A364924 -id:A364924 - cp's OEIS frontend

A364923 G.f. satisfies A(x) = 1 + xA(x)^4 / (1 - 2x*A(x)^3).

1, 1, 6, 48, 442, 4419, 46626, 511032, 5761650, 66394596, 778518552, 9258850440, 111417705702, 1354135251538, 16598001854700, 204945037918800, 2546849778687138, 31828936270676172, 399777371427582024, 5043824569861127808, 63892650400004356776

Offset: 0

Seiichi Manyama, Aug 12 2023

nonn

Cf. A007564, A364924.

Cf. A243659.

A364923 := proc(n)
    add( 3^k*(-2)^(n-k)*binomial(n,k)*binomial(3*n+k+1,n)/(3*n+k+1),k=0..n) ;
end proc:
seq(A364923(n),n=0..80); # R. J. Mathar, Aug 16 2023

a(n) = sum(k=0, n, 3^k*(-2)^(n-k)*binomial(n, k)*binomial(3*n+k+1, n)/(3*n+k+1));

a(n) = Sum_{k=0..n} 3^k * (-2)^(n-k) * binomial(n,k) * binomial(3*n+k+1,n) / (3*n+k+1).
a(n) = (1/n) * Sum_{k=0..n-1} 2^k * binomial(n,k) * binomial(4*n-k,n-1-k) for n > 0.
a(n) = (1/n) * Sum_{k=1..n} 3^(n-k) * binomial(n,k) * binomial(3*n,k-1) for n > 0.
D-finite with recurrence +270*n*(3*n-1)*(3*n+1)*a(n) +(-9463*n^3 -45948*n^2 +88297*n -35478)*a(n-1) +36*(-9017*n^3 +49691*n^2 -90408*n +54354)*a(n-2) +48*(53*n^3 +1724*n^2 -11161*n +16518)*a(n-3) +576*(3*n-10)*(3*n-11) *(n-4)*a(n-4)=0. - R. J. Mathar, Aug 16 2023

A243693 Number of Hyposylvester classes of 3-multiparking functions of length n.

Original entry on oeis.org

1, 1, 5, 32, 233, 1833, 15180, 130392, 1151057, 10378883, 95182445, 885053524, 8324942620, 79071217228, 757310811912, 7305728683824, 70923966744609, 692370887676567, 6792525607165935, 66933512163735000, 662190712902022017, 6574831459429388169, 65494637699437417584

Offset: 0

N. J. A. Sloane, Jun 14 2014

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], 2014-2020. See Fig. 27.
Jun Yan, Results on pattern avoidance in parking functions, arXiv preprint arXiv:2404.07958 [math.CO], 2024. See Theorem 4.1.

Cf. A243694, A243695.

Cf. A007564, A003168, A364923, A364924.

a := proc(n) option remember; if n <= 1 then return 1 fi;
(a(n - 2)*(-800*n^3 + 3024*n^2 - 3184*n + 672) + a(n - 1)*(3275*n^3 - 7467*n^2 +
5038*n - 1008))/(300*n^3 - 234*n^2 - 192*n) end:
seq(a(n), n = 0..22);  # Peter Luschny, Apr 13 2024

a[n_] := 3^(n - Boole[n>0]) Hypergeometric2F1[1 - n, -2 n, 2, 1/3];
Table[a[n], {n, 0, 22}]  (* Peter Luschny, Apr 12 2024 *)

a(n) = sum(k=0, n, 3^k*(-2)^(n-k)*binomial(n, k)*binomial(2*n+k+1, n)/(2*n+k+1)); \\ Seiichi Manyama, Aug 12 2023

From Seiichi Manyama, Aug 12 2023: (Start)
The following statements are equivalent:
The g.f. satisfies A(x) = 1 + x*A(x)^3 / (1 - 2*x*A(x)^2).
a(n) = Sum_{k=0..n} 3^k * (-2)^(n-k) * binomial(n, k) * binomial(2*n+k+1, n) / (2*n + k + 1).
a(n) = (1/n) * Sum_{k=1..n} 3^(n-k) * binomial(n, k) * binomial(2*n, k-1) for n > 0.
a(n) = (1/n) * Sum_{k=0..n-1} 2^k * binomial(n, k)*binomial(3*n-k, n-1-k) for n > 0.
 (End)
The above formula is proved in Theorem 4.1 of the Jun Yan link to be the number of Hyposylvester classes of 3-multiparking functions of length n. - Jun Yan, Apr 12 2024
a(n) ~ 2^(5*n+1) / (sqrt(5*Pi) * n^(3/2) * 3^(n+1)). - Vaclav Kotesovec, Apr 12 2024
a(n) = 3^(n - 1) * hypergeom([1 - n, -2*n], [2], 1/3) for n > 0. - Peter Luschny, Apr 12 2024
G.f. A(x) = 1 + series_reversion( x/((1 + 3*x)*(1 + x)^2) ). - Peter Bala, Sep 10 2024

Name clarified by Jun Yan, Apr 12 2024

cp's OEIS Frontend

A364923 G.f. satisfies A(x) = 1 + xA(x)^4 / (1 - 2x*A(x)^3).

Views

Author

Keywords

Crossrefs

Programs

Maple

PARI

Formula

A243693 Number of Hyposylvester classes of 3-multiparking functions of length n.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

cp's OEIS Frontend

A364923 G.f. satisfies A(x) = 1 + x*A(x)^4 / (1 - 2*x*A(x)^3).

Views

Author

Keywords

Crossrefs

Programs

Maple

PARI

Formula

A243693 Number of Hyposylvester classes of 3-multiparking functions of length n.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A364923 G.f. satisfies A(x) = 1 + xA(x)^4 / (1 - 2x*A(x)^3).