|
%I #14 Oct 14 2019 11:28:55
%S 1,1,2,11,71,515,3997,32488,273009,2352724,20678966,184660333,
%T 1670619561,15279692008,141048655988,1312429249996,12296515232446,
%U 115909188223053,1098444610424929,10459429664510189,100021237512559055,960168745226226195,9249466125601138425
%N G.f. A(x) satisfies: x = Sum_{n>=1} 1/A(x)^(4*n) * Product_{k=1..n} (1 - 1/A(x)^(2*k-1)).
%C Compare the g.f. to the identity:
%C G(x) = Sum_{n>=0} 1/G(x)^(2*n) * Product_{k=1..n} (1 - 1/G(x)^(2*k-1))
%C which holds for all power series G(x) such that G(0)=1.
%H G. C. Greubel, <a href="/A214692/b214692.txt">Table of n, a(n) for n = 0..980</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.
%F G.f. A(x) satisfies:
%F (1) 1+x = A(y) where y = x - 2*x^2 - 3*x^3 - x^4, which is the g.f. of row 2 in triangle A214690.
%F (2) x = Sum_{n>=1} 1/A(x)^(n*(n+4)) * Product_{k=1..n} (A(x)^(2*k-1) - 1).
%F (3) 1+x = A(x)^2 + A(x)^3 - A(x)^4. - _Paul D. Hanna_, Nov 15 2014
%F a(n) ~ sqrt(145/sqrt(41)-21) * ((213+41*sqrt(41))/46)^n / (16 * sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Nov 29 2014
%e G.f.: A(x) = 1 + x + 2*x^2 + 11*x^3 + 71*x^4 + 515*x^5 + 3997*x^6 +...
%e The g.f. satisfies:
%e x = (A(x)-1)/A(x)^5 + (A(x)-1)*(A(x)^3-1)/A(x)^12 + (A(x)-1)*(A(x)^3-1)*(A(x)^5-1)/A(x)^21 + (A(x)-1)*(A(x)^3-1)*(A(x)^5-1)*(A(x)^7-1)/A(x)^32 +
%e (A(x)-1)*(A(x)^3-1)*(A(x)^5-1)*(A(x)^7-1)*(A(x)^9-1)/A(x)^45 +...
%e Related expansions.
%e A(x)^2 = 1 + 2*x + 5*x^2 + 26*x^3 + 168*x^4 + 1216*x^5 + 9429*x^6 +...
%e A(x)^3 = 1 + 3*x + 9*x^2 + 46*x^3 + 297*x^4 + 2148*x^5 + 16649*x^6 +...
%e A(x)^4 = 1 + 4*x + 14*x^2 + 72*x^3 + 465*x^4 + 3364*x^5 + 26078*x^6 +...
%e where 1+x = A(x)^2 + A(x)^3 - A(x)^4.
%t CoefficientList[1+InverseSeries[Series[x - 2*x^2 - 3*x^3 - x^4, {x, 0, 20}], x],x] (* _Vaclav Kotesovec_, Nov 29 2014 *)
%o (PARI) {a(n)=if(n<0, 0, polcoeff(1 + serreverse(x - 2*x^2 - 3*x^3 - x^4 +x^2*O(x^n)), n))}
%o for(n=0, 25, print1(a(n), ", "))
%o (PARI) {a(n)=local(A=[1, 1]); for(i=1, n, A=concat(A, 0); A[#A]=-polcoeff(sum(m=1, #A, 1/Ser(A)^(4*m)*prod(k=1, m, 1-1/Ser(A)^(2*k-1))), #A-1)); A[n+1]}
%o for(n=0, 25, print1(a(n), ", "))
%o (PARI) /* From 1+x = A(x)^2 + A(x)^3 - A(x)^4: */
%o {a(n)=local(A=[1,1]);for(i=1,n,A=concat(A,0);A[#A]=-Vec(Ser(A)^2+Ser(A)^3-Ser(A)^4)[#A]);A[n+1]}
%o for(n=0,25,print1(a(n) ,", "))
%Y Cf. A214690, A214693, A214694, A214695, A001002 (variant).
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jul 26 2012
| 