A249946 G.f.: Sum_{n>=0} x^n/(1-x)^(3*n) * Sum_{k=0..n} C(n,k)^2 * x^k.
1, 1, 5, 20, 81, 335, 1406, 5965, 25517, 109872, 475597, 2067679, 9022210, 39490321, 173311717, 762382740, 3360486897, 14839284335, 65632607150, 290703303277, 1289265151469, 5724578761376, 25445326076925, 113212867808159, 504164051602178, 2247012340118785, 10022342589850853, 44734125313004500
Offset: 0
Keywords
Examples
G.f.: A(x) = 1 + x + 5*x^2 + 20*x^3 + 81*x^4 + 335*x^5 + 1406*x^6 +... where A(x) = 1 + x/(1-x)^3*(1+x) + x^2/(1-x)^6*(1+2^2*x+x^2) + x^3/(1-x)^9*(1+3^2*x+3^2*x^2+x^3) + x^4/(1-x)^12*(1+4^2*x+6^2*x^2+4^2*x^3+x^4) + x^5/(1-x)^15*(1+5^2*x+10^2*x^2+10^2*x^3+5^2*x^4+x^5) +...
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
Programs
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Mathematica
CoefficientList[Series[(1 - x)^2/Sqrt[1 - 6*x + 7*x^2 - 2*x^3 + x^4], {x, 0, 50}], x] (* G. C. Greubel, Feb 05 2017 *) -
PARI
{a(n)=polcoeff( sum(m=0, n, x^m * sum(k=0, m, binomial(m, k)^2 * x^k) / (1-x +x*O(x^n))^(3*m)), n)} for(n=0,30,print1(a(n),", ")) -
PARI
{a(n)=polcoeff( (1-x)^2 / sqrt(1 - 6*x + 7*x^2 - 2*x^3 + x^4 +x*O(x^n)), n)} for(n=0, 30, print1(a(n), ", "))
Formula
G.f.: (1-x)^2 / sqrt(1 - 6*x + 7*x^2 - 2*x^3 + x^4).
a(n) ~ sqrt((21*sqrt(2) + sqrt(14*(88*sqrt(2)-61)))/7)/4 * ((3 + 2*sqrt(2) + sqrt(5+4*sqrt(2)))/2)^n / sqrt(Pi*n). - Vaclav Kotesovec, Nov 09 2014