A323320 - OEIS

%I #5 Jan 10 2019 22:22:01

%S 1,189,136341,165866949,274513563621,564389814803319,

%T 1373687351977035681,3844220718032111632869,

%U 12130905677234774784280281,42569255610714760893622565829,164374338314267349285576891426201,692583662656534583930262265650693159,3162450027762781275258550249673787013761,15558457725978409248029649314240444710279749,82059484588450416190385956503916602281112899421

%N G.f. A(x) satisfies: 1 = Sum_{n>=0} ( (1+x)^n - 19*x*A(x) )^n * 9^n / 10^(n+1).

%F G.f. A(x) satisfies the following identities.

%F (1) 1 = Sum_{n>=0} ( (1+x)^n - 19*x*A(x) )^n * 9^n / 10^(n+1).

%F (2) 1 = Sum_{n>=0} (1+x)^(n^2) * 9^n / (10 + 171*x*A(x)*(1+x)^n)^(n+1).

%e G.f.: A(x) = 1 + 189*x + 136341*x^2 + 165866949*x^3 + 274513563621*x^4 + 564389814803319*x^5 + 1373687351977035681*x^6 + 3844220718032111632869*x^7 + ...

%e such that

%e 1 = 1/10 + ((1+x) - 19*x*A(x))*9/10^2 + ((1+x)^2 - 19*x*A(x))^2*9^2/10^3 + ((1+x)^3 - 19*x*A(x))^3*9^3/10^4 + ((1+x)^4 - 19*x*A(x))^4*9^4/10^5 + ...

%e Also,

%e 1 = 1/(10 + 171*x*A(x)) + (1+x)*9/(10 + 171*x*A(x)*(1+x))^2 + (1+x)^4*9^2/(10 + 171*x*A(x)*(1+x)^2)^3 + (1+x)^9*9^3/(10 + 171*x*A(x)*(1+x)^3)^4 + ...

%o (PARI) \p120

%o {A=vector(1); A[1]=1; for(i=1,20, A = concat(A,0);

%o A[#A] = round( Vec( sum(n=0, 2400, ( (1+x +x*O(x^#A))^n - 19*x*Ser(A) )^n * 9^n/10^(n+1)*1.)/171 ) )[#A+1]); A}

%Y Cf. A301435, A303288, A323314, A323315, A323316, A323317, A323318, A323319, A323321.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jan 10 2019