A244617 - OEIS

%I #13 Jul 02 2014 08:29:21

%S 1,5,26,127,636,3153,15727,78406,391494,1955563,9772721,48847892,

%T 244196337,1220857221,6103941997,30518746918,152591088797,

%U 762948154799,3814720881833,19073550187976,95367603900506,476837621600990,2384187034951204,11920932287085421,59604653684027019

%N G.f.: Sum_{n>=0} (4 + x^n)^n * x^n / (1-x)^(n+1).

%C What is limit ( a(n) - 5^n )^(1/n) ?  (Value is near 2.5659 at n=3000.)

%C Limit is equal to (1+sqrt(17))/2. - _Vaclav Kotesovec_, Jul 02 2014

%H Vaclav Kotesovec, <a href="/A244617/b244617.txt">Table of n, a(n) for n = 0..500</a>

%F G.f.: Sum_{n>=0} x^(n*(n+1)) / (1-x - 4*x^(n+1))^(n+1).

%F G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k) * (4 + x^k)^k.

%F G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k) * (1 + 4*x^k)^(n-k) * x^(k^2).

%F a(n)-5^n ~ n/17 * ((1+sqrt(17))/2)^n. - _Vaclav Kotesovec_, Jul 02 2014

%e G.f.: A(x) = 1 + 5*x + 26*x^2 + 127*x^3 + 636*x^4 + 3153*x^5 + 15727*x^6 +...

%e where we have the series identity:

%e A(x) = 1/(1-x) + (4+x)*x/(1-x)^2 + (4+x^2)^2*x^2/(1-x)^3 + (4+x^3)^3*x^3/(1-x)^4 + (4+x^4)^4*x^4/(1-x)^5 +...+ (4 + x^n)^n * x^n / (1-x)^(n+1) +...

%e A(x) = 1/(1-5*x) + x^2/(1-x-4*x^2)^2 + x^6/(1-x-4*x^3)^3 + x^12/(1-x-4*x^4)^4 + x^20/(1-x-4*x^5)^5 + x^30/(1-x-4*x^6)^6 +...+ x^(n*(n+1)) / (1-x - 4*x^(n+1))^(n+1) +...

%e as well as the binomial identity:

%e A(x) = 1 + x*(1 + (4+x)) + x^2*(1 + 2*(4+x) + (4+x^2)^2) + x^3*(1 + 3*(4+x) + 3*(4+x^2)^2 + (4+x^3)^3) + x^4*(1 + 4*(4+x) + 6*(4+x^2)^2 + 4*(4+x^3)^3 + (4+x^4)^4) + x^5*(1 + 5*(4+x) + 10*(4+x^2)^2 + 10*(4+x^3)^3 + 5*(4+x^4)^4 + (4+x^5)^5) +...+ x^n * Sum_{k=0..n} binomial(n,k) * (4+x^k)^k +...

%e A(x) = 1 + x*(5 + x) + x^2*(5^2 + 2*(1+4*x)*x + x^4) + x^3*(5^3 + 3*(1+4*x)^2*x + 3*(1+4*x^2)*x^4 + x^9) + x^4*(5^4 + 4*(1+4*x)^3*x + 6*(1+4*x^2)^2*x^4 + 4*(1+4*x^3)*x^9 + x^16) + x^5*(5^5 + 5*(1+4*x)^4*x + 10*(1+4*x^2)^3*x^4 + 10*(1+4*x^3)^2*x^9 + 5*(1+4*x^4)*x^16 + x^25) +...+ x^n * Sum_{k=0..n} binomial(n,k) * (1+4*x^k)^(n-k) * x^(k^2) +...

%t Table[SeriesCoefficient[Sum[x^(j*(j+1))/(1-x-4*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, (4 + 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 - 4*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)*(4+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+4*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, A244618, A243919.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jul 02 2014