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A211956 Coefficients of a sequence of polynomials related to the Morgan-Voyce polynomials.

%I A211956 #12 May 13 2025 11:34:24
%S A211956 1,1,1,1,1,2,1,4,2,1,6,4,1,9,12,4,1,12,20,8,1,16,40,32,8,1,20,60,56,
%T A211956 16,1,25,100,140,80,16,1,30,140,224,144,32,1,36,210,448,432,192,32,1,
%U A211956 42,280,672,720,352,64,1,49,392,1176,1680,1232,448,64
%N A211956 Coefficients of a sequence of polynomials related to the Morgan-Voyce polynomials.
%C A211956 The row generating polynomials R(n,x) of A211955 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), where b(n,x) := Sum_{k = 0..n} binomial(n+k,2*k)*x^k are the Morgan-Voyce polynomials of A085478. This triangle lists the coefficients in ascending powers of x of the polynomials P(n,x).
%C A211956 The odd numbered rows of the present triangle produce triangle A123519; the even numbered row entries are recorded separately in A211957 and appear to equal the unsigned and row reversed form of A204021. The even numbered rows with a factor of 2^(k-1) removed from the k-th column entries produce triangle A208513.
%H A211956 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Morgan-VoycePolynomials.html">Morgan-Voyce polynomials</a>
%F A211956 T(n,0) = 1; for k > 0, T(2*n,k) = 2^k * binomial(n+k,2*k) = A123519(n,k);
%F A211956 for k > 0, T(2*n-1,k) = n/(n+k)*(2^k)*binomial(n+k,2*k) = 2^(k-1)*A208513(n,k).
%F A211956 O.g.f.: ((1+t)*(1-t^2)-t^2*x)/((1-t^2)^2-2*t^2*x) = 1 + t + (1+x)*t^2 + (1+2*x)*t^3 + (1+4*x+2*x^2)*t^4 + ....
%F A211956 Row generating polynomials: 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), where b(n,x) := Sum_{k = 0..n} binomial(n+k,2*k)*x^k are the Morgan-Voyce polynomials of A085478.
%F A211956 The product P(n,x)*P(n+1,x) is the n-th row polynomial of A211955.
%F A211956 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).
%F A211956 Other representations for the row polynomials include
%F A211956 P(2*n,x) = 1/2*(1+x+sqrt(x^2+2*x))^n + 1/2*(1+x-sqrt(x^2+2*x))^n;
%F A211956 P(2*n,x) = n*Sum_{k = 0..n}(-1)^(n-k)/(n+k) * binomial(n+k,2*k) * (2*x+4)^k for n >= 1;
%F A211956 P(2*n+1,x) = (2*n+1)*Sum_{k=0..n} (-1)^(n-k)/(n+k+1) * binomial(n+k+1,2*k+1) * (2*x+4)^k.
%F A211956 Recurrence equation: P(n+1,x)*P(n-2,x) - P(n,x)*P(n-1,x) = x.
%F A211956 Row sums A005246(n+2).
%e A211956 Triangle begins
%e A211956 .n\k.|..0....1....2....3....4
%e A211956 = = = = = = = = = = = = = = =
%e A211956 ..0..|..1
%e A211956 ..1..|..1
%e A211956 ..2..|..1....1
%e A211956 ..3..|..1....2
%e A211956 ..4..|..1....4....2
%e A211956 ..5..|..1....6....4
%e A211956 ..6..|..1....9...12....4
%e A211956 ..7..|..1...12...20....8
%e A211956 ..8..|..1...16...40...32....8
%e A211956 ..9..|..1...20...60...56...16
%e A211956 ...
%Y A211956 Cf. A005246 (row sums), A085478, A123519, A204021, A208513, A211955, A211957.
%K A211956 nonn,easy,tabf
%O A211956 0,6
%A A211956 _Peter Bala_, Apr 30 2012