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

Author

N. J. A. Sloane, Jun 12 2014

Keywords

• nonn
• easy

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)
• Paul Barry, The Triple Riordan Group, arXiv:2412.05461 [math.CO], 2024. See pp. 4, 10.
• 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=3 of A336573.

Cf. A003168, A336539.

Cf. A002293, A349331, A364747, A364765.

Programs

Maple

a := proc(n) option remember; if n = 0 then 1 elif n = 1 then 1 else (4*(37604*n^5-158474*n^4+248391*n^3-178459*n^2+58042*n-6720)*a(n-1) - 3*(n-2)*(3*n-4)*(3*n-5)*(119*n^2-85*n+14)*a(n-2) )/ (12*n*(3*n-1)*(3*n+1)*(119*n^2-323*n+218)) fi; end:
seq(a(n), n = 0..20); # Peter Bala, Sep 08 2024

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 *)

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

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

P-recursive: 12*n*(3*n-1)*(3*n+1)*(119*n^2-323*n+218)*a(n) = 4*(37604*n^5-158474*n^4+248391*n^3-178459*n^2+58042*n-6720)*a(n-1) - (3*n-4)*(3*n-5)*(3*n-6)*(119*n^2-85*n+14)*a(n-2) with a(0) = a(1) = 1. - Peter Bala, Sep 08 2024

Extensions

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

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