OFFSET
0,3
COMMENTS
See Novelli-Thibon (2014) for precise definition.
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..865
J.-C. Novelli and J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014. See Eq. (185), p. 47 and Fig. 17.
FORMULA
Novelli-Thibon give an explicit formula in Eq. (182).
From Seiichi Manyama, Jul 26 2020: (Start)
G.f. A(x) satisfies: A(x) = 1 - x * A(x)^4 * (1 - 2 * A(x)).
a(n) = (-1)^n * Sum_{k=0..n} (-2)^k * binomial(n,k) * binomial(4*n+k+1,n)/(4*n+k+1).
a(n) = ( (-1)^n / (4*n+1) ) * Sum_{k=0..n} (-2)^(n-k) * binomial(4*n+1,k) * binomial(5*n-k,n-k). (End)
a(n) ~ 2^(9*n - 15) * sqrt(436289 + 2793997/sqrt(41)) / (sqrt(Pi) * n^(3/2) * (29701 - 4633*sqrt(41))^(n - 1/2)). - Vaclav Kotesovec, Jul 31 2021
a(n) = (1/n) * Sum_{k=0..n-1} binomial(n,k) * binomial(5*n-k,n-1-k) for n > 0. - Seiichi Manyama, Aug 08 2023
MATHEMATICA
P[n_, m_, x_] := 1/(m n + 1) Sum[Binomial[m n + 1, k] Binomial[(m + 1) n - k, n - k] (1 - x)^k x^(n - k), {k, 0, n}];
a[n_] := P[n, 4, 2];
a /@ Range[20] (* Jean-François Alcover, Jan 28 2020 *)
PROG
(PARI) a(n) = my(A=1+x*O(x^n)); for(i=0, n, A=1-x*A^4*(1-2*A)); polcoeff(A, n); \\ Seiichi Manyama, Jul 26 2020
(PARI) a(n) = (-1)^n*sum(k=0, n, (-2)^k*binomial(n, k)*binomial(4*n+k+1, n)/(4*n+k+1)); \\ Seiichi Manyama, Jul 26 2020
(PARI) a(n) = (-1)^n*sum(k=0, n, (-2)^(n-k)*binomial(4*n+1, k)*binomial(5*n-k, n-k))/(4*n+1); \\ Seiichi Manyama, Jul 26 2020
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Jun 14 2014
EXTENSIONS
More terms from Jean-François Alcover, Jan 28 2020
a(0)=1 prepended by Seiichi Manyama, Jul 25 2020
STATUS
approved