A243668 Number of Sylvester classes of 5-packed words of degree n.

1, 1, 7, 69, 793, 9946, 131993, 1822288, 25904165, 376601883, 5573626462, 83692267478, 1271883556731, 19525467196176, 302346907361688, 4716814859429384, 74065892877777885, 1169701519598447641, 18566836447453815317, 296053851068485920563, 4739945317989532651858

Offset: 0

Author

N. J. A. Sloane, Jun 14 2014

Keywords

• nonn

Comments

See Novelli-Thibon (2014) for precise definition.

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..812
• 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.

Crossrefs

Column k=5 of A336573.

Cf. A243667.

Programs

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, 5, 2];
a /@ Range[20] (* Jean-François Alcover, Jan 28 2020 *)

PARI

a(n) = my(A=1+x*O(x^n)); for(i=0, n, A=1-x*A^5*(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(5*n+k+1, n)/(5*n+k+1)); \\ Seiichi Manyama, Jul 26 2020

PARI

a(n) = (-1)^n*sum(k=0, n, (-2)^(n-k)*binomial(5*n+1, k)*binomial(6*n-k, n-k))/(5*n+1); \\ Seiichi Manyama, Jul 26 2020

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)^5 * (1 - 2 * A(x)).

a(n) = (-1)^n * Sum_{k=0..n} (-2)^k * binomial(n,k) * binomial(5*n+k+1,n)/(5*n+k+1).

a(n) = ( (-1)^n / (5*n+1) ) * Sum_{k=0..n} (-2)^(n-k) * binomial(5*n+1,k) * binomial(6*n-k,n-k). (End)

a(n) ~ sqrt(27851068 + 7443921*sqrt(14)) * 5^(5*n - 13/2) / (sqrt(7*Pi) * n^(3/2) * 2^(2*(1 + n)) * (108007 - 28854*sqrt(14))^(n - 1/2)). - Vaclav Kotesovec, Jul 31 2021

a(n) = (1/n) * Sum_{k=0..n-1} binomial(n,k) * binomial(6*n-k,n-1-k) for n > 0. - Seiichi Manyama, Aug 08 2023

Extensions

More terms from Jean-François Alcover, Jan 28 2020

a(0)=1 prepended by Seiichi Manyama, Jul 25 2020