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%I #15 Nov 01 2023 17:16:35
%S 1,1,3,13,67,380,2288,14351,92737,613063,4126289,28179766,194780822,
%T 1360053081,9578997279,67971291791,485464864401,3487203531460,
%U 25176899072984,182598098616625,1329716528758651,9718954060263384,71273846758123552,524279847227139350
%N G.f. satisfies A(x) = 1 + x*A(x)^4 - x^2*A(x)^5.
%C Compare to the g.f. G(x) for the ternary tree numbers (A001764): G(x) = 1 + x*G(x)^4 - x^2*G(x)^6 = 1 + x*G(x)^3.
%H Vaclav Kotesovec, <a href="/A200754/b200754.txt">Table of n, a(n) for n = 0..400</a>
%F Recurrence: 3*n*(n+1)*(3*n - 1)*(3*n + 1)*(1280*n^7 - 17280*n^6 + 94466*n^5 - 266799*n^4 + 407516*n^3 - 311946*n^2 + 81648*n + 12420)*a(n) = n*(281600*n^10 - 3942400*n^9 + 22465080*n^8 - 65757900*n^7 + 98435187*n^6 - 46293273*n^5 - 73831865*n^4 + 131625925*n^3 - 85490202*n^2 + 25470288*n - 2915460)*a(n-1) + (263680*n^11 - 4087040*n^10 + 26682396*n^9 - 96110406*n^8 + 210443037*n^7 - 293037231*n^6 + 270493529*n^5 - 181862299*n^4 + 103208358*n^3 - 47593224*n^2 + 12881700*n - 1360800)*a(n-2) - 5*(640000*n^11 - 10880000*n^10 + 78609000*n^9 - 311686500*n^8 + 721945299*n^7 - 918913929*n^6 + 367548335*n^5 + 642045653*n^4 - 1081692258*n^3 + 697174344*n^2 - 208955268*n + 24222240)*a(n-3) + 5*(5*n - 18)*(5*n - 16)*(5*n - 14)*(5*n - 12)*(1280*n^7 - 8320*n^6 + 17666*n^5 - 8869*n^4 - 15820*n^3 + 22148*n^2 - 9282*n + 1305)*a(n-4). - _Vaclav Kotesovec_, Nov 18 2017
%F a(n) ~ s*sqrt((2*r*s-1) / (2*Pi*(5*r*s-3))) / (2*n^(3/2)*r^n), where r = 0.1272568969777848138753091632571986265610307654216... and s = 1.358291097397172238669759690645074441686961930838... are roots of the system of equations s + r^2*s^5 = 1 + r*s^4, 1 + 5*r^2*s^4 = 4*r*s^3. - _Vaclav Kotesovec_, Nov 18 2017
%F a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n+3*k+1,k) * binomial(k,n-k)/(n+3*k+1). - _Seiichi Manyama_, Nov 01 2023
%e G.f.: A(x) = 1 + x + 3*x^2 + 13*x^3 + 67*x^4 + 380*x^5 + 2288*x^6 +...
%e Related expansions:
%e A(x)^4 = 1 + 4*x + 18*x^2 + 92*x^3 + 515*x^4 + 3068*x^5 + 19092*x^6 +...
%e A(x)^5 = 1 + 5*x + 25*x^2 + 135*x^3 + 780*x^4 + 4741*x^5 + 29915*x^6 +...
%e where a(2) = 4 - 1; a(3) = 18 - 5; a(4) = 92 - 25; a(5) = 515 - 135; ...
%o (PARI) {a(n)=local(A=1+x);for(i=1,n,A=1+x*A^4-x^2*A^5+x*O(x^n));polcoeff(A,n)}
%o (PARI) a(n) = sum(k=0, n, (-1)^(n-k)*binomial(n+3*k+1, k)*binomial(k, n-k)/(n+3*k+1)); \\ _Seiichi Manyama_, Nov 01 2023
%Y Cf. A200753, A200755, A365180.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Nov 21 2011
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