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 A243667 Number of Sylvester classes of 4-packed words of degree n. 4

%I

%S 1,1,6,50,484,5105,56928,660112,7878940,96159476,1194532794,

%T 15053992178,191993403476,2473358617150,32137897641232,

%U 420698195672700,5542894551818268,73447821835338348,978178443083177880,13086377223959022952,175785879063917657688

%N Number of Sylvester classes of 4-packed words of degree n.

%C See Novelli-Thibon (2014) for precise definition.

%H Seiichi Manyama, <a href="/A243667/b243667.txt">Table of n, a(n) for n = 0..865</a>

%H J.-C. Novelli, J.-Y. Thibon, <a href="http://arxiv.org/abs/1403.5962">Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions</a>, arXiv preprint arXiv:1403.5962 [math.CO], 2014. See Eq. (185), p. 47 and Fig. 17.

%F Novelli-Thibon give an explicit formula in Eq. (182).

%F From _Seiichi Manyama_, Jul 26 2020: (Start)

%F G.f. A(x) satisfies: A(x) = 1 - x * A(x)^4 * (1 - 2 * A(x)).

%F a(n) = (-1)^n * Sum_{k=0..n} (-2)^k * binomial(n,k) * binomial(4*n+k+1,n)/(4*n+k+1).

%F 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)

%t 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}];

%t a[n_] := P[n, 4, 2];

%t a /@ Range[20] (* _Jean-François Alcover_, Jan 28 2020 *)

%o (PARI) {a(n) = local(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

%o (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

%o (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

%Y Column k=4 of A336573.

%Y Cf. A243668, A336572.

%K nonn

%O 0,3

%A _N. J. A. Sloane_, Jun 14 2014

%E More terms from _Jean-François Alcover_, Jan 28 2020

%E a(0)=1 prepended by _Seiichi Manyama_, Jul 25 2020

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Last modified January 22 03:32 EST 2021. Contains 340360 sequences. (Running on oeis4.)