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%I #29 Jul 02 2014 08:27:54
%S 1,2,5,10,21,42,85,169,333,655,1286,2519,4935,9675,18982,37285,73346,
%T 144509,285158,563546,1115309,2210243,4385443,8710876,17319387,
%U 34464792,68634821,136771603,272703704,543995341,1085620097,2167267262,4327886353,8644663133,17270784312,34510656589,68969830833
%N G.f.: Sum_{n>=0} (1 + x^n)^n * x^n / (1-x)^(n+1).
%C What is limit ( a(n) - 2^n )^(1/n) ? (Value is near 1.6214 at n=3000.)
%C Limit is equal to (1+sqrt(5))/2. - _Vaclav Kotesovec_, Jul 02 2014
%H Paul D. Hanna, <a href="/A243988/b243988.txt">Table of n, a(n) for n = 0..500</a>
%F G.f.: Sum_{n>=0} x^(n*(n+1)) / (1-x - x^(n+1))^(n+1).
%F G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k) * (1+x^k)^k.
%F G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k) * (1+x^k)^(n-k) * x^(k^2).
%F a(n) ~ 2^n. - _Vaclav Kotesovec_, Jun 18 2014
%F a(n)-2^n ~ n/5 * ((1+sqrt(5))/2)^n. - _Vaclav Kotesovec_, Jul 02 2014
%e G.f.: A(x) = 1 + 2*x + 5*x^2 + 10*x^3 + 21*x^4 + 42*x^5 + 85*x^6 + 169*x^7 +...
%e where we have the series identity:
%e A(x) = 1/(1-x) + (1+x)*x/(1-x)^2 + (1+x^2)^2*x^2/(1-x)^3 + (1+x^3)^3*x^3/(1-x)^4 + (1+x^4)^4*x^4/(1-x)^5 +...+ (1 + x^n)^n * x^n / (1-x)^(n+1) +...
%e A(x) = 1/(1-2*x) + x^2/(1-x-x^2)^2 + x^6/(1-x-x^3)^3 + x^12/(1-x-x^4)^4 + x^20/(1-x-x^5)^5 + x^30/(1-x-x^6)^6 +...+ x^(n*(n+1)) / (1-x - x^(n+1))^(n+1) +...
%e as well as the binomial identity:
%e A(x) = 1 + x*(1 + (1+x)) + x^2*(1 + 2*(1+x) + (1+x^2)^2) + x^3*(1 + 3*(1+x) + 3*(1+x^2)^2 + (1+x^3)^3) + x^4*(1 + 4*(1+x) + 6*(1+x^2)^2 + 4*(1+x^3)^3 + (1+x^4)^4) + x^5*(1 + 5*(1+x) + 10*(1+x^2)^2 + 10*(1+x^3)^3 + 5*(1+x^4)^4 + (1+x^5)^5) +...+ x^n * Sum_{k=0..n} binomial(n,k) * (1+x^k)^k +...
%e A(x) = 1 + x*(2 + x) + x^2*(2^2 + 2*(1+x)*x + x^4) + x^3*(2^3 + 3*(1+x)^2*x + 3*(1+x^2)*x^4 + x^9) + x^4*(2^4 + 4*(1+x)^3*x + 6*(1+x^2)^2*x^4 + 4*(1+x^3)*x^9 + x^16) + x^5*(2^5 + 5*(1+x)^4*x + 10*(1+x^2)^3*x^4 + 10*(1+x^3)^2*x^9 + 5*(1+x^4)*x^16 + x^25) +...+ x^n * Sum_{k=0..n} binomial(n,k) * (1+x^k)^(n-k) * x^(k^2) +...
%t Table[SeriesCoefficient[Sum[x^(j*(j+1))/(1-x-x^(j+1))^(j+1),{j,0,n}],{x,0,n}],{n,0,40}] (* _Vaclav Kotesovec_, Jul 02 2014 *)
%o (PARI) {a(n)=local(A);A=sum(m=0,n, (1 + 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 - 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)*(1+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+x^k)^(m-k)*x^(k^2)) +x*O(x^n)),n)}
%o for(n=0,40,print1(a(n),", "))
%Y Cf. A244615, A244616, A244617, A244618, A243919.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jun 17 2014
%E Name changed by _Paul D. Hanna_, Jul 02 2014
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