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

0,3

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.

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

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 *)

Prog

(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

Crossrefs

Column k=5 of A336573.

Cf. A243667.

Sequence in context: A180911 A084774 A025757 * A265033 A226270 A121351

Adjacent sequences: A243665 A243666 A243667 * A243669 A243670 A243671

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