A295538 - OEIS

%I #22 Apr 04 2024 09:47:38

%S 1,1,4,32,290,2894,30624,337602,3835395,44588657,527903344,6343105788,

%T 77153875396,948150877136,11754481411170,146829606548967,

%U 1846232392749705,23349436820785896,296822925777158448,3790612373731979898,48608130217245939310,625636961746371994680,8079794260209350950338,104667769434155291997329,1359712949654853908780859,17709395639599543591065564

%N G.f. A(x) satisfies A(x)^2 = 1 + x + x*A(x)^9.

%C Note that the function G(x) = 1 + x*G(x)^4 (g.f. of A002293) also satisfies the condition: G(x) = 1/G(-x*G(x)^7).

%H Paul D. Hanna, <a href="/A295538/b295538.txt">Table of n, a(n) for n = 0..500</a>

%F G.f. A(x) satisfies:

%F (1) A(x) = 1 + Series_Reversion( x/(1 + 4*x + 16*x^2 + 34*x^3 + 46*x^4 + 40*x^5 + 22*x^6 + 7*x^7 + x^8) ).

%F (2) F(A(x)) = x such that F(x) = -(1 - x^2)/(1 + x^9).

%F (3) A(x) = 1 / A(-x*A(x)^7).

%F a(n) ~ sqrt((1 + s^9)/(7*Pi)) / (2*n^(3/2)*r^(n - 1/2)), where r = 0.07223758934231429961770532152600550503126361567079... and s = 1.174134228398636389214738979941451774138268651734... are real roots of the system of equations 1 + r + r*s^9 = s^2, 9*r*s^7 = 2. - _Vaclav Kotesovec_, Nov 28 2017

%F From _Seiichi Manyama_, Apr 04 2024: (Start)

%F G.f. A(x) satisfies A(x) = 1 + x * (1 - A(x) + A(x)^2 - A(x)^3 + A(x)^4 - A(x)^5 + A(x)^6 - A(x)^7 + A(x)^8).

%F a(n) = Sum_{k=0..n} binomial(n,k) * binomial(9*k/2+1/2,n)/(9*k+1). (End)

%e G.f.: A(x) = 1 + x + 4*x^2 + 32*x^3 + 290*x^4 + 2894*x^5 + 30624*x^6 + 337602*x^7 + 3835395*x^8 + 44588657*x^9 + 527903344*x^10 + 6343105788*x^11 + 77153875396*x^12 + 948150877136*x^13 + 11754481411170*x^14 + 146829606548967*x^15 +...

%e such that A(x)^2 = 1+x + x*A(x)^9.

%e RELATED SERIES.

%e A(x)^2 = 1 + 2*x + 9*x^2 + 72*x^3 + 660*x^4 + 6624*x^5 + 70380*x^6 + 778164*x^7 + 8860302*x^8 + 103187376*x^9 + 1223410846*x^10 +...

%e A(x)^9 = 1 + 9*x + 72*x^2 + 660*x^3 + 6624*x^4 + 70380*x^5 + 778164*x^6 + 8860302*x^7 + 103187376*x^8 + 1223410846*x^9 + 14717253672*x^10 +...

%e A(-x*A(x)^7) = 1 - x - 3*x^2 - 25*x^3 - 221*x^4 - 2187*x^5 - 22989*x^6 - 252237*x^7 - 2855304*x^8 - 33101152*x^9 - 391010608*x^10 +...

%e which equals 1/A(x).

%o (PARI) {a(n) = my(A=1+x); for(i=1, n, A = sqrt(1+x + x*A^9 +x*O(x^n)) ); polcoeff(A, n)}

%o for(n=0, 30, print1(a(n), ", "))

%Y Cf. A259757, A295537, A366453.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Nov 27 2017