|
%I #9 Aug 28 2023 10:51:19
%S 1,1,1,1,1,2,5,11,21,37,66,127,261,545,1119,2255,4529,9202,18989,
%T 39566,82614,172272,359159,750699,1575649,3319942,7012833,14834345,
%U 31414423,66619692,141526459,301202699,642055773,1370429491,2928418794
%N Expansion of solution of functional equation.
%F G.f. A(x)=y satisfies y=x+(xy)/(1-(xy)^2).
%F Series reversion of g.f. A(x) is -A(-x).
%F D-finite with recurrence 16*(n-1)*(1240223*n -6702246)*(n+1)*a(n) +8*(2480446*n^3 -39153654*n^2 +84032501*n +6702246)*a(n-1) +4*(-31385887*n^3 +335465133*n^2 -849400280*n +599382573)*a(n-2) +2*(29566778*n^3 -194324013*n^2 +26628520*n +491525637)*a(n-3) +6*(6680714*n^3 -167765708*n^2 +1031916951*n -1815562235)*a(n-4) +2*(-155507474*n^3+2303856267*n^2 -11150676133*n +17639612322)*a(n-5) +12*(-41500633*n^3 +711522713*n^2 -3909195761*n +6849836674)*a(n-6) +2*(-122699626*n^3 +2534143032*n^2 -16977163481*n +36816733731)*a(n-7) +(n-9)*(33105709*n^2 -338697405*n +802704794)*a(n-8) +8*(n-10)*(9921784*n-30998433)*(n-8)*a(n-9) +4*(n-11)*(13262141*n-65833637)*(n-9)*a(n-10)=0. - _R. J. Mathar_, Jul 20 2023
%F a(n+1) = Sum_{k=0..floor(n/4)} binomial(n-3*k-1,k) * binomial(n-2*k+1,n-4*k)/(n-2*k+1). - _Seiichi Manyama_, Aug 28 2023
%o (PARI) {a(n)=local(A); if(n<1, 0, A=x+O(x^2); for(k=1,n, A=x+subst(x/(1-x^2),x,x*A)); polcoeff(A,n))}
%K nonn
%O 1,6
%A _Michael Somos_, Sep 20 2005
|
