A295498 - OEIS

%I #11 Nov 23 2017 06:08:51

%S 1,1,1,6,11,51,106,492,1115,5317,12912,62283,159146,767083,2036260,

%T 9765849,26735811,127447531,358219288,1696410364,4879284508,

%U 22946311567,67362378507,314520916727,940422623222,4359165612216,13252603911289,60989336178364,188258217816004,860270701616648,2692815154387672,12220594038311373,38750249291035303,174684318231133053,560585633201038635

%N G.f. A(x) satisfies: A(x) = 1 + x*A(x)^2 - x^2/A(x)^3.

%C Note that G(x) such that G(x) = 1 + x*G(x)^2 - x^2/G(x)^4 has negative coefficients.

%H Paul D. Hanna, <a href="/A295498/b295498.txt">Table of n, a(n) for n = 0..500</a>

%F G.f. A(x) satisfies: x^2 = A(x)^3 - A(x)^4 + x*A(x)^5.

%F a(n) ~ s * sqrt((2*r - s^5) / (Pi*(6*r - s^5))) / (2*n^(3/2)*r^n), where r = 0.2520932825034369933395854430839207204336449133515... and s = 1.957646455287329963796650229699010012363761835805... are real roots of the system of equations 1 + r*s^2 = r^2/s^3 + s, 3*r^2/s^4 + 2*r*s = 1. - _Vaclav Kotesovec_, Nov 23 2017

%e G.f.: A(x) = 1 + x + x^2 + 6*x^3 + 11*x^4 + 51*x^5 + 106*x^6 + 492*x^7 + 1115*x^8 + 5317*x^9 + 12912*x^10 + 62283*x^11 + 159146*x^12 + 767083*x^13 + 2036260*x^14 + 9765849*x^15 + 26735811*x^16 + 127447531*x^17 + 358219288*x^18 + 1696410364*x^19 + 4879284508*x^20 +...

%e such that A(x) = 1 + x*A(x)^2 - x^2/A(x)^3.

%e RELATED SERIES.

%e A(x)^2 = 1 + 2*x + 3*x^2 + 14*x^3 + 35*x^4 + 136*x^5 + 372*x^6 + 1430*x^7 + 4159*x^8 + 16242*x^9 + 49525*x^10 + 196040*x^11 + 618436*x^12 +...

%e A(x)^3 = 1 + 3*x + 6*x^2 + 25*x^3 + 75*x^4 + 276*x^5 + 868*x^6 + 3159*x^7 + 10293*x^8 + 37851*x^9 + 127023*x^10 + 472767*x^11 + 1622387*x^12 +...

%e A(x)^4 = 1 + 4*x + 10*x^2 + 40*x^3 + 135*x^4 + 496*x^5 + 1694*x^6 + 6144*x^7 + 21303*x^8 + 77636*x^9 + 273548*x^10 + 1005368*x^11 + 3591432*x^12 +...

%e A(x)^5 = 1 + 5*x + 15*x^2 + 60*x^3 + 220*x^4 + 826*x^5 + 2985*x^6 + 11010*x^7 + 39785*x^8 + 146525*x^9 + 532601*x^10 + 1969045*x^11 + 7208040*x^12 +...

%e where x^2 = A(x)^3 - A(x)^4 + x*A(x)^5.

%e Let F(x) be the series given by

%e F(x) = (1/x)*Series_Reversion(x*A(x)) = 1 - x + x^2 - 6*x^3 + 21*x^4 - 86*x^5 + 396*x^6 - 1812*x^7 + 8607*x^8 - 41958*x^9 + 207333*x^10 +...+ (-1)^n*A137966(n)*x^n +...

%e then F(x) = 1 - x + x^2*F(x)^6.

%o (PARI) {a(n)=local(A=1+x); for(i=1, n, A = 1 + x*A^2 - x^2/A^3  +x*O(x^n)); polcoeff(G=A, n)}

%o for(n=0, 40, print1(a(n), ", "))

%Y Cf. A137966, A295404, A295504.

%K nonn

%O 0,4

%A _Paul D. Hanna_, Nov 22 2017