A038665 -id:A038665 - cp's OEIS frontend

A000917 a(n) = (2n+3)!/(n!*(n+2)!).

3, 20, 105, 504, 2310, 10296, 45045, 194480, 831402, 3527160, 14872858, 62403600, 260757900, 1085822640, 4508102925, 18668849760, 77138650050, 318107374200, 1309542023790, 5382578744400, 22093039119060, 90567738003600, 370847442355650, 1516927277253024

Offset: 0

N. J. A. Sloane

G.f.: c(x)*(4-c(x))/(1-4*x)^(3/2), c(x) = g.f. for Catalan numbers A000108 (agrees with Hansen, 1975, p. 99, (5.27.9)). Convolution of A038679 with A000984 (central binomial coefficients); also convolution of A038665 with A000302 (powers of 4). - Wolfdieter Lang, Dec 11 1999
Appears as diagonal in A003506. - Zerinvary Lajos, Apr 12 2006
a(n) is the number of double rises in all Grand Dyck paths of semilength n+2. Example: a(0)=3 because in the 6 (=A000984(2)) Grand Dyck paths of semilength 2, namely udud, (uu)dd, uddu, d(uu)d, dudu, dd(uu), we have a total of 3 uu's (shown between parentheses). - Emeric Deutsch, Nov 29 2008

Eldon R. Hansen, A Table of Series and Products, Prentice-Hall, Englewood Cliffs, NJ, 1975, p. 99, (5.27.9).

T. D. Noe, Table of n, a(n) for n = 0..200
Jian Zhou, On Some Mathematics Related to the Interpolating Statistics, arXiv:2108.10514 [math-ph], 2021.

Cf. A001622, A001791, A007054, A038665, A038679, A000108, A000984, A000302, A003506, A073010.

1/beta(n, n+2) in A061928.

[(n+1)*Binomial(2*n+3, n+1): n in [0..25]]; // Vincenzo Librandi, Jun 01 2016

a := proc(n) (n+1)*binomial(2*n+3, n+2) end: seq(a(n), n=0..23); # Zerinvary Lajos, Nov 26 2006
seq((n+1)*binomial(2*n+4, n+2)/2, n=0..23); # Zerinvary Lajos, Feb 28 2007

Table[(2*n + 3)!/(n!*(n + 2)!), {n, 0, 25}] (* T. D. Noe, Jun 20 2012 *)

a(n) = (n+1)*binomial(2*n+3, n+1) = (n+1)*A001700(n+1). - Vincenzo Librandi, Jun 01 2016
a(n) = (2*n+3)*A001791(n+1). - R. J. Mathar, Nov 09 2021
D-finite with recurrence +(n+2)*a(n) +10*(-n-1)*a(n-1) +12*(2*n+1)*a(n-2)=0. - R. J. Mathar, Nov 09 2021
D-finite with recurrence n*(n+2)*a(n) -2*(2*n+3)*(n+1)*a(n-1)=0. - R. J. Mathar, Nov 09 2021
From Amiram Eldar, Jan 24 2022: (Start)
Sum_{n>=0} 1/a(n) = 1 - Pi/(3*sqrt(3)) = 1 - A073010.
Sum_{n>=0} (-1)^n/a(n) = 6*log(phi)/sqrt(5) - 1, where phi is the golden ratio (A001622). (End)

A000777 a(n) = (n+2)*Catalan(n) - 1.

Original entry on oeis.org

1, 2, 7, 24, 83, 293, 1055, 3860, 14299, 53481, 201551, 764217, 2912167, 11143499, 42791039, 164812364, 636438059, 2463251009, 9552773999, 37112526989, 144410649239, 562724141459, 2195581527359, 8576490341249, 33537507830423, 131272552839203, 514285886020255

Offset: 0

N. J. A. Sloane

T. D. Noe, Table of n, a(n) for n = 0..200
Boothby, T.; Burkert, J.; Eichwald, M.; Ernst, D. C.; Green, R. M.; Macauley, M. On the cyclically fully commutative elements of Coxeter groups, J. Algebr. Comb. 36, No. 1, 123-148 (2012), Table 1 type B.
C. K. Fan, Structure of a Hecke algebra quotient, J. Amer. Math. Soc. 10 (1997), no. 1, 139-167.
J. R. Stembridge, Some combinatorial aspects of reduced words in finite Coxeter groups, Trans. Amer. Math. Soc. 349 (1997), no. 4, 1285-1332.

a(n) = A038665(n-1) - 1.

Cf. A000984, A000108.

[(n+2)*Catalan(n)-1: n in [0..30]]; // Vincenzo Librandi, Sep 11 2016

[seq((binomial(2*n,n)/(n+1))*(n+2)-1,n=0..27)]; # Zerinvary Lajos, Jun 25 2006

Table[(n + 2)*CatalanNumber[n] - 1, {n, 0, 20}] (* T. D. Noe, Jun 20 2012 *)

a(n) = (n+2)*binomial(2*n,n)/(n+1) - 1; \\ Michel Marcus, Sep 11 2016

a(n) = (binomial(2*n,n)/(n+1))*(n+2) - 1. - Zerinvary Lajos, Jun 25 2006
G.f.: (1/x)*(1/2 + (6*x-1)/(2*sqrt(1-4*x))-x/(1-x)). - Vladimir Kruchinin, Aug 18 2010
D-finite with recurrence: (n+1)*a(n) + 4*(-3*n+1)*a(n-1) + 5*(9*n-13)*a(n-2) + 2*(-29*n+72)*a(n-3) + 12*(2*n-7)*a(n-4) = 0. - R. J. Mathar, Jun 11 2019

A038679 Convolution of A007054 (Super ballot numbers) with A000302 (powers of 4).

Original entry on oeis.org

3, 14, 59, 242, 982, 3964, 15955, 64106, 257282, 1031780, 4135518, 16569204, 66365964, 265761016, 1064046979, 4259609626, 17050224394, 68241838036, 273110643754, 1092947507356, 4373580244084, 17500703480776

Offset: 0

Wolfdieter Lang

Also: convolution of A038665 with A000984 (central binomial coefficients )

A007054, A000302, A000108, A038665, A000984.

a(n) = 4^(n+1) - C(n+1), C(n): Catalan numbers A000108; G.f. c(x)*(4-c(x))/(1-4*x), where c(x) = g.f. for Catalan numbers.

A038697 Convolution of A000917 with A000984 (central binomial coefficients).

Original entry on oeis.org

3, 26, 163, 894, 4558, 22196, 104739, 483062, 2189530, 9789900, 43295118, 189749676, 825364668, 3567219688, 15332925731, 65591312550, 279415474594, 1185903736412, 5016725589402, 21159849864964, 89012979703940

Offset: 0

Wolfdieter Lang

Also convolution of A007054 (Super ballot numbers) with A002697;

Robert Israel, Table of n, a(n) for n = 0..1652

Cf. A007054, A038665, A038679, A000917, A000108, A000984, A002697.

seq(n*4^(n+1)+binomial(2*n+3,n+1),n=0..30); # Robert Israel, May 22 2019

a(n) = n*4^(n+1)+binomial(2*n+3, n+1).
G.f.: c(x)*(4-c(x))/(1-4*x)^2, where c(x) = g.f. for Catalan numbers A000108.
(160+64*n)*a(n) - (160+48*n)*a(n+1) + (50+12*n)*a(n+2) - (5+n)*a(n+3)=0. - Robert Israel, May 22 2019

A275329 a(n) = (2+[n/2])n!/((1+[n/2])[n/2]!^2).

Original entry on oeis.org

2, 2, 3, 9, 8, 40, 25, 175, 84, 756, 294, 3234, 1056, 13728, 3861, 57915, 14300, 243100, 53482, 1016158, 201552, 4232592, 764218, 17577014, 2912168, 72804200, 11143500, 300874500, 42791040, 1240940160, 164812365, 5109183315, 636438060, 21002455980, 2463251010

Offset: 0

Peter Luschny, Sep 10 2016

nonn

Cf. A038665, A056040, A057977, A097070, A189911.

a := n -> (2+iquo(n,2))*n!/((1+iquo(n,2))*iquo(n, 2)!^2):
seq(a(n), n=0..34);

def A275329():
    x, n, k = 1, 1, 2
    while True:
        yield x * k
        if is_odd(n):
            x *= n
        else:
            k += 1
            x = (x<<2)//(n+2)
        n += 1
a = A275329(); print([next(a) for _ in range(37)])

a(n) = A056040(n)*(2+[n/2])/(1+[n/2]).
a(n) = A057977(n)*A008619(n+2).
a(2*n+1) = (n+2)*binomial(2*n+1, n+1) = A189911(2*n+1).
a(2*n-3) = n*binomial(2*n-3, n-1) = A097070(n) for n>=2.
a(2*n+2) = (n+3)*binomial(2*n+2, n+1)/(n+2) = A038665(n).
Sum_{n>=0} 1/a(n) = 16/3 - 40*Pi/(9*sqrt(3)) + 4*Pi^2/9. - Amiram Eldar, Aug 20 2022

cp's OEIS Frontend

A000917 a(n) = (2n+3)!/(n!*(n+2)!).

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A000777 a(n) = (n+2)*Catalan(n) - 1.

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A038679 Convolution of A007054 (Super ballot numbers) with A000302 (powers of 4).

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A038697 Convolution of A000917 with A000984 (central binomial coefficients).

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A275329 a(n) = (2+[n/2])*n!/((1+[n/2])*[n/2]!^2).

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A275329 a(n) = (2+[n/2])n!/((1+[n/2])[n/2]!^2).