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A244615 G.f.: Sum_{n>=0} (2 + x^n)^n * x^n / (1-x)^(n+1). 5
1, 3, 10, 29, 88, 259, 771, 2284, 6786, 20191, 60189, 179654, 536925, 1606221, 4808601, 14403804, 43163793, 129388755, 387946445, 1163370778, 3489117566, 10465248174, 31391306504, 94164586011, 282474177151, 847381544167, 2542059008323, 7625998475474, 22877623720244 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

What is limit ( a(n) - 3^n )^(1/n) ?  (Value is near 2.0038 at n=3000.)

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

LINKS

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

FORMULA

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

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

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

a(n)-3^n ~ n/9 * 2^n. - Vaclav Kotesovec, Jul 02 2014

EXAMPLE

G.f.: A(x) = 1 + 3*x + 10*x^2 + 29*x^3 + 88*x^4 + 259*x^5 + 771*x^6 +...

where we have the series identity:

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

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

as well as the binomial identity:

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

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

MATHEMATICA

Table[SeriesCoefficient[Sum[x^(j*(j+1))/(1-x-2*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, (2 + 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 - 2*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)*(2+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+2*x^k)^(m-k)*x^(k^2)) +x*O(x^n)), n)}

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

CROSSREFS

Cf. A243988, A244616, A244617, A244618, A243919.

Sequence in context: A096140 A307262 A291393 * A307062 A052976 A147363

Adjacent sequences:  A244612 A244613 A244614 * A244616 A244617 A244618

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jul 02 2014

STATUS

approved

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Last modified April 21 19:16 EDT 2021. Contains 343156 sequences. (Running on oeis4.)