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%I #14 Jul 02 2014 08:28:51
%S 1,4,17,66,265,1046,4165,16561,66017,263377,1051774,4202363,16797551,
%T 67159565,268561326,1074052233,4295730126,17181735211,68724026662,
%U 274888965526,1099538433297,4398111321331,17592342396443,70369120593536,281475881240563,1125902076700152,4503604824386737
%N G.f.: Sum_{n>=0} (3 + x^n)^n * x^n / (1-x)^(n+1).
%C What is limit ( a(n) - 4^n )^(1/n) ? (Value is near 2.3069 at n=3000.)
%C Limit is equal to (1+sqrt(13))/2. - _Vaclav Kotesovec_, Jul 02 2014
%H Vaclav Kotesovec, <a href="/A244616/b244616.txt">Table of n, a(n) for n = 0..500</a>
%F G.f.: Sum_{n>=0} x^(n*(n+1)) / (1-x - 3*x^(n+1))^(n+1).
%F G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k) * (3 + x^k)^k.
%F G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k) * (1 + 3*x^k)^(n-k) * x^(k^2).
%F a(n)-4^n ~ n/13 * ((1+sqrt(13))/2)^n. - _Vaclav Kotesovec_, Jul 02 2014
%e G.f.: A(x) = 1 + 4*x + 17*x^2 + 66*x^3 + 265*x^4 + 1046*x^5 + 4165*x^6 +...
%e where we have the series identity:
%e A(x) = 1/(1-x) + (3+x)*x/(1-x)^2 + (3+x^2)^2*x^2/(1-x)^3 + (3+x^3)^3*x^3/(1-x)^4 + (3+x^4)^4*x^4/(1-x)^5 +...+ (3 + x^n)^n * x^n / (1-x)^(n+1) +...
%e A(x) = 1/(1-4*x) + x^2/(1-x-3*x^2)^2 + x^6/(1-x-3*x^3)^3 + x^12/(1-x-3*x^4)^4 + x^20/(1-x-3*x^5)^5 + x^30/(1-x-3*x^6)^6 +...+ x^(n*(n+1)) / (1-x - 3*x^(n+1))^(n+1) +...
%e as well as the binomial identity:
%e A(x) = 1 + x*(1 + (3+x)) + x^2*(1 + 2*(3+x) + (3+x^2)^2) + x^3*(1 + 3*(3+x) + 3*(3+x^2)^2 + (3+x^3)^3) + x^4*(1 + 4*(3+x) + 6*(3+x^2)^2 + 4*(3+x^3)^3 + (3+x^4)^4) + x^5*(1 + 5*(3+x) + 10*(3+x^2)^2 + 10*(3+x^3)^3 + 5*(3+x^4)^4 + (3+x^5)^5) +...+ x^n * Sum_{k=0..n} binomial(n,k) * (3+x^k)^k +...
%e A(x) = 1 + x*(4 + x) + x^2*(4^2 + 2*(1+3*x)*x + x^4) + x^3*(4^3 + 3*(1+3*x)^2*x + 3*(1+3*x^2)*x^4 + x^9) + x^4*(4^4 + 4*(1+3*x)^3*x + 6*(1+3*x^2)^2*x^4 + 4*(1+3*x^3)*x^9 + x^16) + x^5*(4^5 + 5*(1+3*x)^4*x + 10*(1+3*x^2)^3*x^4 + 10*(1+3*x^3)^2*x^9 + 5*(1+3*x^4)*x^16 + x^25) +...+ x^n * Sum_{k=0..n} binomial(n,k) * (1+3*x^k)^(n-k) * x^(k^2) +...
%t Table[SeriesCoefficient[Sum[x^(j*(j+1))/(1-x-3*x^(j+1))^(j+1),{j,0,n}],{x,0,n}],{n,0,30}] (* _Vaclav Kotesovec_, Jul 02 2014 *)
%o (PARI) {a(n)=local(A); A=sum(m=0, n, (3 + x^m)^m * x^m / (1-x +x*O(x^n) )^(m+1) ); polcoeff(A, n)}
%o for(n=0, 40, print1(a(n), ", "))
%o (PARI) {a(n)=local(A); A=sum(m=0, sqrtint(n+1), x^(m*(m+1)) / (1-x - 3*x^(m+1) +x*O(x^n) )^(m+1) ); polcoeff(A, n)}
%o for(n=0, 40, print1(a(n), ", "))
%o (PARI) {a(n)=polcoeff(sum(m=0, n, x^m*sum(k=0, m, binomial(m, k)*(3+x^k)^k) +x*O(x^n)), n)}
%o for(n=0, 40, print1(a(n), ", "))
%o (PARI) {a(n)=polcoeff(sum(m=0, n, x^m*sum(k=0, m, binomial(m, k)*(1+3*x^k)^(m-k)*x^(k^2)) +x*O(x^n)), n)}
%o for(n=0, 40, print1(a(n), ", "))
%Y Cf. A243988, A244615, A244617, A244618, A243919.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jul 02 2014
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