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%I
%S 1,6,37,218,1309,7810,46797,280345,1681029,10081939,60479790,
%T 362838047,2176908951,13061074143,78365313414,470188412205,
%U 2821120141002,16926689765961,101560046400158,609360004351714,3656159217328421,21936952918704867,131621710507870811,789730242509398308
%N G.f.: Sum_{n>=0} (5 + x^n)^n * x^n / (1-x)^(n+1).
%C What is limit ( a(n) - 6^n )^(1/n) ? (Value is near 2.7959 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="/A244618/b244618.txt">Table of n, a(n) for n = 0..500</a>
%F G.f.: Sum_{n>=0} x^(n*(n+1)) / (1-x - 5*x^(n+1))^(n+1).
%F G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k) * (5 + x^k)^k.
%F G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k) * (1 + 5*x^k)^(n-k) * x^(k^2).
%F a(n)-6^n ~ n/21 * ((1+sqrt(21))/2)^n. - _Vaclav Kotesovec_, Jul 02 2014
%e G.f.: A(x) = 1 + 6*x + 37*x^2 + 218*x^3 + 1309*x^4 + 7810*x^5 +...
%e where we have the series identity:
%e A(x) = 1/(1-x) + (5+x)*x/(1-x)^2 + (5+x^2)^2*x^2/(1-x)^3 + (5+x^3)^3*x^3/(1-x)^4 + (5+x^4)^4*x^4/(1-x)^5 +...+ (5 + x^n)^n * x^n / (1-x)^(n+1) +...
%e A(x) = 1/(1-6*x) + x^2/(1-x-5*x^2)^2 + x^6/(1-x-5*x^3)^3 + x^12/(1-x-5*x^4)^4 + x^20/(1-x-5*x^5)^5 + x^30/(1-x-5*x^6)^6 +...+ x^(n*(n+1)) / (1-x - 5*x^(n+1))^(n+1) +...
%e as well as the binomial identity:
%e A(x) = 1 + x*(1 + (5+x)) + x^2*(1 + 2*(5+x) + (5+x^2)^2) + x^3*(1 + 3*(5+x) + 3*(5+x^2)^2 + (5+x^3)^3) + x^4*(1 + 4*(5+x) + 6*(5+x^2)^2 + 4*(5+x^3)^3 + (5+x^4)^4) + x^5*(1 + 5*(5+x) + 10*(5+x^2)^2 + 10*(5+x^3)^3 + 5*(5+x^4)^4 + (5+x^5)^5) +...+ x^n * Sum_{k=0..n} binomial(n,k) * (5+x^k)^k +...
%e A(x) = 1 + x*(6 + x) + x^2*(6^2 + 2*(1+5*x)*x + x^4) + x^3*(6^3 + 3*(1+5*x)^2*x + 3*(1+5*x^2)*x^4 + x^9) + x^4*(6^4 + 4*(1+5*x)^3*x + 6*(1+5*x^2)^2*x^4 + 4*(1+5*x^3)*x^9 + x^16) + x^5*(6^5 + 5*(1+5*x)^4*x + 10*(1+5*x^2)^3*x^4 + 10*(1+5*x^3)^2*x^9 + 5*(1+5*x^4)*x^16 + x^25) +...+ x^n * Sum_{k=0..n} binomial(n,k) * (1+5*x^k)^(n-k) * x^(k^2) +...
%t Table[SeriesCoefficient[Sum[x^(j*(j+1))/(1-x-5*x^(j+1))^(j+1),{j,0,n}],{x,0,n}],{n,0,25}] (* _Vaclav Kotesovec_, Jul 02 2014 *)
%o (PARI) {a(n)=local(A); A=sum(m=0, n, (5 + 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 - 5*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)*(5+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+5*x^k)^(m-k)*x^(k^2)) +x*O(x^n)), n)}
%o for(n=0, 40, print1(a(n), ", "))
%Y Cf. A243988, A244615, A244616, A244617, A243919.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jul 02 2014
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