A008315 -id:A008315 - cp's OEIS frontend

A026004 a(n) = T(3n+1,n), where T = Catalan triangle (A008315).

1, 3, 14, 75, 429, 2548, 15504, 95931, 600875, 3798795, 24192090, 154969620, 997490844, 6446369400, 41802112192, 271861216539, 1772528290407, 11582393855305, 75831424919250, 497337483739635, 3266814940064445

Offset: 0

Clark Kimberling

nonn

Number of standard tableaux of shape (2n+1,n). Example: a(1)=3 because in the top row we can have 134, 124, or 123 (but not 234). - Emeric Deutsch, May 23 2004

Number of noncrossing forests with n+2 vertices and two components. - Emeric Deutsch, May 31 2004

Vincenzo Librandi, Table of n, a(n) for n = 0..200
P. Flajolet and M. Noy, Analytic combinatorics of noncrossing configurations, Discrete Math. 204 (1999), 203-229.

Cf. A045722.

Table[(n+2)/(2n+2)Binomial[3n+1,n],{n,0,20}] (* Harvey P. Dale, Jun 29 2011 *)

a(n):=sum((k+1)*binomial(n,k)*binomial(2*(n+1),n-k),k,0,n)/(n+1); /* Vladimir Kruchinin, Mar 01 2014 */

a(n) = (n+2)/(2*n+2) * binomial(3*n+1, n); \\ Joerg Arndt, Mar 01 2014

a(n) = (n+2)/(2*n+2) * C(3*n+1, n). - Ralf Stephan, Apr 30 2004
G.f.: ((sqrt(x)*sin(2/3*arcsin((3*sqrt(3)*sqrt(x))/2)))/sqrt(4/3-9*x)-cos(1/3*arccos(1-(27*x)/2))+1)/(3*x). - conjectured by Harvey P. Dale, Jun 30 2011
G.f.: (2*g-1)/((3*g-1)*(g-1)^2) where g*(1-g)^2 = x. - Mark van Hoeij, Nov 09 2011
2*(n+1)*(2*n+1)*a(n) +(-43*n^2-3*n+6)*a(n-1) +12*(3*n-2)*(3*n-4)*a(n-2)=0. - R. J. Mathar, Jun 07 2013
a(n) = sum(k=0..n, (k+1)*binomial(n,k)*binomial(2*(n+1),n-k))/(n+1). - Vladimir Kruchinin, Mar 01 2014
a(n) = [x^n] ((1 - sqrt(1 - 4*x))/(2*x))^(n+2). - Ilya Gutkovskiy, Nov 01 2017

More terms from Ralf Stephan, Apr 30 2004

A026008 a(n) = T(n, floor(n/2)), where T = Catalan triangle (A008315).

Original entry on oeis.org

1, 1, 2, 3, 5, 9, 14, 28, 42, 90, 132, 297, 429, 1001, 1430, 3432, 4862, 11934, 16796, 41990, 58786, 149226, 208012, 534888, 742900, 1931540, 2674440, 7020405, 9694845, 25662825, 35357670, 94287120, 129644790, 347993910, 477638700, 1289624490, 1767263190

Offset: 0

Clark Kimberling

nonn

a(n) is the number of Catalan paths in Quadrant I from (0,0) to (n, gcd(n,2)). - Clark Kimberling, Jun 26 2004

a(2n) = A000108(n+1), a(2n+1) = A000245(n+1).

a(2n) = C(2n+2, n+1)/(n+2), a(2n+1) = 3C(2n+2, n)/(n+3). - Ralf Stephan, Apr 30 2004

Conjecture: (n+5)*a(n) +(n+3)*a(n-1) +(-5*n-9)*a(n-2) -4*n*a(n-3) +4*(n-2)*a(n-4)=0. - R. J. Mathar, Jun 10 2013

A027302 a(n) = Sum_{k=0..floor((n-1)/2)} T(n,k) * T(n,k+1), with T given by A008315.

Original entry on oeis.org

1, 2, 9, 24, 95, 286, 1099, 3536, 13479, 45220, 172150, 594320, 2265003, 7983990, 30487175, 109174560, 417812417, 1514797020, 5810065898, 21275014800, 81775140083, 301892460012, 1162703549474, 4321730134624, 16675372590850, 62340424959176, 240949471232124

Offset: 1

Clark Kimberling

nonn

a(n) is the number of Dyck (n+2)-paths with UU spanning the midpoint. E.g., for n=2 the two Dyck 4-paths are UUDU.UDDD and UDUU.UDDD where dot marks the midpoint. - David Scambler, Feb 11 2011

Apparently also the number of returns to the left of or to the midpoint of all Dyck paths with semilength n+1. - David Scambler, Apr 30 2013

Alon Regev, The central component of a triangulation, arXiv:1210.3349 [math.CO], 2012, see p. 6.
Alon Regev, The Central Component of a Triangulation, J. Int. Seq. 16 (2013) #13.4.1

a[n_] := With[{C = CatalanNumber}, Sum[C[k]*C[n+1-k], {k, 1, (n+1)/2}]]; Array[a, 30] (* Jean-François Alcover, May 01 2017 *)

def C(n): return binomial(2*n,n)/(n+1)  # Catalan numbers
def A027302(n): return add(C(k)*C(n+1-k) for k in (1..(n+1)/2))
[A027302(n) for n in (1..22)]  # Peter Luschny, Jun 27 2013

Conjecture D-finite with recurrence -(n+2)*(13*n-2)*(3+n)^2*a(n) +10*(8*n^2+3*n-8)*(n+2)^2*a(n-1) +8*(12*n^4+47*n^3+52*n^2+67*n+20)*a(n-2) -160*(8*n^2+3*n-8)*(n-1)^2*a(n-3) +128*(7*n+4)*(2*n-5)*(-2+n)^2*a(n-4)=0. - R. J. Mathar, Nov 22 2024

More terms from Sean A. Irvine, Oct 26 2019

A026005 a(n) = T(4*n,n), where T = Catalan triangle (A008315).

Original entry on oeis.org

1, 4, 27, 208, 1700, 14364, 123970, 1085760, 9612108, 85795600, 770755843, 6960408624, 63127818572, 574609830760, 5246348656500, 48027225765120, 440671237120764, 4051508174260272, 37315784743418332

Offset: 0

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

Table[(2 n+2)/(3 n+2) Binomial[4 n+1, n], {n, 0, 20}] (* Vaclav Kotesovec, Dec 02 2016 *)

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

a(n) = (2n+2)/(3n+2) * C(4n+1, n). - Ralf Stephan, Apr 30 2004
a(n) = C(4n,n)-C(4n,n-2)=A039598(2n,n). - Paul Barry, Apr 21 2009
G.f.: (g-2)*g^2/(3*g-4) where g = 1+x*g^4 is the g.f. of A002293. - Mark van Hoeij, Nov 11 2011
Conjecture: 9*n*(3*n+2)*(3*n+1)*a(n) +12*(-55*n^3-59*n^2+65*n-11)*a(n-1) -32*(4*n-5)*(4*n-3)*(2*n-3)*a(n-2)=0. - R. J. Mathar, May 22 2013
a(n) = Sum_{k=0..n}((n+k+1)*binomial(n+k,k)*binomial(3*n-k,n-k))/(2*n+1). - Vladimir Kruchinin, Dec 02 2016
a(n) ~ 2^(8*n+7/2)*3^(-3*n-5/2)/sqrt(Pi*n). - Ilya Gutkovskiy, Dec 02 2016

More terms from Ralf Stephan, Apr 30 2004

A052173 Another version of the Catalan triangle A008315.

Original entry on oeis.org

1, 1, 0, 1, 0, 1, 1, 0, 2, 0, 1, 0, 3, 0, 2, 1, 0, 4, 0, 5, 0, 1, 0, 5, 0, 9, 0, 5, 1, 0, 6, 0, 14, 0, 14, 0, 1, 0, 7, 0, 20, 0, 28, 0, 14, 1, 0, 8, 0, 27, 0, 48, 0, 42, 0, 1, 0, 9, 0, 35, 0, 75, 0, 90, 0, 42, 1, 0, 10, 0, 44, 0, 110, 0, 165, 0, 132, 0, 1, 0, 11, 0, 54, 0, 154, 0, 275, 0, 297, 0

Offset: 0

N. J. A. Sloane, Jan 26 2000

			1;
1 0;
1 0 1;
1 0 2 0;
1 0 3 0 2;
1 0 4 0 5 0;
1 0 5 0 9 0 5;
...

R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6
W. F. Klostermeyer, M. E. Mays, L. Soltes and G. Trapp, A Pascal rhombus, Fibonacci Quarterly, 35 (1997), 318-328.

See A008315 (the main entry for this triangle) for more information.

Reflection of A053121.

a(n, k) = a(n-1, k-2)+a(n-1, k) with a(0, 0)=1 and a(n, k)=0 if k < 0 or k > n.

More terms from Henry Bottomley, Aug 23 2001

A108786 Yet another version of the Catalan triangle A008315.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 3, 2, 1, 4, 5, 1, 5, 9, 5, 1, 6, 14, 14, 1, 7, 20, 28, 14, 1, 8, 27, 48, 42, 1, 9, 35, 75, 90, 42, 1, 10, 44, 110, 165, 132, 1, 11, 54, 154, 275, 297, 132, 1, 12, 65, 208, 429, 572, 429, 1, 13, 77, 273, 637, 1001, 1001, 429, 1, 14, 90, 350, 910, 1638, 2002

Offset: 0

N. J. A. Sloane, Nov 09 2006

			.......|...1
.......|.......1
.......|...1.......1
.......|.......2.......1
.......|...2.......3.......1
.......|.......5.......4.......1
.......|...5.......9.......5.......1
.......|......14......14.......6.......1
.......|..14......28......20.......7.......1
.......|......42......48......27.......8.......1

J. H. Conway and D. A. Smith, On Quaternions and Octonions, A K Peters, Ltd., Natick, MA, 2003. See p. 60. MR1957212 (2004a:17002)

R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6
W. F. Klostermeyer, M. E. Mays, L. Soltes and G. Trapp, A Pascal rhombus, Fibonacci Quarterly, 35 (1997), 318-328.

See A008315 (the main entry for this triangle) for more information.

Cf. A053121, A008313, A052173.

A008315 := proc(n,k)
    binomial(n,k)-binomial(n,k-1) ;
end:
for n from 0 to 30 do
    for k from 0 to n/2 do
        printf("%d, ",A008315(n,k)) ;
    od:
od: # R. J. Mathar, Feb 13 2008

A027301 a(n) = self-convolution of row n of Catalan triangle (A008315).

Original entry on oeis.org

1, 2, 4, 13, 26, 100, 196, 820, 1581, 6954, 13244, 60214, 113620, 528840, 990756, 4692780, 8741876, 41970280, 77824912, 377687453, 697689538, 3415756084, 6289798684, 31018849628, 56964451670, 282658474700, 517885177320

Offset: 0

Clark Kimberling

nonn

A027303 a(n) = Sum_{k=0..floor((n-3)/2)} T(n,k) * T(n,k+2), with T given by A008315.

Original entry on oeis.org

2, 5, 34, 98, 496, 1545, 7010, 22924, 98636, 333463, 1393014, 4824350, 19795200, 69818507, 283195638, 1013208668, 4078132240, 14758843463, 59087809638, 215859015800, 860951003152, 3169965920150, 12609168181324, 46735070397768, 185532421441976, 691582241867227

Offset: 3

Clark Kimberling

nonn

More terms from Sean A. Irvine, Oct 26 2019

A027304 a(n) = Sum_{k=0..floor((n-5)/2)} T(n,k) * T(n,k+1), with T given by A008315.

Original entry on oeis.org

5, 14, 126, 384, 2355, 7568, 39545, 131768, 632177, 2162510, 9853892, 34384064, 151498203, 536917884, 2311543325, 8295107912, 35123988977, 127340511540, 532611748052, 1947506206400, 8069978842770, 29721855687264, 122273332182351, 453128454670472

Offset: 5

Clark Kimberling

nonn

Cf. A008315.

More terms from Sean A. Irvine, Oct 26 2019

A027305 a(n) = Sum_{k=0..floor((n+1)/2)} (k+1) * A008315(n, k).

Original entry on oeis.org

1, 3, 5, 13, 24, 58, 111, 257, 500, 1126, 2210, 4882, 9632, 20980, 41531, 89497, 177564, 379438, 754014, 1600406, 3184016, 6720748, 13382710, 28117498, 56026984, 117254268, 233765636, 487589572, 972504704, 2022568168

Offset: 0

Clark Kimberling

nonn

Cf. A065982.

a(2n) = A065982(n+1)/2 = (n+2)*binomial(2n+2, n+1)/2 - 4^n.

cp's OEIS Frontend

A026004 a(n) = T(3n+1,n), where T = Catalan triangle (A008315).

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Maxima

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A026008 a(n) = T(n, floor(n/2)), where T = Catalan triangle (A008315).

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A027302 a(n) = Sum_{k=0..floor((n-1)/2)} T(n,k) * T(n,k+1), with T given by A008315.

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A026005 a(n) = T(4*n,n), where T = Catalan triangle (A008315).

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PARI

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A052173 Another version of the Catalan triangle A008315.

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A108786 Yet another version of the Catalan triangle A008315.

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Maple

A027301 a(n) = self-convolution of row n of Catalan triangle (A008315).

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A027303 a(n) = Sum_{k=0..floor((n-3)/2)} T(n,k) * T(n,k+2), with T given by A008315.

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A027304 a(n) = Sum_{k=0..floor((n-5)/2)} T(n,k) * T(n,k+1), with T given by A008315.

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Extensions

A027305 a(n) = Sum_{k=0..floor((n+1)/2)} (k+1) * A008315(n, k).

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