A025174 -id:A025174 - cp's OEIS frontend

A013698 a(n) = binomial(3*n+2, n-1).

1, 8, 55, 364, 2380, 15504, 100947, 657800, 4292145, 28048800, 183579396, 1203322288, 7898654920, 51915526432, 341643774795, 2250829575120, 14844575908435, 97997533741800, 647520696018735, 4282083008118300

Offset: 1

Joachim.Rosenthal(AT)nd.edu (Joachim Rosenthal), Emeric Deutsch

Degree of variety K_{2,n}^1. Also number of double-rises (or odd-level peaks) in all generalized {(1,2),(1,-1)}-Dyck paths of length 3(n+1).
Number of dissections of a convex (2n+2)-gon by n-2 noncrossing diagonals into (2j+2)-gons, 1<=j<=n-1.
a(n) is the number of lattice paths from (0,0) to (3n+1,n-1) avoiding two consecutive up-steps. - Shanzhen Gao, Apr 20 2010

Robert Israel, Table of n, a(n) for n = 1..1000
M. S. Ravi et al., Dynamic Pole Assignment and Schubert Calculus, SIAM J. Control Optimization, 34 (1996), 813-832, esp. p. 825.
Daniel W. Stasiuk, An Enumeration Problem for Sequences of n-ary Trees Arising from Algebraic Operads, Master's Thesis, University of Saskatchewan-Saskatoon (2018).

Cf. A013699 (q=2), A013700 (q=3), A013701 (q=4), A013702 (q=5).

A column of triangle A102537. Cf. A001764, A004319, A005809, A006013, A025174, A045721, A117671, A165817, A236194.

List([1..25], n-> Binomial(3*n+2, n-1)) # G. C. Greubel, Mar 21 2019

[Binomial(3*n+2, n-1): n in [1..25]]; // Vincenzo Librandi, Aug 10 2015

seq(binomial(3*n+2,n-1), n=0..30); # Robert Israel, Aug 09 2015

Table[Binomial[3*n+2, n-1], {n, 25}] (* Arkadiusz Wesolowski, Apr 02 2012 *)

first(m)=vector(m,n,binomial(3*n+2, n-1)); /* Anders Hellström, Aug 09 2015 */

[binomial(3*n+2, n-1) for n in (1..25)] # G. C. Greubel, Mar 21 2019

G.f.: g/((g-1)^3*(3*g-1)) where g*(1-g)^2 = x. - Mark van Hoeij, Nov 09 2011
a(n) = Sum_{k=0..n-1} binomial(2*n+k+2,k). - Arkadiusz Wesolowski, Apr 02 2012
D-finite with recurrence 2*(2*n+3)*(n+1)*a(n) -n*(67*n+34)*a(n-1) +30*(3*n-1)*(3*n-2)*a(n-2)=0. - R. J. Mathar, Feb 05 2013
a(n+1) = (3*n+5)*(3*n+4)*(3*n+3)*a(n)/((2*n+5)*(2*n+4)*n). - Robert Israel, Aug 09 2015
With offset 0, the o.g.f. equals f(x)*g(x)^5, 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), A165817 (k = -1), A117671 (k = -2). - Peter Bala, Nov 04 2015

A005156 Number of alternating sign 2n+1 X 2n+1 matrices symmetric about the vertical axis (VSASM's); also 2n X 2n off-diagonally symmetric alternating sign matrices (OSASM's).

Original entry on oeis.org

1, 1, 3, 26, 646, 45885, 9304650, 5382618660, 8878734657276, 41748486581283118, 559463042542694360707, 21363742267675013243931852, 2324392978926652820310084179576, 720494439459132215692530771292602232, 636225819409712640497085074811372777428304

Offset: 0

N. J. A. Sloane

a(n+1) is the Hankel transform of A006013. - Paul Barry, Jan 20 2007

a(n+1) is the Hankel transform of A025174(n+1). - Paul Barry, Apr 14 2008

D. M. Bressoud, Proofs and Confirmations, Camb. Univ. Press, 1999; p. 201, VS(2n+1).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Gheorghe Coserea, Table of n, a(n) for n = 0..66
Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, 2016, #16.3.5.
Paul Barry, Chebyshev moments and Riordan involutions, arXiv:1912.11845 [math.CO], 2019.
Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.
Paul Barry, Extensions of Riordan Arrays and Their Applications, Mathematics (2025) Vol. 13, No. 2, 242. See p. 23.
M. T. Batchelor, J. de Gier and B. Nienhuis, The quantum symmetric XXZ chain at Delta=-1/2, alternating sign matrices and plane partitions, arXiv:cond-mat/0101385 [cond-mat.stat-mech], 2001, (see A_V(2n+1)).
N. T. Cameron, Random walks, trees and extensions of Riordan group techniques, Dissertation, Howard University, 2002.
J. de Gier, Loops, matchings and alternating-sign matrices, arXiv:math/0211285 [math.CO], 2002-2003.
I. Fischer, The number of monotone triangles with prescribed bottom row, arXiv:math/0501102 [math.CO], 2005.
I. Gessel and G. Xin, The generating function of ternary trees and continued fractions, arXiv:math/0505217 [math.CO], 2005.
W. Hebsich and M. Rubey, Extended Rate, More Gfun, arXiv:math/0702086 [math.CO], 2007. [See p. 23.]
G. Kuperberg, Symmetry classes of alternating-sign matrices under one roof, arXiv:math/0008184 [math.CO], 2000-2001, (see A_V(2n+1)).
A. V. Razumov and Yu. G. Stroganov, On refined enumerations of some symmetry classes of alternating sign matrices, arXiv:math-ph/0312071, 2003.
D. P. Robbins, Symmetry classes of alternating sign matrices, arXiv:math/0008045 [math.CO], 2000.
R. P. Stanley, A baker's dozen of conjectures concerning plane partitions, pp. 285-293 of "Combinatoire Enumerative (Montreal 1985)", Lect. Notes Math. 1234, 1986.
R. P. Stanley, A baker's dozen of conjectures concerning plane partitions, pp. 285-293 of "Combinatoire Enumerative (Montreal 1985)", Lect. Notes Math. 1234, 1986. Preprint. [Annotated scanned copy]

Cf. A109074, A134357.

A005156 := proc(n) local i,j,t1; (-3)^(n^2)*mul( mul( (6*j-3*i+1)/(2*j-i+2*n+1), j=1..n ),i=1..2*n+1); end;

Table[1/2^n Product[((6k-2)!(2k-1)!)/((4k-1)!(4k-2)!),{k,n}],{n,0,20}] (* Harvey P. Dale, Jul 07 2011 *)

a(n) = prod(k = 0, n-1, (3*k+2)*(6*k+3)!*(2*k+1)!/((4*k+2)!*(4*k+3)!));
vector(15, n, a(n-1))  \\ Gheorghe Coserea, May 30 2016

The formula for a(n) (see the Maple code) was conjectured by Robbins and proved by Kuperberg.
a(n) = (1/2^n) * Product_{k=1..n} ((6k-2)!(2k-1)!)/((4k-1)!(4k-2)!) (Razumov/Stroganov).
a(n) ~ exp(1/72) * Pi^(1/6) * 3^(3*n^2 + 3*n/2 + 11/72) / (A^(1/6) * GAMMA(1/3)^(1/3) * n^(5/72) * 2^(4*n^2 + 3*n + 1/9)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Mar 01 2015

A333093 a(n) is equal to the n-th order Taylor polynomial (centered at 0) of c(x)^n evaluated at x = 1, where c(x) = (1 - sqrt(1 - 4x))/(2x) is the o.g.f. of the Catalan numbers A000108.

Original entry on oeis.org

1, 2, 8, 41, 232, 1377, 8399, 52138, 327656, 2077934, 13270633, 85226594, 549837391, 3560702069, 23132584742, 150695482041, 984021596136, 6438849555963, 42208999230224, 277144740254566, 1822379123910857, 11998811140766701, 79095365076843134

Offset: 0

Peter Bala, Mar 07 2020

The sequence satisfies the Gauss congruences: a(n*p^k) == a(n*p^(k-1)) ( mod p^k ) for all prime p and positive integers n and k.
We conjecture that the sequence satisfies the stronger supercongruences a(n*p^k) == a(n*p^(k-1)) ( mod p^(3*k) ) for prime p >= 5 and positive integers n and k. Examples of these congruences are given below.
More generally, for each integer m, we conjecture that the sequence
  {a_m(n) : n >= 0}, defined by setting a_m(n) = the n-th order Taylor polynomial of c(x)^(m*n) evaluated at x = 1, satisfies the same supercongruences. For cases, see A099837 (m = -2), A100219 (m = -1), A000012 (m = 0), A333094 (m = 2), A333095 (m = 3), A333096 (m = 4), A333097 (m = 5).

			n-th order Taylor polynomial of c(x)^n:
  n = 0: c(x)^0 = 1 + O(x)
  n = 1: c(x)^1 = 1 + x + O(x^2)
  n = 2: c(x)^2 = 1 + 2*x + 5*x^2 + O(x^3)
  n = 3: c(x)^3 = 1 + 3*x + 9*x^2 + 28*x^3 + O(x^4)
  n = 4: c(x)^4 = 1 + 4*x + 14*x^2 + 48*x^3 + 165*x^4 + O(x^5)
Setting x = 1 gives a(0) = 1, a(1) = 1 + 1 = 2, a(2) = 1 + 2 + 5 = 8, a(3) = 1 + 3 + 9 + 28 = 41 and a(4) = 1 + 4 + 14 + 48 + 165 = 232.
The triangle of coefficients of the n-th order Taylor polynomial of c(x)^n, n >= 0, in descending powers of x begins
                                        row sums
  n = 0 |   1                               1
  n = 1 |   1   1                           2
  n = 2 |   5   2    1                      8
  n = 3 |  28   9    3   1                 41
  n = 4 | 165  48   14   4   1            232
   ...
This is a Riordan array belonging to the Hitting time subgroup of the Riordan group. The first column sequence [1, 1, 5, 28, 165, ...] = [x^n] c(x)^n = A025174(n).
Examples of supercongruences:
a(13) - a(1) = 3560702069 - 2 = (3^2)*(13^3)*31*37*157 == 0 ( mod 13^3 ).
a(3*7) - a(3) = 11998811140766701 - 41 = (2^2)*5*(7^4)*32213*7756841 == 0 ( mod 7^3 ).
a(5^2) - a(5) = 22794614296746579502 - 1377 = (5^6)*7*53*6491*605796421 == 0 ( mod 5^6 ).

Peter Bala, Notes on A333093

Cf. A000108, A001006, A001764, A005554, A025174, A333090 through A333097, A372214, A372215.

seq(add(n/(n+k)*binomial(n+2*k-1,k), k = 0..n), n = 1..25);
#alternative program
c:= x -> (1/2)*(1-sqrt(1-4*x))/x:
G := (x,n) -> series(c(x)^n, x, 51):
seq(add(coeff(G(x, n), x, k), k = 0..n), n = 0..25);

Table[SeriesCoefficient[((1 + x)^2 * (1 - Sqrt[(1 - 3*x)/(1 + x)]) / (2*x))^n, {x, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, Mar 28 2020 *)

a(n) = Sum_{k = 0..n} n/(n+k)*binomial(n+2*k-1,k) for n >= 1.
a(n) = [x^n] ( (1 + x)*c(x/(1 + x)) )^n = [x^n] ( (1 + x)*(1 + x*M(x)) )^n, where M(x) = ( 1 - x - sqrt(1 - 2*x - 3*x^2) ) / (2*x^2) is the o.g.f. of the Motzkin numbers A001006.
O.g.f.: ( 1 + x*f'(x)/f(x) )/( 1 - x*f(x) ), where f(x) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + ... = (1/x)*Revert( x/c(x) ) is the o.g.f. of A001764.
Row sums of the Riordan array ( 1 + x*f'(x)/f(x), x*f(x) ) belonging to the Hitting time subgroup of the Riordan group.
a(n) ~ 3^(3*n + 3/2) / (7 * sqrt(Pi*n) * 2^(2*n+1)). - Vaclav Kotesovec, Mar 28 2020
a(n) = Sum_{k = 0..n} n/(n+2*k)*binomial(n+2*k, k) for n >= 1. - Peter Bala, Apr 20 2024
D-finite with recurrence 2*n*(2*n-1)*(3991*n -21664)*a(n) +(-1329757*n^3 +9119565*n^2 -18270518*n +10657440)*a(n-1) +10*(947050*n^3 -6943257*n^2 +15944396*n -11260008)*a(n-2) +12*(-787878*n^3 +5778161*n^2 -13283386*n +9383340)*a(n-3) +9*(3*n-10)*(3*n-8)*(100503*n -141587)*a(n-4)=0, n>=5. - R. J. Mathar, Nov 22 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.

A060543 Triangle, read by antidiagonals, where T(n,k) = C(n+nk+k, nk+k).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 10, 5, 1, 1, 35, 28, 7, 1, 1, 126, 165, 55, 9, 1, 1, 462, 1001, 455, 91, 11, 1, 1, 1716, 6188, 3876, 969, 136, 13, 1, 1, 6435, 38760, 33649, 10626, 1771, 190, 15, 1, 1, 24310, 245157, 296010, 118755, 23751, 2925, 253, 17, 1, 1, 92378, 1562275

Offset: 0

Henry Bottomley, Apr 02 2001

Main diagonal is A108288. Antidiagonal sums is A108289. Inverse binomial transforms of each row give triangle A108290. G.f. of row n multiplied by (1-x)^(n+1) equals g.f. of row n of triangle A108267 (rows sums of A108267 equal (n+1)^n).

			row 1: (2*n+1)/1!
row 2: (3*n+1)*(3*n+2)/2!
row 3: (4*n+1)*(4*n+2)*(4*n+3)/3!
row 4: (5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)/4!
row 5: (6*n+1)*(6*n+2)*(6*n+3)*(6*n+4)*(6*n+5)/5!.
Table begins:
1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...
1,3,5,7,9,11,13,15,17,19,21,23,25,27,...
1,10,28,55,91,136,190,253,325,406,496,...
1,35,165,455,969,1771,2925,4495,6545,...
1,126,1001,3876,10626,23751,46376,82251,...
1,462,6188,33649,118755,324632,749398,...
1,1716,38760,296010,1344904,4496388,...

Harry J. Smith, Table of n, a(n) for n=0..252

Cf. A108290, A108267, A108288, A108289, A060544 (row 2), A015219 (row 3).

Rows include A000012, A001700, A025174. Columns include A000012, A005408, A060544. Main diagonal is A060545.

PARI
```
T(n,k)=binomial(n+n*k+k,n*k+k)
```

{ i=0; write("b060543.txt", "0 1"); for (m=0, 20, for (k=0, m + 1, n=m - k + 1; write("b060543.txt", i++, " ", binomial(n + n*k + k, n*k + k))); ) } \\ Harry J. Smith, Jul 06 2009

a(n) = A060539(n, k)/n = A007318(nk, k)/n = A060540(n, k)/A060540(n-1, k).

Entry revised by Paul D. Hanna, May 31 2005

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 17 2007

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)

A104978 Triangle read by rows, where the g.f. satisfies A(x, y) = 1 + xA(x, y)^2 + xy*A(x, y)^3.

Original entry on oeis.org

1, 1, 1, 2, 5, 3, 5, 21, 28, 12, 14, 84, 180, 165, 55, 42, 330, 990, 1430, 1001, 273, 132, 1287, 5005, 10010, 10920, 6188, 1428, 429, 5005, 24024, 61880, 92820, 81396, 38760, 7752, 1430, 19448, 111384, 352716, 678300, 813960, 596904, 245157, 43263, 4862, 75582, 503880, 1899240, 4476780, 6864396, 6864396, 4326300, 1562275, 246675

Offset: 0

Paul D. Hanna, Mar 30 2005

			The triangle T(n, k) begins:
  [0]    1;
  [1]    1,     1;
  [2]    2,     5,      3;
  [3]    5,    21,     28,     12;
  [4]   14,    84,    180,    165,     55;
  [5]   42,   330,    990,   1430,   1001,    273;
  [6]  132,  1287,   5005,  10010,  10920,   6188,   1428;
  [7]  429,  5005,  24024,  61880,  92820,  81396,  38760,   7752;
  [8] 1430, 19448, 111384, 352716, 678300, 813960, 596904, 245157, 43263;
  ...
The array A(n, k) begins:
  [0]   1,    1,      3,      12,       55,       273,       1428, ...  [A001764]
  [1]   1,    5,     28,     165,     1001,      6188,      38760, ...  [A025174]
  [2]   2,   21,    180,    1430,    10920,     81396,     596904, ...  [A383450]
  [3]   5,   84,    990,   10010,    92820,    813960,    6864396, ...  [A383451]
  [4]  14,  330,   5005,   61880,   678300,   6864396,   65615550, ...
  [5]  42, 1287,  24024,  352716,  4476780,  51482970,  551170620, ...
  [6] 132, 5005, 111384, 1899240, 27457584, 354323970, 4206302100, ...
  [A000108]  |  [A074922][A383452]
         [A002054]

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
N. J. Wildberger and Dean Rubine, A Hyper-Catalan Series Solution to Polynomial Equations, and the Geode, Amer. Math. Monthly (2025). See sections 8 and 12.
Jian Zhou, Fat and Thin Emergent Geometries of Hermitian One-Matrix Models, arXiv:1810.03883 [math-ph], 2018.

Columns of array: A000108, A002054, A074922, A383452.
Rows of array: A001764, A025174, A383450, A383451.
Cf. A001002 (antidiagonal sums), A001764 (semidiagonal sums), A027307 (row sums), A104979, A383439 (central terms).

[Binomial(2*n+k, n+2*k)*Binomial(n+2*k, k)/(n+k+1): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 08 2021

From Peter Luschny, May 04 2025:  (Start)
T := (n, k) -> (k + 2*n)!/(k!*(n - k)!*(n + k + 1)!):
seq(print(seq(T(n, k), k = 0..n)), n = 0..10);
# Alternatively the array:
A := (n, k) -> (3*k + 2*n)!/(k!*n!*(n + 2*k + 1)!);
for n from 0 to 8 do seq(A(n, k), k = 0..7) od;  (End)

T[n_, k_]:= Binomial[2n+k, n+2k]*Binomial[n+2k, k]/(n+k+1);
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* Jean-François Alcover, Jan 27 2019 *)

T(n,k) = local(A=1+x+x*y+x*O(x^n)+y*O(y^k)); for(i=1,n,A=1+x*A^2+x*y*A^3); polcoeff(polcoeff(A,n,x),k,y)
for(n=0, 10, for(k=0, n, print1(T(n,k),", ")); print(""))

Dy(n, F)=local(D=F); for(i=1, n, D=deriv(D,y)); D
T(n,k)=local(A=1); A=1+sum(m=1, n+1, x^m/y^(m+1) * Dy(m-1, (y^2+y^3)^m/m!)) +x*O(x^n)+y*O(y^k); polcoeff(polcoeff(A, n,x),k,y)
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print()) \\ Paul D. Hanna, Jun 22 2012

x='x; y='y; z='z; Fxyz = 1 - z + x*z^2 + x*y*z^3;
seq(N) = {
  my(z0 = 1 + O((x*y)^N), z1 = 0);
  for (k = 1, N^2,
    z1 = z0 - subst(Fxyz, z, z0)/subst(deriv(Fxyz, z), z, z0);
    if (z0 == z1, break()); z0 = z1);
  vector(N, n, Vecrev(polcoeff(z0, n-1, 'x)));
};
concat(seq(9)) \\ Gheorghe Coserea, Nov 30 2016

flatten([[binomial(2*n+k, n+2*k)*binomial(n+2*k, k)/(n+k+1) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 08 2021

T(n, k) = binomial(2*n+k, n+2*k)*binomial(n+2*k, k)/(n+k+1).
G.f.: A(x, y) = Sum_{n>=0} x^n/y^(n+1) * d^(n-1)/dy^(n-1) (y^2 + y^3)^n / n!. - Paul D. Hanna, Jun 22 2012
G.f. of row n: 1/y^(n+1) * d^(n-1)/dy^(n-1) (y^2+y^3)^n / n!. - Paul D. Hanna, Jun 22 2012
A(n, k) = T(n + k, k) = (3*k + 2*n)! / (k!*n!*(n + 2*k + 1)!). - Peter Luschny, May 04 2025

A108767 Triangle read by rows: T(n,k) is number of paths from (0,0) to (3n,0) that stay in the first quadrant (but may touch the horizontal axis), consisting of steps u=(1,1), d=(1,-2) and have k peaks (i.e., ud's).

Original entry on oeis.org

1, 1, 2, 1, 6, 5, 1, 12, 28, 14, 1, 20, 90, 120, 42, 1, 30, 220, 550, 495, 132, 1, 42, 455, 1820, 3003, 2002, 429, 1, 56, 840, 4900, 12740, 15288, 8008, 1430, 1, 72, 1428, 11424, 42840, 79968, 74256, 31824, 4862, 1, 90, 2280, 23940, 122094, 325584, 465120

Offset: 1

Emeric Deutsch, Jun 24 2005

Row sums yield A001764.
From Peter Bala, Sep 16 2012: (Start)
The number of 2-Dyck paths of order n with k peaks (Cigler). A 2-Dyck path of order n is a lattice path from (0,0) to (2*n,n) with steps (0,1) (North) and (1,0) (East) that never goes below the diagonal {2*i,i} 0 <= i <= n. A peak is a consecutive North East pair.
Row reverse of A120986. Described in A173020 as generalized Runyon numbers R_{n,k}^(2).
(End)
From Alexander Burstein, Jun 15 2020: (Start)
T(n,k) is the number of paths from (0,0) to (3n,0) that stay on or above the horizontal axis, with unit steps u=(1,1) and d=(1,-2), that have n+1-k peaks at even height.
T(n,k) is also the number of paths from (0,0) to (3n,0) that stay on or above the horizontal axis, with unit steps u=(1,1) and d=(1,-2), that have n-k peaks at odd height. (End)
An apparent refinement is A338135. - Tom Copeland, Oct 12 2020

			T(3,2)=6 because we have uuduuuudd, uuuduuudd, uuuuduudd, uuuudduud, uuuuududd and uuuuuddud.
Triangle starts:
  1;
  1,  2;
  1,  6,   5;
  1, 12,  28,   14;
  1, 20,  90,  120,   42;
  1, 30, 220,  550,  495,  132;
  1, 42, 455, 1820, 3003, 2002, 429;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275
Alexander Burstein, Distribution of peak heights modulo k and double descents on k-Dyck paths, arXiv preprint arXiv:2009.00760 [math.CO], 2020.
Frédéric Chapoton and Philippe Nadeau, Combinatorics of the categories of noncrossing partitions, Séminaire Lotharingien de Combinatoire 78B (2017), Article #37.
J. Cigler, Some remarks on Catalan families, European J. Combin., 8(3): 261-267.
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. 5.
Chao-Jen Wang, Applications of the Goulden-Jackson cluster method to counting Dyck paths by occurrences of subwords, Thesis, 2011.
Tad White, Quota Trees, arXiv:2401.01462 [math.CO], 2024. See p. 20.

Cf. A000108, A001764, A025174, A120986.

Runyon numbers R_{n,k}^(m): A010054 (m=0), A001263 (m=1), this sequence (m=2), A173020 (m=3).

A108767:= func< n,k,m | Binomial(n,k)*Binomial(m*n,k-1)/n >;
[A108767(n,k,2): k in [1..n], n in [1..12]]; // G. C. Greubel, Feb 20 2021

T:=(n,k)->binomial(n,k)*binomial(2*n,k-1)/n: for n from 1 to 10 do seq(T(n,k),k=1..n) od; # yields sequence in triangular form

T[n_, k_] := Binomial[n, k]*Binomial[2*n, k - 1]/n;
Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Nov 11 2017, from Maple *)

T(n,k) = binomial(n, k)*binomial(2*n, k-1)/n; \\ Andrew Howroyd, Nov 06 2017

from sympy import binomial
def T(n, k): return binomial(n, k)*binomial(2*n, k - 1)//n
for n in range(1, 21): print([T(n, k) for k in range(1, n + 1)]) # Indranil Ghosh, Nov 07 2017

T <- function(n, k) {
  choose(n, k)*choose(2*n, k - 1)/n
} # Indranil Ghosh, Nov 07 2017

def A108767(n,k,m): return binomial(n,k)*binomial(m*n,k-1)/n
flatten([[A108767(n,k,2) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Feb 20 2021

T(n, k) = binomial(n, k)*binomial(2*n, k-1)/n.
T(n, n) = A000108(n) (the Catalan numbers).
Sum_{k=1..n} k*T(n,k) = A025174(n).
G.f.: T-1, where T = T(t, z) satisfies T = 1 + z*T^2*(T - 1 + t).
From Peter Bala, Oct 22 2008: (Start)
Define a functional I on formal power series of the form f(x) = 1 + ax + bx^2 + ... by the following iterative process. Define inductively f^(1)(x) = f(x) and f^(n+1)(x) = f(x*f^(n)(x)) for n >= 1. Then set I(f(x)) = Limit_{n -> oo} f^(n)(x) in the x-adic topology on the ring of formal power series; the operator I may also be defined by I(f(x)) := 1/x*series reversion of x/f(x).
The o.g.f. for the array of Narayana numbers A001263 is I(1 + t*x + t*x^2 + t*x^3 + ...) = 1 + t*x + (t + t^2)*x^2 + (t + 3*t^2 + t^3)*x^3 + ... . The o.g.f. for the current array is IoI(1 + t*x + t*x^2 + t*x^3 + ...) = 1 + t*x + (t + 2*t^2)*x^2 + (t + 6*t^2 + 5*t^3)*x^3 + ... . Cf. A132081 and A141618. Alternatively, the o.g.f. of this array can be obtained from a single application of I, namely, form I(1 + t*x^2 + t*x^4 + t*x^6 + ...) = 1 + t*x^2 + (t + 2*t^2)*x^4 + (t + 6*t^2 + 5*t^3)*x^6 + ... and then replace x by sqrt(x). This is a particular case of the general result that forming the n-fold composition I^(n)(f(x)) and then replacing x with x^n produces the same result as I(f(x^n)). (End)
O.g.f. is series reversion with respect to x of x/((1+x)*(1+x*u)^2). Cf. A173020. - Peter Bala, Sep 12 2012
n-th row polynomial = x * hypergeom([1 - n, -2*n], [2], x). - Peter Bala, Aug 30 2023

A119301 Triangle read by rows: T(n,k) = binomial(3*n-k,n-k).

Original entry on oeis.org

1, 3, 1, 15, 5, 1, 84, 28, 7, 1, 495, 165, 45, 9, 1, 3003, 1001, 286, 66, 11, 1, 18564, 6188, 1820, 455, 91, 13, 1, 116280, 38760, 11628, 3060, 680, 120, 15, 1, 735471, 245157, 74613, 20349, 4845, 969, 153, 17, 1, 4686825, 1562275, 480700, 134596, 33649, 7315

Offset: 0

Paul Barry, May 13 2006

First column is A005809. Second column is A025174.

Row sums are A045721. Inverse is Riordan array (1-3x,x(1-x)^2), A119302.

			Triangle begins
  1,
  3, 1,
  15, 5, 1,
  84, 28, 7, 1,
  495, 165, 45, 9, 1,
  3003, 1001, 286, 66, 11, 1,
  18564, 6188, 1820, 455, 91, 13, 1,
  116280, 38760, 11628, 3060, 680, 120, 15, 1
  ...
Horizontal recurrence: T(4,1) = 1*84 + 2*28 + 3*7 + 4*1 = 165. - _Peter Bala_, Dec 29 2014

Cf. A092392, A119304.

Cf. A005809, A025174, A045721, A119302.

T := proc(n,k) option remember;
`if`(n = 0, 1, add(i*T(n-1,k-2+i),i=1..n+1-k)) end:
for n from 0 to 9 do print(seq(T(n,k),k=0..n)) od; # Peter Luschny, Dec 30 2014

Flatten[Table[Binomial[3n-k,n-k],{n,0,10},{k,0,n}]] (* Harvey P. Dale, Jul 28 2012 *)

G.f. g(x) = 2*sin(arcsin(3*sqrt(3*x)/2)/3)/sqrt(3*x) satisfies g(x) = 1/(1-x*g(x)^2).
Riordan array (1/(1-3*x*g(x)^2),x*g(x)^2) where g(x)=1+x*g(x)^3.
'Horizontal' recurrence equation: T(n,0) = binomial(3*n,n) and for k >= 1, T(n,k) = Sum_{i = 1..n+1-k} i*T(n-1,k-2+i). - Peter Bala, Dec 28 2014
T(n, k) = Sum_{j = 0..n} binomial(n+j-1, j)*binomial(2*n-k-j, n). - Peter Bala, Jun 04 2024

cp's OEIS Frontend

A013698 a(n) = binomial(3*n+2, n-1).

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A005156 Number of alternating sign 2n+1 X 2n+1 matrices symmetric about the vertical axis (VSASM's); also 2n X 2n off-diagonally symmetric alternating sign matrices (OSASM's).

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A333093 a(n) is equal to the n-th order Taylor polynomial (centered at 0) of c(x)^n evaluated at x = 1, where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f. of the Catalan numbers A000108.

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

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

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A060543 Triangle, read by antidiagonals, where T(n,k) = C(n+n*k+k, n*k+k).

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A333093 a(n) is equal to the n-th order Taylor polynomial (centered at 0) of c(x)^n evaluated at x = 1, where c(x) = (1 - sqrt(1 - 4x))/(2x) is the o.g.f. of the Catalan numbers A000108.

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

A060543 Triangle, read by antidiagonals, where T(n,k) = C(n+nk+k, nk+k).

A104978 Triangle read by rows, where the g.f. satisfies A(x, y) = 1 + xA(x, y)^2 + xy*A(x, y)^3.