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A323318 G.f. A(x) satisfies: 1 = Sum_{n>=0} ( (1+x)^n - 15*x*A(x) )^n * 7^n / 8^(n+1). 9

%I

%S 1,119,53473,40508503,41741036561,53428266259151,80958982980046129,

%T 141048455946249441191,277099512762218200167617,

%U 605370915659340921493495687,1455268739680049030318517763457,3817384299846582450604884256739951,10851817459553385455156107655677525601,33237713019302068995081812342685224005719,109138923772997447194531532072327732171764385

%N G.f. A(x) satisfies: 1 = Sum_{n>=0} ( (1+x)^n - 15*x*A(x) )^n * 7^n / 8^(n+1).

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

%F (1) 1 = Sum_{n>=0} ( (1+x)^n - 15*x*A(x) )^n * 7^n / 8^(n+1).

%F (2) 1 = Sum_{n>=0} (1+x)^(n^2) * 7^n / (8 + 105*x*A(x)*(1+x)^n)^(n+1).

%e G.f.: A(x) = 1 + 119*x + 53473*x^2 + 40508503*x^3 + 41741036561*x^4 + 53428266259151*x^5 + 80958982980046129*x^6 + 141048455946249441191*x^7 + ...

%e such that

%e 1 = 1/8 + ((1+x) - 15*x*A(x))*7/8^2 + ((1+x)^2 - 15*x*A(x))^2*7^2/8^3 + ((1+x)^3 - 15*x*A(x))^3*7^3/8^4 + ((1+x)^4 - 15*x*A(x))^4*7^4/8^5 + ...

%e Also,

%e 1 = 1/(8 + 105*x*A(x)) + (1+x)*7/(8 + 105*x*A(x)*(1+x))^2 + (1+x)^4*7^2/(8 + 105*x*A(x)*(1+x)^2)^3 + (1+x)^9*7^3/(8 + 105*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, 2000, ( (1+x +x*O(x^#A))^n - 15*x*Ser(A) )^n * 7^n/8^(n+1)*1.)/105 ) )[#A+1]); A}

%Y Cf. A301435, A303288, A323314, A323315, A323316, A323317, A323319, A323320, A323321.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jan 10 2019

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Last modified October 1 09:13 EDT 2022. Contains 357140 sequences. (Running on oeis4.)