A100089 -id:A100089 - cp's OEIS frontend

A337725 a(n) = (3n+1)! Sum_{k=0..n} 1 / (3*k+1)!.

1, 25, 5251, 3780721, 6487717237, 21798729916321, 126737815733490295, 1171057417377450325801, 16160592359808814496053801, 317652603424402057734433512457, 8567090714356123497097671830965291, 307592825008242258039794809418977808065

Offset: 0

Ilya Gutkovskiy, Sep 17 2020

nonn

Cf. A000522, A051396, A051397, A087350, A100089, A143820, A330044, A337726, A337727, A337728, A337729, A337730.

Table[(3 n + 1)! Sum[1/(3 k + 1)!, {k, 0, n}], {n, 0, 11}]
Table[(3 n + 1)! SeriesCoefficient[(Exp[3 x/2] - 2 Sin[Pi/6 - Sqrt[3] x/2])/(3 Exp[x/2] (1 - x^3)), {x, 0, 3 n + 1}], {n, 0, 11}]
Table[Floor[(Exp[3/2] + 2 Sin[(3 Sqrt[3] - Pi)/6])/(3 Sqrt[Exp[1]]) (3 n + 1)!], {n, 0, 11}]

a(n) = (3*n+1)!*sum(k=0, n, 1/(3*k+1)!); \\ Michel Marcus, Sep 17 2020

E.g.f.: (exp(3*x/2) - 2 * sin(Pi/6 - sqrt(3)*x/2)) / (3*exp(x/2) * (1 - x^3)) = x + 25*x^4/4! + 5251*x^7/7! + 3780721*x^10/10! + ...

a(n) = floor(c * (3*n+1)!), where c = (exp(3/2) + 2 * sin((3 * sqrt(3) - Pi) / 6))/(3 * sqrt(exp(1))) = A143820.

A217360 a(n) = 2^nbinomial(4n, n)/(3*n+1).

Original entry on oeis.org

1, 2, 16, 176, 2240, 31008, 453376, 6888960, 107707392, 1721477120, 28000141312, 461964898304, 7712495058944, 130050777006080, 2211737871974400, 37892693797109760, 653389823437701120, 11330548232319664128, 197475886172892823552

Offset: 0

R. J. Mathar, Oct 01 2012

Old name was: Series reversion of x - 2*x^4.
Regular blocks of 2 intermediate zeros have been removed from the sequence: If y = x - 2*x^4, then x = y + 2*y^4 + 16*y^7 + 176*y^10 + 2240*y^13 + 31008*y^16 + ...
a(n) is the number of lattice paths (Schroeder paths) from (0,0) to (n,4n) with unit steps N=(0,1), E=(1,0) and D=(1,1) staying weakly above the line y = 4x with the total number of occurrences of NE and D equal to n. - Michael D. Weiner, Jul 25 2019

Jinyuan Wang, Table of n, a(n) for n = 0..100
D. Birmajer, J. B. Gil, J. D. Gil and M. D. Weiner, Schröder Coloring and Applications, arXiv:1908.08103 [math.CO], 2019.

Cf. A153231 (x+2*x^3).

[2^n*Binomial(4*n, n)/(3*n+1): n in [0..25]]; // Vincenzo Librandi, Jul 26 2019

A100089 := proc(n)
    (3*n+1)! ;
end proc:
A060706 := proc(n)
    (4*n)!/n!/4^n ;
end proc:
A217360 := proc(n)
    8^(n)*A060706(n)/A100089(n) ;
end proc:
seq(A217360(n), n=0..20);

Table[2^n Binomial[4 n, n] / (3 n + 1), {n, 0, 20}] (* Vincenzo Librandi, Jul 26 2019 *)

D-finite with recurrence 3*n*(3*n-1)*(3*n+1)*a(n)- 8*(4*n-1)*(4*n-3)*(4*n-2)*a(n-1) = 0, so a(n) = 8^n*A060706(n)/A100089(n) = 2^n*A002293(n).
a(n) = [x^(3*n)](f(x)/x) where f(x) is the reversion of x - 2*x^4.
G.f.: F([1/4, 1/2, 3/4], [2/3, 4/3], 512*x/27), where F is the generalized hypergeometric function. - Stefano Spezia, Aug 18 2019
G.f. A(x) satisfies: A(x) = 1 / (1 - 2 * x * A(x)^3). - Ilya Gutkovskiy, Nov 12 2021

Offset decreased by 1 and name changed by Michael D. Weiner, Jul 25 2019

cp's OEIS Frontend

A337725 a(n) = (3n+1)! Sum_{k=0..n} 1 / (3*k+1)!.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula

A217360 a(n) = 2^nbinomial(4n, n)/(3*n+1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Formula

Extensions

cp's OEIS Frontend

A337725 a(n) = (3*n+1)! * Sum_{k=0..n} 1 / (3*k+1)!.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula

A217360 a(n) = 2^n*binomial(4*n, n)/(3*n+1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Formula

Extensions

A337725 a(n) = (3n+1)! Sum_{k=0..n} 1 / (3*k+1)!.

A217360 a(n) = 2^nbinomial(4n, n)/(3*n+1).