k,2*k)*x^k are listed in A085478. The rational functions 1/2*(b(2*n,2*x) + 1)/b(n,2*x) turn out to be integer polynomials. Their coefficients are listed in this triangle. These polynomials occur as factors of the row polynomials R(n,x) of A211955.

This triangle appears to be the row reverse of the unsigned triangle |A204021|.

LINKS

Table of n, a(n) for n=0..65.

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

FORMULA

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

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

n-th row polynomial R(n,x) = 1/2*(b(2*n,2*x) + 1)/b(n,2*x) = T(2*n,u), where u = sqrt((x+2)/2) and T(n,u) denotes the Chebyshev polynomial of the first kind.

T(n,k) = 2*T(n-1,k)+2*T(n-1,k-1)-T(n-2,k), T(0,0)=T(1,0)=T(1,1)=1, T(n,k)=0 if k<0 or if k>n. - Philippe Deléham, Nov 16 2013

EXAMPLE

Triangle begins

.n\k.|..0....1....2....3....4....5....6....7

= = = = = = = = = = = = = = = = = = = = = = =

..0..|..1

..1..|..1....1

..2..|..1....4....2

..3..|..1....9...12....4

..4..|..1...16...40...32....8

..5..|..1...25..100..140...80...16

..6..|..1...36..210..448..432..192...32

..7..|..1...49..392.1176.1680.1232..448...64

CROSSREFS

A085478, A111125, A204021, A211955, A211956.

Sequence in context: A160905 A208612 A183157 * A063983 A259985 A144084

Adjacent sequences:  A211954 A211955 A211956 * A211958 A211959 A211960

KEYWORD

nonn,easy,tabl

AUTHOR

Peter Bala, Apr 30 2012

STATUS

approved

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Last modified November 14 22:17 EST 2019. Contains 329134 sequences. (Running on oeis4.)