A112805 Expansion of solution of functional equation.
1, 1, 1, 1, 1, 2, 5, 11, 21, 37, 66, 127, 261, 545, 1119, 2255, 4529, 9202, 18989, 39566, 82614, 172272, 359159, 750699, 1575649, 3319942, 7012833, 14834345, 31414423, 66619692, 141526459, 301202699, 642055773, 1370429491, 2928418794
Offset: 1
Keywords
Programs
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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))}
Formula
G.f. A(x)=y satisfies y=x+(xy)/(1-(xy)^2).
Series reversion of g.f. A(x) is -A(-x).
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
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