A276314 - OEIS

%I #24 Oct 15 2019 12:22:18

%S 1,1,5,20,104,546,3066,17655,104555,630773,3867617,24020932,150827740,

%T 955808680,6105327912,39268000188,254093573088,1652984379150,

%U 10804631902350,70925539707330,467373389649870,3090558380977020,20501504119375500,136392970090612950

%N G.f. A(x) satisfies: x = A(x)-A(x)^2-3*A(x)^3.

%H Michael De Vlieger, <a href="/A276314/b276314.txt">Table of n, a(n) for n = 1..1181</a>

%H Elżbieta Liszewska, Wojciech Młotkowski, <a href="https://arxiv.org/abs/1907.10725">Some relatives of the Catalan sequence</a>, arXiv:1907.10725 [math.CO], 2019.

%H Thomas M. Richardson, <a href="http://arxiv.org/abs/1609.01193">The three 'R's and the Riordan dual</a>, arXiv:1609.01193 [math.CO], 2016.

%F G.f.: Series_Reversion(x-x^2-3*x^3)

%F Conjecture: +169*n*(n+2)*(n-1)*a(n) +13*(n-1) *(13*n^2+26*n-220) *a(n-1) +(-7277*n^3+13423*n^2+43814*n-81700) *a(n-2) -27*(3*n-10) *(3*n-8) *(71*n+197)*a(n-3)=0. - _R. J. Mathar_, Sep 17 2016

%F a(n) ~ (29 + 20*sqrt(10))^(n - 1/2) / (2^(5/4) * 5^(1/4) * sqrt(Pi) * n^(3/2) * 13^(n - 1/2)). - _Vaclav Kotesovec_, Aug 22 2017

%e G.f.: A(x) = x+x^2+5*x^3+20*x^4+104*x^5+546*x^6+3066*x^7+... Related Expansions:

%e A(x)^2=x^2+2*x^3+11*x^4+50*x^5+273*x^6+1500*x^7+8664*x^8+...

%e A(x)^3=x^3+3*x^4+18*x^5+91*x^6+522*x^7+2997*x^8+17831*x^9+...

%t Rest[CoefficientList[InverseSeries[Series[x - x^2 - 3*x^3, {x, 0, 20}], x],x]] (* _Vaclav Kotesovec_, Aug 22 2017 *)

%o (PARI) {a(n)=polcoeff(serreverse(x - x^2 - 3*x^3 + x^2*O(x^n)), n)}

%o for(n=1, 30, print1(a(n), ", "))

%Y Cf. A250886, A276310, A276315, A276316.

%K nonn,easy

%O 1,3

%A _Tom Richardson_, Aug 29 2016