A243988 G.f.: Sum_{n>=0} (1 + x^n)^n * x^n / (1-x)^(n+1).

1, 2, 5, 10, 21, 42, 85, 169, 333, 655, 1286, 2519, 4935, 9675, 18982, 37285, 73346, 144509, 285158, 563546, 1115309, 2210243, 4385443, 8710876, 17319387, 34464792, 68634821, 136771603, 272703704, 543995341, 1085620097, 2167267262, 4327886353, 8644663133, 17270784312, 34510656589, 68969830833

Offset

0,2

Comments

What is limit ( a(n) - 2^n )^(1/n) ? (Value is near 1.6214 at n=3000.)

Limit is equal to (1+sqrt(5))/2. - Vaclav Kotesovec, Jul 02 2014

Links

Paul D. Hanna, Table of n, a(n) for n = 0..500

Formula

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

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

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

a(n) ~ 2^n. - Vaclav Kotesovec, Jun 18 2014

a(n)-2^n ~ n/5 * ((1+sqrt(5))/2)^n. - Vaclav Kotesovec, Jul 02 2014

Example

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 +...
where we have the series identity:
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) +...
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) +...
as well as the binomial identity:
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 +...
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) +...

Mathematica

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 *)

Prog

(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)}
for(n=0, 40, print1(a(n), ", "))
(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)}
for(n=0, 40, print1(a(n), ", "))
(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)}
for(n=0, 40, print1(a(n), ", "))
(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)}
for(n=0, 40, print1(a(n), ", "))

Crossrefs

Cf. A244615, A244616, A244617, A244618, A243919.

Sequence in context: A116385 A267515 A215411 * A279811 A279751 A000975

Adjacent sequences: A243985 A243986 A243987 * A243989 A243990 A243991

Keyword

nonn

Author

Paul D. Hanna, Jun 17 2014

Extensions

Name changed by Paul D. Hanna, Jul 02 2014

Status

approved