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A112805
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Expansion of solution of functional equation.
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3
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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
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OFFSET
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1,6
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LINKS
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FORMULA
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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
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PROG
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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))}
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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