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
R. J. Mathar, Recurrence for the Atkinson-Steenwijk Integrals for Resistors in the Infinite Triangular Lattice, vixra:2208.0111
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
KEYWORD
AUTHOR
Peter Bala, Apr 30 2012
STATUS
approved