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A301312 G.f.: Sum_{n>=0} ( (1+x)^n + (1+2*x)^n )^n / 3^(n+1). 3

%I

%S 1,15,818,75237,9704172,1610219061,326647152627,78322576680405,

%T 21671112063131181,6796070676619188507,2382079473846244973676,

%U 922852837565535061715211,391585126460697011112411576,180608810765895380940822677853,89966485333525032479641295540142,48135006154108946423894447901622257,27530139433795469892705229664410072563,16761420605156187498505881406969283279001

%N G.f.: Sum_{n>=0} ( (1+x)^n + (1+2*x)^n )^n / 3^(n+1).

%H Paul D. Hanna, <a href="/A301312/b301312.txt">Table of n, a(n) for n = 0..100</a>

%F G.f.: Sum_{n>=0} Sum_{k=0..n} binomial(n,k) * (1+x)^(n*(n-k)) * (1+2*x)^(n*k) / 3^(n+1).

%F G.f.: Sum_{n>=0} [ Sum_{k=0..n} binomial(n,k) * (1+2^k)*x^k ]^n / 3^(n+1).

%F a(n) ~ c * d^n * n! / sqrt(n), where d = 36.9010493125536756798917509741716959... and c = 0.4484222753815457836094869794957853521... - _Vaclav Kotesovec_, Oct 10 2020

%e G.f.: A(x) = 1 + 15*x + 818*x^2 + 75237*x^3 + 9704172*x^4 + 1610219061*x^5 + 326647152627*x^6 + 78322576680405*x^7 + 21671112063131181*x^8 + ...

%e such that

%e A(x) = 1/3 + ((1+x) + (1+2*x))/3^2 + ((1+x)^2 + (1+2*x)^2)^2/3^3 + ((1+x)^3 + (1+2*x)^3)^3/3^4 + ((1+x)^4 + (1+2*x)^4)^4/3^5 + ((1+x)^5 + (1+2*x)^5)^5/3^6 + ... + ((1+x)^n + (1+2*x)^n)^n / 3^(n+1) + ...

%e Equivalently,

%e A(x) = 1/3 + (2 + 3*x)/3^2 + (2 + 6*x + 5*x^2)^2/3^3 + (2 + 9*x + 15*x^2 + 9*x^3)^3/3^4 + (2 + 12*x + 30*x^2 + 36*x^3 + 17*x^4)^4/3^5 + (2 + 15*x + 50*x^2 + 90*x^3 + 85*x^4 + 33*x^5)^5/3^6 + ...

%Y Cf. A301307, A301310.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Mar 18 2018

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