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A211957 Triangle of coefficients of a polynomial sequence related to the Morgan-Voyce polynomials A085478.

%I A211957 #10 Feb 16 2025 08:33:17
%S A211957 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,
%T A211957 432,192,32,1,49,392,1176,1680,1232,448,64,1,64,672,2688,5280,5632,
%U A211957 3328,1024,128,1,81,1080,5544,14256,20592,17472,8640,2304,256,1,100,1650,10560,34320,64064,72800,51200,21760,5120,512
%N A211957 Triangle of coefficients of a polynomial sequence related to the Morgan-Voyce polynomials A085478.
%C A211957 Triangle formed from the even numbered rows of A211956.
%C A211957 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.
%C A211957 This triangle appears to be the row reverse of the unsigned triangle |A204021|.
%H A211957 R. J. Mathar, <a href="https://vixra.org/abs/2208.0111">Recurrence for the Atkinson-Steenwijk Integrals for Resistors in the Infinite Triangular Lattice</a>, vixra:2208.0111
%H A211957 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Morgan-VoycePolynomials.html">Morgan-Voyce polynomials</a>
%F A211957 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).
%F A211957 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 + ....
%F A211957 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.
%F A211957 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
%e A211957 Triangle begins
%e A211957 .n\k.|..0....1....2....3....4....5....6....7
%e A211957 = = = = = = = = = = = = = = = = = = = = = = =
%e A211957 ..0..|..1
%e A211957 ..1..|..1....1
%e A211957 ..2..|..1....4....2
%e A211957 ..3..|..1....9...12....4
%e A211957 ..4..|..1...16...40...32....8
%e A211957 ..5..|..1...25..100..140...80...16
%e A211957 ..6..|..1...36..210..448..432..192...32
%e A211957 ..7..|..1...49..392.1176.1680.1232..448...64
%Y A211957 A085478, A111125, A204021, A211955, A211956.
%K A211957 nonn,easy,tabl
%O A211957 0,5
%A A211957 _Peter Bala_, Apr 30 2012