A244616 G.f.: Sum_{n>=0} (3 + x^n)^n * x^n / (1-x)^(n+1).

1, 4, 17, 66, 265, 1046, 4165, 16561, 66017, 263377, 1051774, 4202363, 16797551, 67159565, 268561326, 1074052233, 4295730126, 17181735211, 68724026662, 274888965526, 1099538433297, 4398111321331, 17592342396443, 70369120593536, 281475881240563, 1125902076700152, 4503604824386737

Offset

0,2

Comments

What is limit ( a(n) - 4^n )^(1/n) ? (Value is near 2.3069 at n=3000.)

Limit is equal to (1+sqrt(13))/2. - Vaclav Kotesovec, Jul 02 2014

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..500

Formula

G.f.: Sum_{n>=0} x^(n*(n+1)) / (1-x - 3*x^(n+1))^(n+1).

G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k) * (3 + x^k)^k.

G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k) * (1 + 3*x^k)^(n-k) * x^(k^2).

a(n)-4^n ~ n/13 * ((1+sqrt(13))/2)^n. - Vaclav Kotesovec, Jul 02 2014

Example

G.f.: A(x) = 1 + 4*x + 17*x^2 + 66*x^3 + 265*x^4 + 1046*x^5 + 4165*x^6 +...
where we have the series identity:
A(x) = 1/(1-x) + (3+x)*x/(1-x)^2 + (3+x^2)^2*x^2/(1-x)^3 + (3+x^3)^3*x^3/(1-x)^4 + (3+x^4)^4*x^4/(1-x)^5 +...+ (3 + x^n)^n * x^n / (1-x)^(n+1) +...
A(x) = 1/(1-4*x) + x^2/(1-x-3*x^2)^2 + x^6/(1-x-3*x^3)^3 + x^12/(1-x-3*x^4)^4 + x^20/(1-x-3*x^5)^5 + x^30/(1-x-3*x^6)^6 +...+ x^(n*(n+1)) / (1-x - 3*x^(n+1))^(n+1) +...
as well as the binomial identity:
A(x) = 1 + x*(1 + (3+x)) + x^2*(1 + 2*(3+x) + (3+x^2)^2) + x^3*(1 + 3*(3+x) + 3*(3+x^2)^2 + (3+x^3)^3) + x^4*(1 + 4*(3+x) + 6*(3+x^2)^2 + 4*(3+x^3)^3 + (3+x^4)^4) + x^5*(1 + 5*(3+x) + 10*(3+x^2)^2 + 10*(3+x^3)^3 + 5*(3+x^4)^4 + (3+x^5)^5) +...+ x^n * Sum_{k=0..n} binomial(n,k) * (3+x^k)^k +...
A(x) = 1 + x*(4 + x) + x^2*(4^2 + 2*(1+3*x)*x + x^4) + x^3*(4^3 + 3*(1+3*x)^2*x + 3*(1+3*x^2)*x^4 + x^9) + x^4*(4^4 + 4*(1+3*x)^3*x + 6*(1+3*x^2)^2*x^4 + 4*(1+3*x^3)*x^9 + x^16) + x^5*(4^5 + 5*(1+3*x)^4*x + 10*(1+3*x^2)^3*x^4 + 10*(1+3*x^3)^2*x^9 + 5*(1+3*x^4)*x^16 + x^25) +...+ x^n * Sum_{k=0..n} binomial(n,k) * (1+3*x^k)^(n-k) * x^(k^2) +...

Mathematica

Table[SeriesCoefficient[Sum[x^(j*(j+1))/(1-x-3*x^(j+1))^(j+1), {j, 0, n}], {x, 0, n}], {n, 0, 30}] (* Vaclav Kotesovec, Jul 02 2014 *)

Prog

(PARI) {a(n)=local(A); A=sum(m=0, n, (3 + 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 - 3*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)*(3+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+3*x^k)^(m-k)*x^(k^2)) +x*O(x^n)), n)}
for(n=0, 40, print1(a(n), ", "))

Crossrefs

Cf. A243988, A244615, A244617, A244618, A243919.

Sequence in context: A217539 A046723 A291394 * A030529 A266862 A239845

Adjacent sequences: A244613 A244614 A244615 * A244617 A244618 A244619

Keyword

nonn

Author

Paul D. Hanna, Jul 02 2014

Status

approved