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A119020 Eigenvector of triangle A055151 of Motzkin polynomial coefficients, where A055151(n,k) = n!/((n-2k)!*k!*(k+1)!) for 0<=k<=[n/2], n>=0.

%I A119020 #7 Jan 25 2025 02:11:34
%S A119020 1,1,2,4,11,31,96,302,1023,3607,13318,50348,195361,772565,3112630,
%T A119020 12715692,52648847,220705119,937145214,4028239116,17522172021,
%U A119020 77071709841,342583183572,1537550150766,6961838925069,31774593686661
%N A119020 Eigenvector of triangle A055151 of Motzkin polynomial coefficients, where A055151(n,k) = n!/((n-2k)!*k!*(k+1)!) for 0<=k<=[n/2], n>=0.
%C A119020 Binomial transform is A119021. Inverse binomial transform is A119022.
%F A119020 Eigenvector: a(n) = Sum_{k=0..[n/2]} n!/((n-2k)!*k!*(k+1)!)*a(k), for n>=0, with a(0)=1.
%F A119020 G.f. satisfies: A(x) = A(-x/(1-2*x))/(1-2*x); i.e., 2nd inverse binomial transform equals A(-x).
%F A119020 G.f. satisfies: A(x/(1-x))/(1-x) = A(-x/(1-3*x))/(1-3*x).
%F A119020 G.f. of inverse binomial transform: A(x/(1+x))/(1+x) = B(x^2) where [x^n] B(x) = a(n)*C(2*n,n)/(n+1) = a(n)*A000108(n) and A000108=Catalan.
%e A119020 A(x) = 1 + x + 2*x^2 + 4*x^3 + 11*x^4 + 31*x^5 + 96*x^6 +...
%e A119020 A(x/(1+x))/(1+x) = 1 + x^2 + 2*2*x^4 + 4*5*x^6 + 11*14*x^8 +...+ a(n)*A000108(n)*x^(2n) +...
%o A119020 (PARI) {a(n)=if(n==0,1,sum(k=0,n\2,n!/((n-2*k)!*k!*(k+1)!)*a(k)))}
%Y A119020 Cf. A055151 (Motzkin polynomials), A119021 (binomial), A119022 (inverse binomial).
%K A119020 nonn
%O A119020 0,3
%A A119020 _Paul D. Hanna_, May 09 2006