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%I #19 Jul 26 2023 18:02:25
%S 1,1,2,7,27,112,492,2243,10513,50353,245353,1212398,6061225,30601910,
%T 155808915,799096655,4124491215,21408066097,111672838857,585128521777,
%U 3078178384457,16252057372887,86089680204939,457400940705274,2436895852070559,13015917111573039
%N G.f. satisfies A(x) = 1 + x*A(x) + x^2*A(x)^5.
%C Compare to a g.f. C(x) of Catalan numbers: C(x) = 1 + x*C(x) + x^2*C(x)^3.
%H Vaclav Kotesovec, <a href="/A182454/b182454.txt">Table of n, a(n) for n = 0..400</a>
%F G.f.: A(x) = sqrt( (1/x)*Series_Reversion( (1 + 2*x - 2*x^3 - sqrt(1 + 4*x - 4*x^3))/(2*x) ) ).
%F G.f. satisfies: A(x) = G(x*A(x)) where G(x) = A(x/G(x)) is the g.f. of A019497 (number of ternary search trees on n keys).
%F G.f. satisfies: A(x) = (1 + x*A(x)) * (1 + x*A(x)^4) / (1 + x*A(x)^3).
%F Recurrence: 64*(n-1)*n*(2*n - 1)*(2*n + 1)*(5*n - 17)*(5*n - 14)*(5*n - 12)*(5*n - 11)*(5*n - 9)*(5*n - 7)*a(n) = 32*(n-1)*(2*n - 1)*(5*n - 17)*(5*n - 14)*(5*n - 12)*(2500*n^5 - 16000*n^4 + 37400*n^3 - 38660*n^2 + 16767*n - 2304)*a(n-1) + (5*n - 17)*(5*n - 8)*(353125*n^8 - 4590625*n^7 + 26079625*n^6 - 84463075*n^5 + 169363570*n^4 - 212446228*n^3 + 159705192*n^2 - 64147968*n + 10184832)*a(n-2) + 8*(5*n - 2)*(1000000*n^9 - 19100000*n^8 + 158791250*n^7 - 752940875*n^6 + 2239835525*n^5 - 4325771435*n^4 + 5410989493*n^3 - 4216402206*n^2 + 1852118136*n - 348425280)*a(n-3) - 8*(5*n - 7)*(5*n - 4)*(5*n - 2)*(20000*n^7 - 368000*n^6 + 2847450*n^5 - 11988080*n^4 + 29592479*n^3 - 42711795*n^2 + 33256206*n - 10724400)*a(n-4) + 8*(n-4)*(2*n - 5)*(4*n - 19)*(4*n - 13)*(5*n - 12)*(5*n - 9)*(5*n - 7)*(5*n - 6)*(5*n - 4)*(5*n - 2)*a(n-5). - _Vaclav Kotesovec_, Nov 18 2017
%F a(n) ~ sqrt((1 + 2*r*s^4) / (10*Pi)) / (2*s * n^(3/2) * r^(n + 1/2)), where r = 0.1762643878022406506907195466376048222228890731329... and s = 1.517477187449684643254531724911215527841313263152... are roots of the system of equations 1 + r*s + r^2*s^5 = s, r + 5*r^2*s^4 = 1. - _Vaclav Kotesovec_, Nov 18 2017
%F a(n) = Sum_{k=0..floor(n/2)} binomial(n+3*k,k) * binomial(n+2*k,n-2*k) / (4*k+1). - _Seiichi Manyama_, Jul 26 2023
%e G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 27*x^4 + 112*x^5 + 492*x^6 +..
%e Related expansions:
%e A(x)^3 = 1 + 3*x + 9*x^2 + 34*x^3 + 141*x^4 + 615*x^5 + 2792*x^6 +...
%e A(x)^4 = 1 + 4*x + 14*x^2 + 56*x^3 + 241*x^4 + 1080*x^5 + 4998*x^6 +...
%e A(x)^5 = 1 + 5*x + 20*x^2 + 85*x^3 + 380*x^4 + 1751*x^5 + 8270*x^6 +...
%o (PARI) {a(n)=local(A=1+x);for(i=1,n,A=1+x*A+x^2*A^5+x*O(x^n));polcoeff(A,n)}
%o (PARI) {a(n)=local(A=1+x); for(i=1, n, A=(1+x*A)*(1+x*A^4)/((1+x*A^3)*1+x*O(x^n))); polcoeff(A, n)}
%o (PARI) {a(n)=polcoeff((1/x)*serreverse(x^2/serreverse((sqrt(1+4*x-4*x^3+x^2*O(x^n))-1)/2)),n)}
%o (PARI) {a(n)=polcoeff(sqrt((1/x)*serreverse((1+2*x-2*x^3-sqrt(1+4*x-4*x^3+x^3*O(x^n)))/(2*x))),n)}
%o for(n=0, 40, print1(a(n), ", "))
%Y Cf. A019497.
%Y Cf. A000045, A000108, A001006, A186996.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Apr 29 2012
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