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%I #17 Jul 02 2014 08:28:25
%S 1,3,10,29,88,259,771,2284,6786,20191,60189,179654,536925,1606221,
%T 4808601,14403804,43163793,129388755,387946445,1163370778,3489117566,
%U 10465248174,31391306504,94164586011,282474177151,847381544167,2542059008323,7625998475474,22877623720244
%N G.f.: Sum_{n>=0} (2 + x^n)^n * x^n / (1-x)^(n+1).
%C What is limit ( a(n) - 3^n )^(1/n) ? (Value is near 2.0038 at n=3000.)
%C In general, for g.f.: Sum_{n>=0} (k + x^n)^n * x^n / (1-x)^(n+1), we have a(n)-(k+1)^n ~ n/(4*k+1) * ((1+sqrt(4*k+1))/2)^n, and limit is equal to (1+sqrt(4*k+1))/2. - Vaclav Kotesovec, Jul 02 2014
%H Vaclav Kotesovec, <a href="/A244615/b244615.txt">Table of n, a(n) for n = 0..500</a>
%F G.f.: Sum_{n>=0} x^(n*(n+1)) / (1-x - 2*x^(n+1))^(n+1).
%F G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k) * (2 + x^k)^k.
%F G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k) * (1 + 2*x^k)^(n-k) * x^(k^2).
%F a(n)-3^n ~ n/9 * 2^n. - _Vaclav Kotesovec_, Jul 02 2014
%e G.f.: A(x) = 1 + 3*x + 10*x^2 + 29*x^3 + 88*x^4 + 259*x^5 + 771*x^6 +...
%e where we have the series identity:
%e A(x) = 1/(1-x) + (2+x)*x/(1-x)^2 + (2+x^2)^2*x^2/(1-x)^3 + (2+x^3)^3*x^3/(1-x)^4 + (2+x^4)^4*x^4/(1-x)^5 +...+ (2 + x^n)^n * x^n / (1-x)^(n+1) +...
%e A(x) = 1/(1-3*x) + x^2/(1-x-2*x^2)^2 + x^6/(1-x-2*x^3)^3 + x^12/(1-x-2*x^4)^4 + x^20/(1-x-2*x^5)^5 + x^30/(1-x-2*x^6)^6 +...+ x^(n*(n+1)) / (1-x - 2*x^(n+1))^(n+1) +...
%e as well as the binomial identity:
%e A(x) = 1 + x*(1 + (2+x)) + x^2*(1 + 2*(2+x) + (2+x^2)^2) + x^3*(1 + 3*(2+x) + 3*(2+x^2)^2 + (2+x^3)^3) + x^4*(1 + 4*(2+x) + 6*(2+x^2)^2 + 4*(2+x^3)^3 + (2+x^4)^4) + x^5*(1 + 5*(2+x) + 10*(2+x^2)^2 + 10*(2+x^3)^3 + 5*(2+x^4)^4 + (2+x^5)^5) +...+ x^n * Sum_{k=0..n} binomial(n,k) * (2+x^k)^k +...
%e A(x) = 1 + x*(3 + x) + x^2*(3^2 + 2*(1+2*x)*x + x^4) + x^3*(3^3 + 3*(1+2*x)^2*x + 3*(1+2*x^2)*x^4 + x^9) + x^4*(3^4 + 4*(1+2*x)^3*x + 6*(1+2*x^2)^2*x^4 + 4*(1+2*x^3)*x^9 + x^16) + x^5*(3^5 + 5*(1+2*x)^4*x + 10*(1+2*x^2)^3*x^4 + 10*(1+2*x^3)^2*x^9 + 5*(1+2*x^4)*x^16 + x^25) +...+ x^n * Sum_{k=0..n} binomial(n,k) * (1+2*x^k)^(n-k) * x^(k^2) +...
%t Table[SeriesCoefficient[Sum[x^(j*(j+1))/(1-x-2*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, (2 + 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 - 2*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)*(2+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+2*x^k)^(m-k)*x^(k^2)) +x*O(x^n)), n)}
%o for(n=0, 40, print1(a(n), ", "))
%Y Cf. A243988, A244616, A244617, A244618, A243919.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jul 02 2014
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