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URL: https://oeis.org/A243667

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A243667
Number of Sylvester classes of 4-packed words of degree n.
14
1, 1, 6, 50, 484, 5105, 56928, 660112, 7878940, 96159476, 1194532794, 15053992178, 191993403476, 2473358617150, 32137897641232, 420698195672700, 5542894551818268, 73447821835338348, 978178443083177880, 13086377223959022952, 175785879063917657688
OFFSET
0,3
COMMENTS
See Novelli-Thibon (2014) for precise definition.
LINKS
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
G.f.: 1 + Series_Reversion( x / ((1+x)^4 * (1+2*x)) ). - Seiichi Manyama, Sep 28 2025
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
Column k=4 of A336573.
Sequence in context: A180910 A199680 A039742 * A365189 A303562 A125558
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