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

1,6

Links

Table of n, a(n) for n=1..35.

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

Prog

(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))}

Crossrefs

Sequence in context: A183929 A365252 A003522 * A119970 A326509 A082775

Adjacent sequences: A112802 A112803 A112804 * A112806 A112807 A112808

Keyword

nonn

Author

Michael Somos, Sep 20 2005

Status

approved