

A211957


Triangle of coefficients of a polynomial sequence related to the MorganVoyce 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 MorganVoyce 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, MorganVoyce polynomials


FORMULA

T(n,0) = 1 and for k > 0, T(n,k) = n/k*2^(k1)*binomial(n+k1,2*k1) = 2^(k1)*A208513(n,k).
O.g.f.: ((1t)t*x)/((1t)^22*t*x) = 1 + (1+x)*t + (1+4*x+2*x^2)*t^2 + ....
nth 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(n1,k)+2*T(n1,k1)T(n2,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 * A338397 A063983 A259985
Adjacent sequences: A211954 A211955 A211956 * A211958 A211959 A211960


KEYWORD

nonn,easy,tabl


AUTHOR

Peter Bala, Apr 30 2012


STATUS

approved



