A243659 Number of Sylvester classes of 3-packed words of degree n.

1, 1, 5, 34, 267, 2279, 20540, 192350, 1853255, 18252079, 182924645, 1859546968, 19127944500, 198725331588, 2082256791048, 21979169545670, 233495834018591, 2494624746580655, 26786319835972799, 288915128642169250, 3128814683222599331, 34007373443388857999

Offset

0,3

Comments

See Novelli-Thibon (2014) for precise definition.

Links

Seiichi Manyama, Table of n, a(n) for n = 0..941 (terms 1..100 from Lars Blomberg)

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).

a(0) = 1 and a(n) = (1/n) * Sum_{i=1..n} ( Sum_{j=0..3*i} (2^(j-2*i)*(-1)^(i-j) * binomial(i,3*i-j)*binomial(i+j-1,i-1)) *a(n-i) ) for n > 0. - Vladimir Kruchinin, Apr 09 2017

From Seiichi Manyama, Jul 26 2020: (Start)

G.f. A(x) satisfies: A(x) = 1 - x * A(x)^3 * (1 - 2 * A(x)).

a(n) = (-1)^n * Sum_{k=0..n} (-2)^k * binomial(n,k) * binomial(3*n+k+1,n)/(3*n+k+1).

a(n) = ( (-1)^n / (3*n+1) ) * Sum_{k=0..n} (-2)^(n-k) * binomial(3*n+1,k) * binomial(4*n-k,n-k). (End)

a(n) ~ sqrt(24388 + 9221*sqrt(7)) * (316 + 119*sqrt(7))^(n - 1/2) / (sqrt(7*Pi) * n^(3/2) * 2^(n + 3/2) * 3^(3*n + 3/2)). - Vaclav Kotesovec, Jul 31 2021

a(n) = (1/n) * Sum_{k=0..n-1} binomial(n,k) * binomial(4*n-k,n-1-k) for n > 0. - Seiichi Manyama, Aug 08 2023

Mathematica

b[0] = 1; b[n_] := b[n] = 1/n Sum[Sum[2^(j-2i)(-1)^(i-j) Binomial[i, 3i-j] Binomial[i+j-1, i-1], {j, 0, 3i}] b[n-i], {i, 1, n}];
a[n_] := b[n+1];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jul 27 2018, after Vladimir Kruchinin *)

Prog

(Maxima)
a(n):=if n=0 then 1 else 1/n*sum(sum(2^(j-2*i)*(-1)^(i-j)*binomial(i, 3*i-j)*binomial(i+j-1, i-1), j, 0, 3*i)*a(n-i), i, 1, n); /* Vladimir Kruchinin, Apr 07 2017 */
(PARI) a(n) = if(n==0, 1, sum(i=1, n, a(n-i)*sum(j=0, 3*i, 2^(j-2*i)*(-1)^(i-j)*binomial(i, 3*i-j)*binomial(i+j-1, i-1)))/n); \\ Seiichi Manyama, Jul 26 2020
(PARI) a(n) = my(A=1+x*O(x^n)); for(i=0, n, A=1-x*A^3*(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(3*n+k+1, n)/(3*n+k+1)); \\ Seiichi Manyama, Jul 26 2020
(PARI) a(n) = (-1)^n*sum(k=0, n, (-2)^(n-k)*binomial(3*n+1, k)*binomial(4*n-k, n-k))/(3*n+1); \\ Seiichi Manyama, Jul 26 2020

Crossrefs

Column k=3 of A336573.

Cf. A003168, A336539.

Cf. A002293, A349331, A364747, A364765.

Sequence in context: A081342 A248055 A365218 * A365183 A371391 A058248

Adjacent sequences: A243656 A243657 A243658 * A243660 A243661 A243662

Keyword

nonn

Author

N. J. A. Sloane, Jun 12 2014

Extensions

a(9)-a(21) from Lars Blomberg, Jul 12 2017

a(0)=1 inserted by Seiichi Manyama, Jul 26 2020

Status

approved