A178807 G.f.: Sum_{n>=0} [Sum_{k>=0} C(n+k-1,k)^2*x^k]^2 * x^n.

1, 1, 3, 12, 57, 295, 1620, 9261, 54459, 326952, 1994085, 12313323, 76795452, 482903473, 3057550167, 19472859840, 124645558881, 801367484155, 5172056157960, 33494998326741, 217581335698815, 1417270332517704, 9254594901658689, 60567020813141991

Offset

0,3

Comments

Compare g.f. to a g.f. of the Central binomial coefficients (A000984):

Sum_{n>=0} [Sum_{k>=0} C(n+k,k)^2*x^k] * x^n.

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Formula

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

G.f.: 1 + x / AGM(1-7*x+x^2, 1+x+x^2), where AGM(x,y) = AGM((x+y)/2,sqrt(x*y)) is the arithmetic-geometric mean. - Paul D. Hanna, Aug 31 2014

a(n) ~ phi^(4*n) / (4*Pi*n), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jan 24 2019

Example

G.f.: A(x) = 1 + x + 3*x^2 + 12*x^3 + 57*x^4 + 295*x^5 + 1620*x^6 +...
which equals the sum of the series:
A(x) = 1 + (1 + x + x^2 + x^3 + x^4 + x^5 +...)^2*x
+ (1 + 2^2*x + 3^2*x^2 + 4^2*x^3 + 5^2*x^4 + 6^2*x^5 +...)^2*x^2
+ (1 + 3^2*x + 6^2*x^2 + 10^2*x^3 + 15^2*x^4 + 21^2*x^5 +...)^2*x^3
+ (1 + 4^2*x + 10^2*x^2 + 20^2*x^3 + 35^2*x^4 + 56^2*x^5 +...)^2*x^4
+ (1 + 5^2*x + 15^2*x^2 + 35^2*x^3 + 70^2*x^4 + 126^2*x^5 +...)^2*x^5
+ (1 + 6^2*x + 21^2*x^2 + 56^2*x^3 + 126^2*x^4 + 252^2*x^5 +...)^2*x^6 +...
The g.f. can also be expressed as:
A(x) = 1 + x/(1-x)^2 + (1+x)^2*x^2/(1-x)^6
+ (1 + 2^2*x + x^2)^2*x^3/(1-x)^10
+ (1 + 3^2*x + 3^2*x^2 + x^3)^2*x^4/(1-x)^14
+ (1 + 4^2*x + 6^2*x^2 + 4^2*x^3 + x^4)^2*x^5/(1-x)^18
+ (1 + 5^2*x + 10^2*x^2 + 10^2*x^3 + 5^2*x^4 + x^5)^2*x^6/(1-x)^22
+ (1 + 6^2*x + 15^2*x^2 + 20^2*x^3 + 15^2*x^4 + 6^2*x^5 + x^6)^2*x^7/(1-x)^26 +...

Mathematica

With[{m=30}, CoefficientList[Series[1 +x*Sum[Sum[Binomial[n, k]^2*x^k, {k, 0, n}]^2* x^n/(1-x)^(4*n+2), {n, 0, m}], {x, 0, m}], x]] (* G. C. Greubel, Jan 23 2019 *)

Prog

(PARI) {a(n)=polcoeff(sum(m=0, n, sum(k=0, n, binomial(m+k-1, k)^2*x^k)^2*x^m)+x*O(x^n), n)}
for(n=0, 25, print1(a(n), ", "))
(PARI) {a(n)=polcoeff(1+x*sum(m=0, n, sum(k=0, m, binomial(m, k)^2*x^k)^2*x^m/(1-x+x*O(x^n))^(4*m+2)), n)}
for(n=0, 25, print1(a(n), ", "))
(PARI) {a(n)=polcoeff( 1 + x / agm(1-7*x+x^2, 1+x+x^2 +x*O(x^n))), n)}
for(n=0, 25, print1(a(n), ", "))

Crossrefs

Sequence in context: A133158 A328295 A194089 * A361844 A047891 A166991

Adjacent sequences: A178804 A178805 A178806 * A178808 A178809 A178810

Keyword

nonn

Author

Paul D. Hanna, Dec 26 2010

Status

approved