OFFSET
0,2
COMMENTS
In general, if m > 1, s >= 1 then Sum_{k=0..n} m^k * binomial(n,k) * binomial(s*n,k) ~ m^(s*n) * (1-r)^((s-1)*n) * s^(s*n + 1/2) / (sqrt(2*Pi*(2*s - r*(s+1))*n) * r^(2*s*n + 1/2)), where r = (m*(s+1) - sqrt(m*(m*(s-1)^2 + 4*s))) / (2*(m-1)). - Vaclav Kotesovec, Sep 21 2025
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..500
FORMULA
a(n) = [x^n] (1-2*x)^n/(1-3*x)^(4*n+1).
a(n) = Sum_{k=0..n} 3^k * (-2)^(n-k) * binomial(n,k) * binomial(4*n+k,k).
a(n) = Sum_{k=0..n} 2^(n-k) * binomial(n,k) * binomial(4*n+k,n).
a(n) = hypergeom([-4*n, -n], [1], 3). - Stefano Spezia, Sep 13 2025
a(n) = [x^n] ((1+x)^4 * (3+x))^n. - Seiichi Manyama, Sep 21 2025
a(n) ~ (7781 + 731*sqrt(129))^n / (sqrt((59*sqrt(129) - 645)*Pi*n) * 2^(9*n-2)). - Vaclav Kotesovec, Sep 21 2025
MATHEMATICA
Table[Sum[3^k*Binomial[n, k]*Binomial[4*n, k], {k, 0, n}], {n, 0, 25}] (* Vincenzo Librandi, Sep 19 2025 *)
PROG
(PARI) a(n) = sum(k=0, n, 3^k*binomial(n, k)*binomial(4*n, k));
(Magma) [&+[3^k*Binomial(n, k)*Binomial(4*n, k): k in [0..n]]: n in [0..20]]; // Vincenzo Librandi, Sep 19 2025
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Sep 13 2025
STATUS
approved