A013698 -id:A013698 - cp's OEIS frontend

A025174 a(n) = binomial(3n-1, n-1).

0, 1, 5, 28, 165, 1001, 6188, 38760, 245157, 1562275, 10015005, 64512240, 417225900, 2707475148, 17620076360, 114955808528, 751616304549, 4923689695575, 32308782859535, 212327989773900, 1397281501935165, 9206478467454345, 60727722660586800, 400978991944396320

Offset: 0

Wouter Meeussen, Emeric Deutsch

Number of standard tableaux of shape (2n-1,n). Example: a(2)=5 because in the top row we can have 123, 124, 125, 134, or 135. - Emeric Deutsch, May 23 2004
Number of peaks in all generalized {(1,2),(1,-1)}-Dyck paths of length 3n.
Positive terms in this sequence are the numbers k such that k and 2k are consecutive terms in a row of Pascal's triangle. 1001 is the only k such that k, 2k, and 3k are consecutive terms in a row of Pascal's triangle. - J. Lowell, Mar 11 2023

			L.g.f.: L(x) = x + 5*x^2/2 + 28*x^3/3 + 165*x^4/4 + 1001*x^5/5 + 6188*x^6/6 + ...
where G(x) = exp(L(x)) satisfies G(x) = 1 + x*G(x)^3, and begins:
exp(L(x)) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + ... + A001764(n)*x^n + ...

B. C. Berndt, Ramanujan's Notebooks Part I, Springer-Verlag, see Entry 14, Corollary 1, p. 71.

Robert Israel, Table of n, a(n) for n = 0..1190
Paul Barry, On the Central Antecedents of Integer (and Other) Sequences, J. Int. Seq., Vol. 23 (2020), Article 20.8.3.
D. Kruchinin and V. Kruchinin, A Generating Function for the Diagonal T2n,n in Triangles, Journal of Integer Sequence, Vol. 18 (2015), article 15.4.6.
W. Mlotkowski and K. A. Penson, Probability distributions with binomial moments, arXiv preprint arXiv:1309.0595 [math.PR], 2013.
Emanuele Munarini, Shifting Property for Riordan, Sheffer and Connection Constants Matrices, Journal of Integer Sequences, Vol. 20 (2017), Article 17.8.2.
Stepan Orevkov, Asymptotics of the number of lattice triangulations of rectangles of width 4 and 5, arXiv:2412.17065 [math.CO], 2024. See p. 16.
N. J. Wildberger and Dean Rubine, A Hyper-Catalan Series Solution to Polynomial Equations, and the Geode, Amer. Math. Monthly (2025). See section 12.

Cf. A001764 (binomial(3n,n)/(2n+1)), A117671 (C(3n+1,n+1)), A004319, A005809, A006013, A013698, A045721, A117671, A165817, A224274, A236194.

Cf. A000108, A001045, A005156, A224274.

[Binomial(3*n-1,n-1): n in [0..30]]; // Vincenzo Librandi, Nov 12 2014

with(combinat):seq(numbcomp(3*i,i), i=0..20); # Zerinvary Lajos, Jun 16 2007

Table[ GegenbauerC[ n, n, 1 ]/2, {n, 0, 24} ]
Join[{0},Table[Binomial[3n-1,n-1],{n,20}]] (* Harvey P. Dale, Oct 19 2022 *)
nmax=23; CoefficientList[Series[(2+HypergeometricPFQ[{1/3,2/3},{1/2,1},27x/4])/3-1,{x,0,nmax}],x]Range[0,nmax]! (* Stefano Spezia, Dec 31 2024 *)

vector(30, n, n--; binomial(3*n-1, n-1)) \\ Altug Alkan, Nov 04 2015

G.f.: z*g^2/(1-3*z*g^2), where g=g(z) is given by g=1+z*g^3, g(0)=1, that is, (in Maple command) g := 2*sin(arcsin(3*sqrt(3*z)/2)/3)/sqrt(3*z). - Emeric Deutsch, May 22 2003
a(n) = Sum_{k=0..n} ((3k+1)/(2n+k+1))C(3n, 2n+k)*A001045(k). - Paul Barry, Oct 07 2005
Hankel transform of a(n+1) is A005156(n+1). - Paul Barry, Apr 14 2008
G.f.: x*B'(x)/B(x) where B(x) is the g.f. of A001764. - Vladimir Kruchinin Feb 03 2013
D-finite with recurrence: 2*n*(2*n-1)*a(n) -3*(3*n-1)*(3*n-2)*a(n-1)=0. - R. J. Mathar, Feb 05 2013
Logarithmic derivative of A001764; g.f. of A001764 satisfies G(x) = 1 + x*G(x)^3. - Paul D. Hanna, Jul 14 2013
G.f.: (2*cos((1/3)*arcsin((3/2)*sqrt(3*x)))-sqrt(4-27*x))/(3*sqrt(4-27*x)). - Emanuele Munarini, Oct 14 2014
a(n) = Sum_{k=1..n} binomial(n-1,n-k)*binomial(2*n,n-k). - Vladimir Kruchinin, Nov 12 2014
a(n) = [x^n] C(x)^n for n >= 1, where C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the generating function for A000108 (Ramanujan). - Peter Bala, Jun 24 2015
From Peter Bala, Nov 04 2015: (Start)
Without the initial term 0, the o.g.f. equals f(x)*g(x)^2, where f(x) is the o.g.f. for A005809 and g(x) is the o.g.f. for A001764. g(x)^2 is the o.g..f for A006013. More generally, f(x)*g(x)^k is the o.g.f. for the sequence binomial(3*n + k,n). Cf. A045721 (k = 1), A004319 (k = 3), A236194 (k = 4), A013698 (k = 5), A165817 (k = -1), A117671 (k = -2). (End)
G.f.: ( 2F1(1/3,2/3;1/2;27*x/4)-1)/3. - R. J. Mathar, Jan 27 2020
O.g.f. without the initial term 0, in the form g(x)=(2*cos(arcsin((3*sqrt(3)*sqrt(x))/2)/3)/sqrt(4-27*x)-1)/(3*x), satisfies the following algebraic equation: 1+(9*x-1)*g(x)+x*(27*x-4)*g(x)^2+x^2*(27*x-4)*g(x)^3=0. - Karol A. Penson, Oct 11 2021
O.g.f. equals f(x)/(1 - 2*f(x)), where f(x) = series reversion (x/(1 + x)^3) = x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + ... is the o.g.f. of A001764 with the initial term omitted. Cf. A224274. - Peter Bala, Feb 03 2022
Right-hand side of the identities (1/2)*Sum_{k = 0..n} (-1)^(n+k)*C(x*n,n-k)*C((x+2)*n+k-1,k) = C(3*n-1,n-1) and (1/3)*Sum_{k = 0..n} (-1)^k* C(x*n,n-k)*C((x-3)*n+k-1,k) = C(3*n-1,n-1), both valid for n >= 1 and x arbitrary. - Peter Bala, Feb 28 2022
a(n) ~ 2^(-2*n)*3^(3*n)/(2*sqrt(3*n*Pi)). - Stefano Spezia, Apr 25 2024
a(n) = Sum_{k = 0..n-1} binomial(2*n+k-1, k) = Sum_{k = 0..n-1} (-1)^(n+k+1)* binomial(3*n, k). - Peter Bala, Jul 21 2024
E.g.f.: (2 + hypergeom([1/3, 2/3], [1/2, 1], 27*x/4))/3 - 1. - Stefano Spezia, Dec 31 2024

A045721 a(n) = binomial(3*n+1,n).

Original entry on oeis.org

1, 4, 21, 120, 715, 4368, 27132, 170544, 1081575, 6906900, 44352165, 286097760, 1852482996, 12033222880, 78378960360, 511738760544, 3348108992991, 21945588357420, 144079707346575, 947309492837400, 6236646703759395

Offset: 0

Emeric Deutsch

Number of leaves in all noncrossing rooted trees on n nodes on a circle.
Number of standard tableaux of shape (n-1,1^(2n-3)). - Emeric Deutsch, May 25 2004
a(n) = number of Dyck (2n-3)-paths with exactly one descent of odd length. For example, a(3) counts all 5 Dyck 3-paths except UDUDUD. - David Callan, Jul 25 2005
a(n+2) gives row sums of A119301. - Paul Barry, May 13 2006
a(n) is the number of paths avoiding UU from (0,0) to (3n,n) and taking steps from {U,H}. - Shanzhen Gao, Apr 15 2010
Central coefficients of triangle A078812. - Vladimir Kruchinin, May 10 2012
Row sums of A252501. - L. Edson Jeffery, Dec 18 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Milan Janjic, Two Enumerative Functions
D. Kruchinin and V. Kruchinin, A Method for Obtaining Generating Function for Central Coefficients of Triangles, Journal of Integer Sequence, Vol. 15 (2012), article 12.9.3.
W. Mlotkowski and K. A. Penson, Probability distributions with binomial moments, arXiv preprint arXiv:1309.0595 [math.PR], 2013.
Index entries for sequences related to rooted trees

Cf. A252501, A263134 (partial sums), A001764, A004319, A005809, A006013, A013698, A025174, A117671, A165817, A236194.

[Binomial(3*n+1, n): n in [0..20]]; // Vincenzo Librandi, Aug 07 2014

[seq( binomial(3*n+1,n),n=0..40)]; # N. J. A. Sloane, Jun 09 2007

Table[Binomial[3 n + 1, n], {n, 0, 20}] (* Vincenzo Librandi, Aug 07 2014 *)

a(n)=binomial(3*n+1,n) \\ Charles R Greathouse IV, Mar 18 2014

a(n) is asymptotic to c/sqrt(n)*(27/4)^n with c=0.73... - Benoit Cloitre, Jan 27 2003
G.f.: gz^2/(1-3zg^2), where g=g(z) is given by g=1+zg^3, g(0)=1, i.e. (in Maple command) g := 2*sin(arcsin(3*sqrt(3*z)/2)/3)/sqrt(3*z). - Emeric Deutsch, May 22 2003
a(n+2) = C(3n+1,n) = Sum_{k=0..n} C(3n-k,n-k). - Paul Barry, May 13 2006
a(n+2) = C(3n+1,2n+1) = A078812(2n,n). - Paul Barry, Nov 09 2006
G.f.: A(x)=(2*cos(asin((3^(3/2)*sqrt(x))/2)/3)* sin(asin((3^(3/2)* sqrt(x))/2)/3))/(sqrt(3)*sqrt(1-(27*x)/4)*sqrt(x)). - Vladimir Kruchinin, Jun 10 2012
From Peter Luschny, Sep 04 2012: (Start)
O.g.f.: hypergeometric2F1([2/3, 4/3], [3/2], x*27/4).
a(n) = (n+1)*hypergeometric2F1([-2*n, -n], [2], 1).  (End)
D-finite with recurrence 2*n*(2*n+1)*a(n) - 3*(3*n-1)*(3*n+1)*a(n-1) = 0. - R. J. Mathar, Feb 05 2013
a(n) = Sum_{r=0..n} C(n,r) * C(2*n+1,r). - J. M. Bergot, Mar 18 2014
From Peter Bala, Nov 04 2015: (Start)
a(n) = Sum_{k = 0..n} 1/(2*k + 1)*binomial(3*n - 3*k,n - k)*binomial(3*k, k).
O.g.f. equals f(x)*g(x), where f(x) is the o.g.f. for A005809 and g(x) is the o.g.f. for A001764. More generally, f(x)*g(x)^k is the o.g.f. for the sequence binomial(3*n + k,n). Cf. A025174 (k = 2), A004319 (k = 3), A236194 (k = 4), A013698 (k = 5), A165817 (k = -1), A117671 (k = -2). (End)
a(n) = [x^n] 1/(1 - x)^(2*(n+1)). - Ilya Gutkovskiy, Oct 10 2017
O.g.f.: (i/24)*((4*sqrt(4 - 27*z) + 12*i*sqrt(3)*sqrt(z))^(2/3) - (4*sqrt(4 - 27*z) - 12*i*sqrt(3)*sqrt(z))^(2/3))*sqrt(3)*8^(1/3)*sqrt(4 - 27*z)/(sqrt(z)*(-4 + 27*z)), where i = sqrt(-1). - Karol A. Penson, Dec 13 2023
a(n-1) = (1/(4*n))*binomial(2*n, n)^2 * (1 - 3*((n - 1)/(n + 1))^3 + 5*((n - 1)*(n - 2)/((n + 1)*(n + 2)))^3 - 7*((n - 1)*(n - 2)*(n - 3)/((n + 1)*(n + 2)*(n + 3)))^3 + - ...) for n >= 1. Cf. A112029. - Peter Bala, Aug 08 2024

Simpler definition from Ira M. Gessel, May 26 2007. This change means that most of the offsets in the comments will now need to be changed too.

A092392 Triangle read by rows: T(n,k) = C(2*n - k,n), 0 <= k <= n.

Original entry on oeis.org

1, 2, 1, 6, 3, 1, 20, 10, 4, 1, 70, 35, 15, 5, 1, 252, 126, 56, 21, 6, 1, 924, 462, 210, 84, 28, 7, 1, 3432, 1716, 792, 330, 120, 36, 8, 1, 12870, 6435, 3003, 1287, 495, 165, 45, 9, 1, 48620, 24310, 11440, 5005, 2002, 715, 220, 55, 10, 1, 184756, 92378, 43758, 19448, 8008, 3003, 1001, 286, 66, 11, 1

Offset: 0

Ralf Stephan, Mar 21 2004

First column is C(2*n,n) or A000984. Central coefficients are C(3*n,n) or A005809. - Paul Barry, Oct 14 2009
T(n,k) = A046899(n,n-k), k = 0..n-1. - Reinhard Zumkeller, Jul 27 2012
From  Peter Bala, Nov 03 2015: (Start)
Viewed as the square array [binomial (2*n + k, n + k)]n,k>=0  this is the generalized Riordan array ( 1/sqrt(1 - 4*x),c(x) ) in the sense of the Bala link, where c(x) is the o.g.f. for A000108.
The square array factorizes as ( 1/(2 - c(x)),x*c(x) ) * ( 1/(1 - x),1/(1 - x) ), which equals the matrix product of A100100 with the square Pascal matrix [binomial (n + k,k)]n,k>=0. See the example below. (End)

			From _Paul Barry_, Oct 14 2009: (Start)
Triangle begins
  1,
  2, 1,
  6, 3, 1,
  20, 10, 4, 1,
  70, 35, 15, 5, 1,
  252, 126, 56, 21, 6, 1,
  924, 462, 210, 84, 28, 7, 1,
  3432, 1716, 792, 330, 120, 36, 8, 1
Production array is
  2, 1,
  2, 1, 1,
  2, 1, 1, 1,
  2, 1, 1, 1, 1,
  2, 1, 1, 1, 1, 1,
  2, 1, 1, 1, 1, 1, 1,
  2, 1, 1, 1, 1, 1, 1, 1,
  2, 1, 1, 1, 1, 1, 1, 1, 1,
  2, 1, 1, 1, 1, 1, 1, 1, 1, 1 (End)
As a square array = A100100 * square Pascal matrix:
  /1   1  1  1 ...\   / 1          \/1 1  1  1 ...\
  |2   3  4  5 ...|   | 1 1        ||1 2  3  4 ...|
  |6  10 15 21 ...| = | 3 2 1      ||1 3  6 10 ...|
  |20 35 56 84 ...|   |10 6 3 1    ||1 4 10 20 ...|
  |70 ...         |   |35 ...      ||1 ...        |
- _Peter Bala_, Nov 03 2015

Reinhard Zumkeller, Rows n=0..149 of triangle, flattened
P. Bala, Notes on generalized Riordan arrays
P. Barry, On the Central Coefficients of Riordan Matrices, Journal of Integer Sequences, 16 (2013), #13.5.1.
Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, 2016, #16.3.5.
Ik-Pyo Kim, Michael J. Tsatsomeros, Inverse Relations in Shapiro's Open Questions, arXiv:1707.06590 [math.CO], 2017. See p. 7.

Rows 0-14 are A000984, A001700, A001791, A002054, A002694, A003516, A002696, A030053, A004310, A030054, A004311, A030055, A004312, A030056, A004313.
Columns are A000217, A000292, A000332, A000389, A000579.
Diagonals are A005809, A045721, A025174, A004319, A013698, A003408.
Cf. A100100.

a092392 n k = a092392_tabl !! (n-1) !! (k-1)
a092392_row n = a092392_tabl !! (n-1)
a092392_tabl = map reverse a046899_tabl
-- Reinhard Zumkeller, Jul 27 2012

/* As a triangle */ [[Binomial(2*n-k, n): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Nov 22 2017

A092392 := proc(n,k)
    binomial(2*n-k,n-k) ;
end proc:
seq(seq(A092392(n,k),k=0..n),n=0..10) ; # R. J. Mathar, Feb 06 2015

Table[Binomial[2 n - k, n], {n, 0, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, Mar 19 2016 *)

C(x):=(1-sqrt(1-4*x))/2;
A(x,y):=(1/sqrt(1-4*x))/(1-y*C(x));
taylor(A(x,y),y,0,10,x,0,10); /* Vladimir Kruchinin, Mar 19 2016 */

for(n=0,10, for(k=0,n, print1(binomial(2*n - k,n), ", "))) \\ G. C. Greubel, Nov 22 2017

As a number triangle, this is T(n, k) = if(k <= n, C(2*n - k, n), 0). Its row sums are C(2*n + 1, n + 1) = A001700. Its diagonal sums are A176287. - Paul Barry, Apr 23 2005
G.f. of column k: 2^k/[sqrt(1 - 4*x)*(1 + sqrt(1 - 4*x))^k].
As a number triangle, this is the Riordan array (1/sqrt(1 - 4*x), x*c(x)), c(x) the g.f. of A000108. - Paul Barry, Jun 24 2005
G.f.: A(x,y)=1/sqrt(1 - 4*x)/(1-y*x*C(x)), where C(x) is g.f. of Catalan numbers. - Vladimir Kruchinin, Mar 19 2016

Diagonal sums comment corrected by Paul Barry, Apr 14 2010

Offset corrected by R. J. Mathar, Feb 08 2013

A165817 Number of compositions (= ordered integer partitions) of n into 2n parts.

Original entry on oeis.org

1, 2, 10, 56, 330, 2002, 12376, 77520, 490314, 3124550, 20030010, 129024480, 834451800, 5414950296, 35240152720, 229911617056, 1503232609098, 9847379391150, 64617565719070, 424655979547800, 2794563003870330, 18412956934908690, 121455445321173600

Offset: 0

Thomas Wieder, Sep 29 2009

Number of ways to put n indistinguishable balls into 2*n distinguishable boxes.

Number of rankings of n unlabeled elements for 2*n levels.

			Let [1,1,1], [1,2] and [3] be the integer partitions of n=3.
Then [0,0,0,1,1,1], [0,0,0,0,1,2] and [0,0,0,0,0,3] are the corresponding partitions occupying 2*n = 6 positions.
We have to take into account the multiplicities of the parts including the multiplicities of the zeros.
Then
  [0,0,0,1,1,1] --> 6!/(3!*3!) = 20
  [0,0,0,0,1,2] --> 6!/(4!*1!*1!) = 30
  [0,0,0,0,0,3] --> 6!/(5!*1!) = 6
and thus a(3) = 20+30+6=56.
a(2)=10, since we have 10 ordered partitions of n=2 where the parts are distributed over 2*n=4 boxes:
  [0, 0, 0, 2]
  [0, 0, 1, 1]
  [0, 0, 2, 0]
  [0, 1, 0, 1]
  [0, 1, 1, 0]
  [0, 2, 0, 0]
  [1, 0, 0, 1]
  [1, 0, 1, 0]
  [1, 1, 0, 0]
  [2, 0, 0, 0].

Alois P. Heinz, Table of n, a(n) for n = 0..400
W. Mlotkowski and K. A. Penson, Probability distributions with binomial moments, arXiv preprint arXiv:1309.0595 [math.PR], 2013.

Cf. A000079, A001700, A059481, A081204, A001764, A004319, A006013, A005809, A013698, A025174, A045721, A117671, A236194.

[Binomial(3*n-1, n): n in [0..30]]; // Vincenzo Librandi, Aug 07 2014

for n from 0 to 16 do
a[n] := 9*sqrt(3)*GAMMA(n+5/3)*GAMMA(n+4/3)*27^n/(Pi*GAMMA(2*n+3))
end do;

Table[Binomial[3 n - 1, n], {n, 0, 20}] (* Vincenzo Librandi, Aug 07 2014 *)

vector(30, n, n--; binomial(3*n-1, n)) \\ Altug Alkan, Nov 04 2015

from math import comb
def A165817(n): return comb(3*n-1,n) if n else 1 # Chai Wah Wu, Oct 11 2023

def A165817(n):
    return rising_factorial(2*n,n)/falling_factorial(n,n)
[A165817(n) for n in (0..22)]   # Peter Luschny, Nov 21 2012

a(n) = 9*sqrt(3)*GAMMA(n+5/3)*GAMMA(n+4/3)*27^n/(Pi*GAMMA(2*n+3)).
a(n) = binomial(3*n-1, n);
Let denote P(n) = the number of integer partitions of n,
p(i) = the number of parts of the i-th partition of n,
d(i) = the number of different parts of the i-th partition of n,
m(i,j) = multiplicity of the j-th part of the i-th partition of n.
Then one has:
a(n) = Sum_{i=1..P(n)} (2*n)!/((2*n-p(i))!*(Prod_{j=1..d(i)} m(i,j)!)).
a(n) = rf(2*n,n)/ff(n,n), where rf is the rising factorial and ff the falling factorial. - Peter Luschny, Nov 21 2012
G.f.: A(x) = x*B'(x)/B(x), where B(x) satisfies B(x)^3-2*B(x)^2+B(x)=x, B(x)=A006013(x). - Vladimir Kruchinin, Feb 06 2013
G.f.: A(x) = sqrt(3*x)*cot(asin((3^(3/2)*sqrt(x))/2)/3)/(sqrt(4-27*x)). - Vladimir Kruchinin, Mar 18 2015
a(n) = Sum_{k=0..n} binomial(n-1,n-k)*binomial(2*n,k). - Vladimir Kruchinin, Oct 06 2015
From Peter Bala, Nov 04 2015: (Start)
The o.g.f. equals f(x)/g(x), where f(x) is the o.g.f. for A005809 and g(x) is the o.g.f. for A001764. More generally, f(x)*g(x)^k is the o.g.f. for the sequence binomial(3*n + k,n). Cf. A045721 (k = 1), A025174 (k = 2), A004319 (k = 3), A236194 (k = 4), A013698 (k = 5) and A117671 (k = -2). (End)
a(n) = [x^n] 1/(1 - x)^(2*n). - Ilya Gutkovskiy, Oct 03 2017
a(n) = A059481(2n,n). - Alois P. Heinz, Oct 17 2022
From Peter Bala, Feb 14 2024: (Start)
a(n) = (-1)^n * binomial(-2*n, n).
a(n) = hypergeom([1 - 2*n, -n], [1], 1).
The g.f. A(x) satisfies A(x/(1 + x)^3) = 1/(1 - 2*x).
Sum_{n >= 0} a(n)/9^n = (1 + 4*cos(Pi/9))/3.
Sum_{n >= 0} a(n)/27^n = (3 + 4*sqrt(3)*cos(Pi/18))/9.
Sum_{n >= 0} a(n)*(2/27)^n = (2 + sqrt(3))/3. (End)
From Peter Bala, Sep 16 2024: (Start)
a(n) = Sum_{k = 0..n} binomial(n+k-1, k)*binomial(2*n-k-1, n-k).
More generally, a(n) = Sum_{k = 0..n} (-1)^k*binomial(x*n, k)*binomial((x+3)*n-k-1, n-k) for arbitrary x.
a(n) = (2/3) * Sum_{k = 0..n} (-1)^k*binomial(x*n+k-1, k)*binomial((x+3)*n, n-k) for n >= 1 and arbitrary x. (End)
G.f.: 1/(3-2*g) where g = 1+x*g^3 is the g.f. of A001764. - Seiichi Manyama, Aug 17 2025

a(0) prepended and more terms from Alois P. Heinz, Apr 04 2012

A004319 a(n) = binomial(3*n, n - 1).

Original entry on oeis.org

1, 6, 36, 220, 1365, 8568, 54264, 346104, 2220075, 14307150, 92561040, 600805296, 3910797436, 25518731280, 166871334960, 1093260079344, 7174519270695, 47153358767970, 310325523515700, 2044802197953900, 13488561475572645, 89067326568860640, 588671286046028640

Offset: 1

N. J. A. Sloane

Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.

Seiichi Manyama, Table of n, a(n) for n = 1..1000
Milan Janjic, Two Enumerative Functions.
Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Emanuele Munarini, Shifting Property for Riordan, Sheffer and Connection Constants Matrices, Journal of Integer Sequences, Vol. 20 (2017), Article 17.8.2.
Franck Ramaharo, Statistics on some classes of knot shadows, arXiv:1802.07701 [math.CO], 2018.

Cf. A045721 (k=1), A025174 (k=2), A236194 (k=4), A013698 (k=5), A165817 (k=-1), A117671 (k=-2).

Cf. A000139, A001764, A005809, A006013, A236194, A001791, A004331, A004343, A004356, A004369, A004382.

A004319 := proc(n)
binomial(3*n,n-1);
end proc: # R. J. Mathar, Aug 10 2015

Table[Binomial[3n, n - 1], {n, 20}] (* Harvey P. Dale, Sep 21 2011 *)

a(n):=sum((binomial(3*i-1,2*i-1)*binomial(3*n-3*i-3,2*n-2*i-2))/(2*n-2*i-1),i,1,n-1)/2; /* Vladimir Kruchinin, May 15 2013 */

vector(30, n, binomial(3*n, n-1)) \\ Altug Alkan, Nov 04 2015

G.f.: (g-1)/(1-3*z*g^2), where g = g(z) is given by g = 1 + z*g^3, g(0) = 1, i.e. (in Maple notation), g := 2*sin(arcsin(3*sqrt(3*z)/2)/3)/sqrt(3*z). - Emeric Deutsch, May 22 2003
a(n) = Sum_{i=0..n-1} binomial(i+2*n, i). - Ralf Stephan, Jun 03 2005
D-finite with recurrence -2*(2*n+1)*(n-1)*a(n) + 3*(3*n-1)*(3*n-2)*a(n-1) = 0. - R. J. Mathar, Feb 05 2013
a(n) = (1/2) * Sum_{i=1..n-1} binomial(3*i - 1, 2*i - 1)*binomial(3*n - 3*i - 3, 2*n - 2*i - 2)/(2*n - 2*i - 1). - Vladimir Kruchinin, May 15 2013
G.f.: x*hypergeom2F1(5/3, 4/3; 5/2; 27x/4). - R. J. Mathar, Aug 10 2015
a(n) = n*A001764(n). - R. J. Mathar, Aug 10 2015
From Peter Bala, Nov 04 2015: (Start)
With offset 0, the o.g.f. equals f(x)*g(x)^3, where f(x) is the o.g.f. for A005809 and g(x) is the o.g.f. for A001764. More generally, f(x)*g(x)^k is the o.g.f. for the sequence binomial(3*n + k, n). See the cross-references. (End)
G.f.: cos(t)/(2*sqrt(1 - (27*x)/4)) - sin(t)/(sqrt(3)*sqrt(x)), where t = arcsin((sqrt(27*x))/2)/3. - Vladimir Kruchinin, May 13 2016
a(n) = [x^(2*n+1)] 1/(1 - x)^n. - Ilya Gutkovskiy, Oct 10 2017
a(n) = binomial(n+1, 2) * A000139(n). - F. Chapoton, Feb 23 2024

A117671 a(n) = binomial(3*n+1, n+1).

Original entry on oeis.org

1, 6, 35, 210, 1287, 8008, 50388, 319770, 2042975, 13123110, 84672315, 548354040, 3562467300, 23206929840, 151532656696, 991493848554, 6499270398159, 42671977361650, 280576272201225

Offset: 0

Zerinvary Lajos, Apr 12 2006

a(n) = A258993(2*n+1, n). - Reinhard Zumkeller, Jun 22 2015

			if n=0 then C(3*0+1,0+1) = C(1,1) = 1.
if n=10 then C(3*10+1,10+1) = C(31,11) = 84672315.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Milan Janjic, Two Enumerative Functions

Cf. A025174: binomial(3n-1,n-1), A006013.

Cf. A258993, A001764, A004319, A005809, A013698, A045721, A165817, A236194.

a117671 n = a258993 (2 * n + 1) n  -- Reinhard Zumkeller, Jun 22 2015

seq(binomial(3*n+1,n+1),n=0..30); # Robert Israel, Oct 10 2017

Table[Binomial[3n+1,n+1],{n,0,20}] (* Harvey P. Dale, Jul 19 2011 *)

vector(30, n, n--; binomial(3*n+1, n+1)) \\ Altug Alkan, Nov 04 2015

G.f.: (2*(-1+Hypergeometric2F1[-(1/3),1/3,-(1/2),(27*x)/4]))/(3*x). - Harvey P. Dale, Jul 19 2011
G.f.: A(x) = B'(x)/B(x)-B'(x)-1/x, where B(x) = 4/3*sin(1/3*asin(sqrt((27*x)/4)))^2. - Vladimir Kruchinin, Nov 26 2014
From Peter Bala, Nov 04 2015: (Start)
With an extra initial term equal to 1, the o.g.f. equals f(x)/g(x)^2, where f(x) is the o.g.f. for A005809 and g(x) is the o.g.f. for A001764.
More generally, f(x)*g(x)^k is the o.g.f. for the sequence binomial(3*n + k,n). Cf. A045721 (k = 1), A025174 (k = 2), A004319 (k = 3), A236194 (k = 4), A013698 (k = 5), A165817 (k = -1). (End)
a(n) = [x^(2*n)] 1/(1 - x)^(n+2). - Ilya Gutkovskiy, Oct 10 2017
a(n+1) = 3*(3*n+2)*(3*n+4)*a(n)/(2*(n+2)*(2*n+1)). - Robert Israel, Oct 10 2017

A264772 Triangle T(n,k) = binomial(3n - 2k, 2*n - k), 0 <= k <= n.

Original entry on oeis.org

1, 3, 1, 15, 4, 1, 84, 21, 5, 1, 495, 120, 28, 6, 1, 3003, 715, 165, 36, 7, 1, 18564, 4368, 1001, 220, 45, 8, 1, 116280, 27132, 6188, 1365, 286, 55, 9, 1, 735471, 170544, 38760, 8568, 1820, 364, 66, 10, 1, 4686825, 1081575, 245157, 54264, 11628, 2380, 455, 78, 11, 1

Offset: 0

Peter Bala, Nov 24 2015

Riordan array (f(x), x*g(x)), where g(x) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + ... is the o.g.f. for A001764 and f(x) = g(x)/(3 - 2*g(x)) = 1 + 3*x + 15*x^2 + 84*x^3 + 495*x^4 + ... is the o.g.f. for A005809.
The even numbered columns give the Riordan array A119301, the odd numbered columns give the Riordan array A144484. A159841 is the array formed from columns 1,4,7,10,....
More generally, if R = (R(n,k))n,k>=0 is a proper Riordan array, m is a nonnegative integer and a > b are integers then the array with (n,k)-th element R((m + 1)*n - a*k, m*n - b*k) is also a Riordan array (not necessarily proper). Here we take R as Pascal's triangle and m = a = 2, b = 1. See A092392, A264773, A264774 and A113139 for further examples.

			Triangle begins
.n\k.|......0.....1....2....3...4..5...6..7...
----------------------------------------------
..0..|      1
..1..|      3     1
..2..|     15     4    1
..3..|     84    21    5    1
..4..|    495   120   28    6   1
..5..|   3003   715  165   36   7  1
..6..|  18564  4368 1001  220  45  8  1
..7..| 116280 27132 6188 1365 286 55  9  1
...

Michael De Vlieger, Table of n, a(n) for n = 0..11475
Peter Bala, A 4-parameter family of embedded Riordan arrays
Paul Barry, On the halves of a Riordan array and their antecedents, arXiv:1906.06373 [math.CO], 2019.
E. Lebensztayn, On the asymptotic enumeration of accessible automata, Discrete Mathematics and Theoretical Computer Science, Vol. 12, No.3, 2010, 75-80, Section 2.
R. Sprugnoli, An Introduction to Mathematical Methods in Combinatorics, CreateSpace Independent Publishing Platform 2006, Section 5.6, ISBN-13: 978-1502925244.

Cf. A005809 (column 0), A045721 (column 1), A025174 (column 2), A004319 (column 3), A236194 (column 4), A013698 (column 5). Cf. A001764, A007318, A092392, A119301 (C(3n-k,2n)), A144484 (C(3n+1-k,2n+1)), A159841 (C(3n+1,2n+k+1)), A264773, A264774.

/* As triangle */ [[Binomial(3*n-2*k, n-k): k in [0..n]]: n in [0.. 10]]; // Vincenzo Librandi, Dec 02 2015

A264772:= proc(n,k) binomial(3*n - 2*k, 2*n - k); end proc:
seq(seq(A264772(n,k), k = 0..n), n = 0..10);

Table[Binomial[3 n - 2 k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* Michael De Vlieger, Dec 01 2015 *)

T(n,k) = binomial(3*n - 2*k, n - k).

O.g.f.: f(x)/(1 - t*x*g(x)), where f(x) = Sum_{n >= 0} binomial(3*n,n)*x^n and g(x) = Sum_{n >= 0} 1/(2*n + 1)*binomial(3*n,n)*x^n.

A236194 a(n) = binomial(3n+1, n-1).

Original entry on oeis.org

1, 7, 45, 286, 1820, 11628, 74613, 480700, 3108105, 20160075, 131128140, 854992152, 5586853480, 36576848168, 239877544005, 1575580702584, 10363194502115, 68248282427325, 449972009097765, 2969831763694950, 19619725782651120, 129728497393775280

Offset: 1

Bruno Berselli, Jan 20 2014

This sequence is related to A006013 by a(n)/n = A006013(n)/2.

Bruno Berselli, Table of n, a(n) for n = 1..100
Emanuele Munarini, Shifting Property for Riordan, Sheffer and Connection Constants Matrices, Journal of Integer Sequences, Vol. 20 (2017), Article 17.8.2.

Cf. A006013; A025174: C(3n-1, n-1); A117671: C(3n+1, n+1).
Second column of the triangle A159841.
Third column of the triangle A119301.
Cf. A001764, A004319, A005809, A013698, A045721, A165817.
Cf. A045721, A117671.

Magma
```
[Binomial(3*n+1,n-1): n in [1..30]];
```
Mathematica
```
Table[Binomial[3n+1, n-1], {n, 30}]
```

makelist(binomial(3*n+4,n),n,0,40); /* Emanuele Munarini, Oct 14 2014 */

vector(30, n, binomial(3*n+1, n-1)) \\ Altug Alkan, Nov 04 2015

[binomial(3*n+1,n-1) for n in range(1,31)] # G. C. Greubel, Nov 09 2022

G.f.: (sqrt(4-27*x)*cos((2/3)*arcsin((3/2)*sqrt(3*x))) + sqrt(3*x)*sin((2/3)*arcsin((3/2)*sqrt(3*x))) - sqrt(4-27*x))/(3*sqrt(4-27*x)*x^2). - Emanuele Munarini, Oct 14 2014
From Peter Bala, Nov 04 2015: (Start)
With offset 0, the o.g.f. equals f(x)*g(x)^4, where f(x) is the o.g.f. for A005809 and g(x) is the o.g.f. for A001764.
More generally, f(x)*g(x)^k is the o.g.f. for the sequence binomial(3*n + k,n). Cf. A045721 (k = 1), A025174 (k = 2), A004319 (k = 3), A013698 (k = 5), A165817 (k = -1), A117671 (k = -2). (End)
a(n) = [x^n] x/(1 - x)^(2*n+3). - Ilya Gutkovskiy, Oct 10 2017
From Karol A. Penson, Mar 02 2024: (Start)
G.f.: ((sqrt(3)*sqrt(x)*i + sqrt(4 - 27*x))*(4*sqrt(4 - 27*x) - 12*i*sqrt(3)*sqrt(x))^(2/3) + (-sqrt(3)*sqrt(x)*i + sqrt(4 - 27*x))*(4*sqrt(4 - 27*x) + 12*i*sqrt(3)*sqrt(x))^(2/3) - 8*sqrt(4 - 27*x))/(24*sqrt(4 - 27*x)*x^2), where i is the imaginary unit, i=sqrt(-1).
G.f.: hypergeometric3F2([5/3,2,7/3],[5/2,3],27*x/4).
G.f. = G satisfies the algebraic equation: 1 + (7*z-1)*G + (27*z-4)*z^2*G^2 + (27*z-4)*z^4*G^3 = 0. (End)

A102537 Triangle T(n,k) read by rows: (1/n) * C(2n+k,k-1) * C(n,k); n, k >= 1.

Original entry on oeis.org

1, 1, 3, 1, 8, 12, 1, 15, 55, 55, 1, 24, 156, 364, 273, 1, 35, 350, 1400, 2380, 1428, 1, 48, 680, 4080, 11628, 15504, 7752, 1, 63, 1197, 9975, 41895, 92169, 100947, 43263, 1, 80, 1960, 21560, 123970, 396704, 708400, 657800, 246675, 1, 99, 3036, 42504

Offset: 1

Ralf Stephan, Jan 14 2005

Number of dissections of a convex (2n+2)-gon by k-1 noncrossing diagonals into (2j+2)-gons, 1 <= j <= n-1.
Apparently, a signed, refined version of this array is given on page 65 of the Einziger link, related to the antipode of a Hopf algebra. - Tom Copeland, May 19 2015
The f-vectors of the simplicial noncrossing hypertree complexes of McCammond (p. 15). The reduced Euler characteristics are the signed Catalan numbers A000108. - Tom Copeland, May 19 2017
The rows seem to give (up to sign) the coefficients in the expansion of the integer-valued polynomial ((x+1)*(x+2)*...*(x+2n+1))*((x+n+2)*(x+n+3)*...*(x+2n)) / ((2n+1)!*(n)!) in the basis made of the binomial(x+i,i). - F. Chapoton, Nov 01 2022
Chapoton's observation above is correct: the precise expansion is ((x+1)*(x+2)*...*(x+2n+1))*((x+n+2)*(x+n+3)*...*(x+2n)) / ((2n+1)!*n!) = Sum_{k = 1..n} (-1)^(k+1)*T(n,n+1-k)*binomial(x+3*n+1-k, 3*n+1-k), as can be verified using the WZ algorithm. For example, n = 3 gives (x+1)*(x+2)*(x+3)*(x+4)*(x+5)*(x+6)*(x+7)*(x+5)(x+6)/(7!*3!) = 12*binomial(x+9,9) - 8*binomial(x+8,8) + binomial(x+7,7). - Peter Bala, Jun 25 2023

			Triangle begins
  1;
  1,  3;
  1,  8,   12;
  1, 15,   55,    55;
  1, 24,  156,   364,    273;
  1, 35,  350,  1400,   2380,   1428;
  1, 48,  680,  4080,  11628,  15504,   7752;
  1, 63, 1197,  9975,  41895,  92169, 100947,  43263;
  1, 80, 1960, 21560, 123970, 396704, 708400, 657800, 246675;

Michael De Vlieger, Table of n, a(n) for n = 1..11325 (rows 1 <= n <= 150).
H. Einziger, Incidence Hopf algebras: Antipodes, forest formulas, and noncrossing partitions, Dissertation (2010), George Washington University.
J. McCammond, Noncrossing Hypertrees, 2015.
Jean-Christophe Novelli and Jean-Yves Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014.
Jean-Christophe Novelli and Jean-Yves Thibon, Noncommutative Symmetric Functions and Lagrange Inversion II: Noncrossing partitions and the Farahat-Higman algebra, arXiv:2106.08257 [math.CO], 2021-2022.
E. Tzanaki, Polygon dissections and some generalizations of cluster complexes, arXiv:math/0501100 [math.CO], 2005.

Left-hand columns include A005563. Right-hand columns include essentially A001764 and A013698.
Row sums are in A003168.
Cf. A000108, A120986.
Cf. A243662 for rows reversed.

[[1/n * Binomial(2*n+k,k-1) * Binomial(n,k): k in [1..n]]: n in [1.. 15]]; // Vincenzo Librandi, May 20 2015

Table[1/n*Binomial[2 n + k, k - 1] Binomial[n, k], {n, 10}, {k, n}] // Flatten (* Michael De Vlieger, May 20 2017 *)

A013702 Degree of variety K_{2,n}^5.

Original entry on oeis.org

1, 2048, 832040, 193710244, 34673583028, 5372862566400, 763562937059280, 102703589621825280, 13319075453502743045, 1684658996331320739600, 209381188023138094045641, 25698882980999585848747760

Offset: 1

Joachim.Rosenthal(AT)nd.edu (Joachim Rosenthal)

M. S. Ravi et al., Dynamic pole assignment and Schubert calculus, SIAM J. Control Optimization, 34 (1996), 813-832, esp. p. 825.

Cf. A013698 (q=1), A013699 (q=2), A013700 (q=3), A013701 (q=4).

K(n,q=5)=(2*n+n*q+2*q)!*sum(j=0,q,((q-2*j)*(n+2)+1)/(n+j*(n+2))!/(n+1+(q-j)*(n+2))!)

Edited by Ralf Stephan, May 13 2003

cp's OEIS Frontend

A025174 a(n) = binomial(3n-1, n-1).

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A045721 a(n) = binomial(3*n+1,n).

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A092392 Triangle read by rows: T(n,k) = C(2*n - k,n), 0 <= k <= n.

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A165817 Number of compositions (= ordered integer partitions) of n into 2n parts.

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A004319 a(n) = binomial(3*n, n - 1).

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A117671 a(n) = binomial(3*n+1, n+1).

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A264772 Triangle T(n,k) = binomial(3n - 2k, 2*n - k), 0 <= k <= n.