cp's OEIS Frontend

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

%I A301312 #19 Oct 10 2020 07:33:56
%S A301312 1,15,818,75237,9704172,1610219061,326647152627,78322576680405,
%T A301312 21671112063131181,6796070676619188507,2382079473846244973676,
%U A301312 922852837565535061715211,391585126460697011112411576,180608810765895380940822677853,89966485333525032479641295540142,48135006154108946423894447901622257,27530139433795469892705229664410072563,16761420605156187498505881406969283279001
%N A301312 G.f.: Sum_{n>=0} ( (1+x)^n + (1+2*x)^n )^n / 3^(n+1).
%H A301312 Paul D. Hanna, <a href="/A301312/b301312.txt">Table of n, a(n) for n = 0..100</a>
%F A301312 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 A301312 G.f.: Sum_{n>=0} [ Sum_{k=0..n} binomial(n,k) * (1+2^k)*x^k ]^n / 3^(n+1).
%F A301312 a(n) ~ c * d^n * n! / sqrt(n), where d = 36.9010493125536756798917509741716959... and c = 0.4484222753815457836094869794957853521... - _Vaclav Kotesovec_, Oct 10 2020
%e A301312 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 A301312 such that
%e A301312 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 A301312 Equivalently,
%e A301312 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 A301312 Cf. A301307, A301310.
%K A301312 nonn
%O A301312 0,2
%A A301312 _Paul D. Hanna_, Mar 18 2018