OFFSET
0,3
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..1000 (terms 0..199 from Ludovic Schwob)
FORMULA
a(n) = Sum_{k=0..n} (C(n,k)/(n-k+1)) * C(n+4*k-1,n-k).
a(n) ~ sqrt((1 - r*s)*(1 + 4*r*s) / (5*Pi*(5*s - 2))) / (2 * n^(3/2) * r^(n+1)), where r = 0.124910212976238209867004924637837518925706044646... and s = 1.72708330560542094133450070142549940430523638921... are real roots of the system of equations s*(1 - r/(1 - r*s)^5) = 1, 5*r^2*s^2 = (1 - r*s)^6. - Vaclav Kotesovec, Nov 21 2018
MAPLE
eq:= a - 1/(1-x/(1-x*a)^5):
S:= series(RootOf(numer(eq), a), x, 31):
seq(coeff(S, x, j), j=0..30); # Robert Israel, Dec 10 2018
MATHEMATICA
a[n_]:=Sum[ Binomial[n, k]/(n-k+1)*Binomial[n+4*k-1, n-k], {k, 0, n}]; Array[a, 20, 0] (* Stefano Spezia, Nov 19 2018 *)
A[_] = 0; Do[A[x_] = 1/(1-x/(1-x*A[x])^5)+O[x]^25, {25}];
CoefficientList[A[x], x] (* Jean-François Alcover, Dec 30 2018 *)
PROG
(PARI) a(n) = sum(k=0, n, binomial(n, k)*binomial(n+4*k-1, n-k)/(n-k+1)); \\ Michel Marcus, Nov 19 2018
(GAP) List([0..25], n->Sum([0..n], k->Binomial(n, k)/(n-k+1)*Binomial(n+4*k-1, n-k))); # Muniru A Asiru, Nov 24 2018
(Magma) [1] cat [&+[(Binomial(n, k)/(n-k+1)) * Binomial(n+4*k-1, n-k): k in [0..n]]: n in [1.. 25]]; // Vincenzo Librandi, Dec 08 2018
(Sage) [sum(binomial(n, k)*binomial(n+4*k-1, n-k)/(n-k+1) for k in (0..n)) for n in range(25)] # G. C. Greubel, Dec 14 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Ludovic Schwob, Nov 19 2018
STATUS
approved