A000344 -id:A000344 - cp's OEIS frontend

A052722 Expansion of e.g.f. (1 - 2x - sqrt(1-4x))^2 * (1 - sqrt(1-4*x))/8.

0, 0, 0, 0, 0, 120, 3600, 100800, 3024000, 99792000, 3632428800, 145297152000, 6351561216000, 301699157760000, 15487223431680000, 854894733428736000, 50516506975334400000, 3182539939446067200000, 212985365178313728000000

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

G. C. Greubel, Table of n, a(n) for n = 0..350
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 678

Cf. A052711, A052712, A052713, A052714, A052715, A052716, A052717, A052718, A052719, A052720, A052721, A052723.

Cf. A000108, A000344.

spec := [S,{C=Union(B,Z),B=Prod(C,C),S=Prod(B,B,C)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

With[{nn=20},CoefficientList[Series[((1-2x-Sqrt[1-4x])^2 (1-Sqrt[1-4x]))/8,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Aug 30 2021 *)
Table[If[n<5, 0, 10*(n-2)!*Binomial[n-3,2]*CatalanNumber[n-3]], {n,0,30}] (* G. C. Greubel, May 28 2022 *)

def A052722(n):
    if (n<5): return 0
    else: return 10*factorial(n-2)*binomial(n-3,2)*catalan_number(n-3)
[A052722(n) for n in (0..30)] # G. C. Greubel, May 28 2022

D-finite with recurrence: a(0) = a(1) = a(2) = a(3) = a(4) = 0, a(5)=120, a(n+3) = (9+7*n)*a(n+2) + (14 - 19*n - 13*n^2)*a(n+1) - (20 + 22*n - 2*n^2 - 4*n^3)*a(n).
a(n) = n!*A000344(n-3). - R. J. Mathar, Oct 18 2013
From G. C. Greubel, May 28 2022: (Start)
G.f.: 5!*x^5*hypergeometric2F0([5/2, 3], [], 4*x).
E.g.f.: (1/2)*(1 - 5*x + 5*x^2 - (1 - 3*x + x^2)*sqrt(1-4*x)). (End)

A115144 Fifth convolution of A115140.

Original entry on oeis.org

1, -5, 5, 0, 0, -1, -5, -20, -75, -275, -1001, -3640, -13260, -48450, -177650, -653752, -2414425, -8947575, -33266625, -124062000, -463991880, -1739969550, -6541168950, -24647883000, -93078189750, -352207870014, -1335293573130, -5071418015120, -19293438101000

Offset: 0

Wolfdieter Lang, Jan 13 2006

Seiichi Manyama, Table of n, a(n) for n = 0..1669

Cf. A000344, A115139 - A115143, A115145 - A115149.

m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!( (1-5*x+5*x^2 +(1-3*x+x^2)*Sqrt(1-4*x))/2 )); // G. C. Greubel, Feb 12 2019

CoefficientList[Series[(1-5*x+5*x^2 +(1-3*x+x^2)*Sqrt[1-4*x])/2, {x, 0, 30}], x] (* G. C. Greubel, Feb 12 2019 *)

my(x='x+O('x^30)); Vec((1-5*x+5*x^2 +(1-3*x+x^2)*sqrt(1-4*x))/2) \\ G. C. Greubel, Feb 12 2019

((1-5*x+5*x^2 +(1-3*x+x^2)*sqrt(1-4*x))/2).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Feb 12 2019

O.g.f.: 1/c(x)^5 = P(6, x) - x*P(5, x)*c(x) with the o.g.f. c(x) = (1-sqrt(1-4*x))/(2*x) of A000108 (Catalan numbers) and the polynomials P(n, x) defined in A115139. Here P(6, x)=1-4*x+3*x^2 and P(5, x)=1-3*x+x^2.
a(n) = -C5(n-5), n>=5, with C5(n) = A000344(n+2) (fifth convolution of Catalan numbers). a(0)=1, a(1)=-5, a(2)=5, a(3)=0=a(4). [1, -5, 5] is row n=5 of signed A034807 (signed Lucas polynomials). See A115149 and A034807 for comments.
D-finite with recurrence +n*(n-5)*a(n) -2*(n-3)*(2*n-7)*a(n-1)=0. - R. J. Mathar, Sep 23 2021
From Peter Bala, Mar 05 2023: (Start)
a(n) = binomial(2*n - 6, n) - binomial(2*n - 6, n + 1).
a(n) = = -5/(n - 5)*binomial(2*n - 6, n) for n != 5.
a(n) = -A000344(n-3) for n >= 5. (End)

A143955 Sum of the altitudes of the leftmost valleys of all Dyck paths of semilength n (if path has no valley, then this altitude is taken to be 0).

Original entry on oeis.org

0, 0, 0, 1, 6, 26, 101, 376, 1377, 5017, 18277, 66727, 244377, 898129, 3312554, 12260129, 45526754, 169588754, 633580634, 2373550184, 8914719134, 33562602134, 126640791884, 478848661898, 1814142235028, 6885560250148

Offset: 0

Emeric Deutsch, Oct 14 2008

nonn

The positive terms form the partial sums of A000344.

			a(4)=6 because the Dyck paths of semilength 4 with leftmost valley at a positive altitude are UUDUDDUD, UUDUDUDD, UUDUUDDD, UUUDDUDD and UUUDUDDD, where U=(1,1) and D=(1,-1); these altitudes are 1, 1, 1, 1 and 2, respectively.

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A000108, A000344, A097607, A323224.

C:=((1-sqrt(1-4*z))*1/2)/z: G:=z^3*C^5/(1-z): Gser:=series(G,z=0,32): seq(coeff(Gser,z,n),n=0..27);

CoefficientList[Series[x^3 ((1 - (1 - 4 x)^(1/2))/(2 x))^5/(1 - x), {x, 0, 40}], x] (* Vaclav Kotesovec, Mar 21 2014 *)

a(n):=5*sum(binomial(2*k,k-2)/(k+3),k,2,n-1); /* Vladimir Kruchinin, Mar 15 2016 */

from functools import cache
@cache
def B(n, k):
    if n <= 0 or k <= 0: return 0
    if n == k: return 1
    return B(n - 1, k) + B(n, k - 1)
def A143955(k):
    return B(k + 3, k - 2)
print([A143955(n) for n in range(26)]) # Peter Luschny, May 15 2022

a(n) = Sum_{k>=0} k*A097607(n,k).
G.f.: z^3*C^5/(1-z), where C=(1-sqrt(1-4*z))/(2*z) is the generating function of the Catalan numbers (A000108).
Conjecture: (n+2)*a(n) -4*(2*n+1)*a(n-1) +2*(10*n-9)*a(n-2) +17*(2-n)*a(n-3) +2*(2*n-7)*a(n-4)=0. - R. J. Mathar, Jul 24 2012
a(n) ~ 5*4^n/(3*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 21 2014
a(n) = 5*Sum_{k=2..n-1}(binomial(2*k,k-2)/(k+3)). - Vladimir Kruchinin, Mar 15 2016

A026029 Number of (s(0), s(1), ..., s(2n)) such that s(i) is a nonnegative integer and |s(i) - s(i-1)| = 1 for i = 1,2,...,n, s(0) = 3, s(2n) = 3. Also T(2n,n), where T is defined in A026022.

Original entry on oeis.org

1, 2, 6, 20, 69, 242, 858, 3068, 11050, 40052, 145996, 534888, 1968685, 7276050, 26993490, 100490220, 375287550, 1405622460, 5278838100, 19873977240, 74994427170, 283595947284, 1074568266756, 4079184055640, 15511924233204, 59083160374952, 225384613313944

Offset: 0

Clark Kimberling

nonn

Hankel transform is A008619(n+1). - Paul Barry, May 11 2009

Michael De Vlieger, Table of n, a(n) for n = 0..1000
Andrei Asinowski and Cyril Banderier, From geometry to generating functions: rectangulations and permutations, arXiv:2401.05558 [cs.DM], 2024. See page 2.
Arturo Merino and Torsten Mütze, Combinatorial generation via permutation languages. III. Rectangulations, arXiv:2103.09333 [math.CO], 2021.

CoefficientList[Series[(1 - 2*x)*(-1 + Sqrt[1 - 4*x] + 2*x)^2 / (4*x^4), {x, 0, 20}], x] (* Vaclav Kotesovec, Sep 03 2019 *)

Expansion of (1+x^2*C^4)*C^2, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108.
a(n) = Sum_{k=0..n} C(n, k)*Sum_{i=0..k} C(k, 2i)*A000108(i+1). - Paul Barry, Jul 18 2003
a(n) = Sum_{k=0..3} A039599(n,k) = A000108(n) + A000245(n) + A000344(n) + A000588(n) = A026012(n) + A000588(n). - Philippe Deléham, Nov 12 2008
a(n) = C(2n,n) - C(2n,n-4). - Paul Barry, May 11 2009
Conjecture: (n+4)*a(n) + 6*(-n-2)*a(n-1) + 4*(2*n-1)*a(n-2) = 0. - R. J. Mathar, Nov 24 2012
a(n) ~ 4^(n+2) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Sep 03 2019
E.g.f.: exp(2*x)*(BesselI(0, 2*x) - BesselI(4, 2*x)). - Stefano Spezia, Jan 17 2024

A050155 Triangle T(n,k), k>=0 and n>=1, read by rows defined by: T(n,k) = (2k+3)*binomial(2n,n-k-1)/(n+k+2).

Original entry on oeis.org

1, 3, 1, 9, 5, 1, 28, 20, 7, 1, 90, 75, 35, 9, 1, 297, 275, 154, 54, 11, 1, 1001, 1001, 637, 273, 77, 13, 1, 3432, 3640, 2548, 1260, 440, 104, 15, 1, 11934, 13260, 9996, 5508, 2244, 663, 135, 17, 1, 41990, 48450, 38760, 23256, 10659, 3705, 950, 170, 19, 1

Offset: 1

Clark Kimberling

T(n-2k-1,k) = number of n-th generation vertices in the tree of sequences with unit increase labeled by 2k+2 (cf. Zoran Sunic reference) . - Benoit Cloitre, Oct 07 2003
Number of lattice paths from (0,0) to (n,n) with steps E=(1,0) and N=(0,1) which touch but do not cross the line x-y=k+1 . - Herbert Kociemba, May 24 2004
Number of standard tableaux of shape (n+k+1, n-k-1). - Emeric Deutsch, May 30 2004
Riordan array (c(x)^3,xc(x)^2) where c(x) is the g.f. of A000108. Inverse array is A109954. - Paul Barry, Jul 06 2005

			    1;
    3,   1;
    9,   5,   1;
   28,  20,   7,  1;
   90,  75,  35,  9,  1;
  297, 275, 154, 54, 11, 1;
  ...

Alois P. Heinz, Rows n = 1..141, flattened
R. K. Guy, Catwalks, Sansteps and Pascal Pyramids, J. Integer Seq., Vol. 3 (2000), #00.1.6
V. E. Hoggatt, Jr. and M. Bicknell, Catalan and related sequences arising from inverses of Pascal's triangle matrices, Fib. Quart., 14 (1976), 395-405.
A. Papoulis, A new method of inversion of the Laplace transform, Quart. Appl. Math 14 (1957), 405-414.
J. Riordan, The distribution of crossings of chords joining pairs of 2n points on a circle, Math. Comp., 29 (1975), 215-222.
_Zoran Sunic_, Self-Describing Sequences and the Catalan Family Tree, Electronic Journal of Combinatorics, 10 (2003) #N5.

First columns: A000245, A000344, A000588, A001392, A000589, A000590.

Cf. A000108, A001791 (row sums), A050144.

T:= (n, k)->  (2*k+3)*binomial(2*n, n-k-1)/(n+k+2):
seq(seq(T(n, k), k=0..n-1), n=1..10);  # Alois P. Heinz, Jan 19 2013

T[n_, k_] :=  (2*k + 3)*Binomial[2*n, n - k - 1]/(n + k + 2);
Table[T[n, k], {n, 1, 10}, {k, 0, n - 1}] // Flatten (* Jean-François Alcover, May 21 2016 *)

Sum_{ k = 0, .., n-1} T(n, k) = binomial(2n, n-1) = A001791(n).
G.f. of column k: x^(k+1)*C^(2*k+3) where C = (1-(1-4*x)^(1/2))/(2*x) is the g.f. of Catalan numbers A000108. - Philippe Deléham, Feb 03 2004
T(n, k) = A039599(n, k+1) = A009766(n+k+1, n-k-1) = A033184(n+k+2, 2k+3) . - Philippe Deléham, May 28 2005
Sum_{k>= 0} T(m, k)*T(n, k) = A000108(m+n) - A000108(m)*A000108(n). - Philippe Deléham, May 28 2005
T(n, k)=(2k+3)binomial(2n+2, n+k+2)/(n+k+3)=C(2n+2, n+k+2)-C(2n+2, n+k+3) [offset (0, 0)]. - Paul Barry, Jul 06 2005

Edited by Philippe Deléham, May 22 2005

A071718 Expansion of (1+x^2C)C^3, where C = (1-(1-4x)^(1/2))/(2x) is g.f. for Catalan numbers, A000108.

Original entry on oeis.org

1, 3, 10, 32, 104, 345, 1166, 4004, 13936, 49062, 174420, 625328, 2258416, 8209045, 30008790, 110255100, 406923360, 1507973610, 5608843020, 20931740640, 78354322800, 294127079610, 1106939020044, 4175827174152, 15787544777504

Offset: 0

N. J. A. Sloane, Jun 06 2002

nonn

a(n)=number of Dyck (n+3)-paths whose third from last upstep initiates a long ascent, n>=1. A long ascent is one consisting of 2 or more upsteps. For example, a(1)=3 counts UDuUUDDD, UDuUDUDD, UDuUDDUD (third from last upstep in small type). - David Callan, Dec 08 2004
For n>0 a(n)=number of Dyck (n+3)-paths whose 5th and 6th steps are DU. For example, a(1)=3 counts UDUUduDD, UUDUduDD, UUUDduDD. - David Scambler, Feb 14 2011
Let X_n be the set of all noncrossing set partitions of an n-element set which either do not contain {n-1,n} as a block, or which do not contain the block {n} whenever 1 and n-1 are in the same block. a(n) is the cardinality of X_{n+2}. For example, a(1)=3 counts 1|2|3, 13|2, 123. - Henri Mühle, Jan 10 2017

M. Bruce, M. Dougherty, M. Hlavacek, R. Kudo, I. Nicolas, A Decomposition of Parking Functions by Undesired Spaces, The Electronic Journal of Combinatorics 23(3), 2016.
H. Mühle, Two Posets of Noncrossing Partitions Coming From Undesired Parking Spaces, arXiv:1701.02109 [math.CO], 2017.

{1, 3}~Join~Table[(5/(n + 3) + 9/(n - 1))*Binomial[2 n, n - 2], {n, 2, 24}] (* Michael De Vlieger, Jan 10 2017 *)

For n>1, a(n) = 3*A000245(n) + A000344(n) = (5/(n+3) + 9/(n-1))*binomial(2n,n-2).

D-finite with recurrence (n+3)*a(n) + 2*(-2*n-3)*a(n-1) + 2*(-n+1)*a(n-2) + 4*(2*n-5)*a(n-3) = 0. - R. J. Mathar, Aug 25 2013

A119245 Triangle, read by rows, defined by: T(n,k) = (4k+1)binomial(2n+1, n-2k)/(2n+1) for n >= 2k >= 0.

Original entry on oeis.org

1, 1, 2, 1, 5, 5, 14, 20, 1, 42, 75, 9, 132, 275, 54, 1, 429, 1001, 273, 13, 1430, 3640, 1260, 104, 1, 4862, 13260, 5508, 663, 17, 16796, 48450, 23256, 3705, 170, 1, 58786, 177650, 95931, 19019, 1309, 21, 208012, 653752, 389367, 92092, 8602, 252, 1

Offset: 0

Paul D. Hanna, May 10 2006

Closely related to triangle A118919.
Row n contains 1+floor(n/2) terms.
From Peter Bala, Mar 20 2009: (Start)
Combinatorial interpretations of T(n,k):
1) The number of standard tableaux of shape (n-2*k,n+2*k).
2) The entries in column k are (with an offset of 2*k) the number of n-th generation vertices in the tree of sequences with unit increase labeled by 4*k. See [Sunik, Theorem 4]. (End)

			Triangle begins:
     1;
     1;
     2,     1;
     5,     5;
    14,    20,    1;
    42,    75,    9;
   132,   275,   54,   1;
   429,  1001,  273,  13;
  1430,  3640, 1260, 104,  1;
  4862, 13260, 5508, 663, 17; ...

Zoran Sunic, Self describing sequences and the Catalan family tree, Elect. J. Combin., 10 (No. 1, 2003). - _Peter Bala_, Mar 20 2009

Cf. A119244 (eigenvector), A088218, A000108, A000344, A001392; A118919 (variant), A158483; A002057, A002894.

f1 = (1-Sqrt[1-4*x])/(2*x);
DeleteCases[CoefficientList[Normal@Series[f1/(1 - x^2*y*f1^4),{x,0,10},{y,0,5}],{x,y}],0,Infinity]//TableForm  (* Bradley Klee, Feb 26 2018 *)
Table[(1+4*k)/(n+1+2*k)*Binomial[2*n,n+2*k],{n,0,10},{k,0,Floor[n/2]}]//TableForm (* Bradley Klee, Feb 26 2018 *)

T(n,k)=(4*k+1)*binomial(2*n+1,n-2*k)/(2*n+1)

G.f.: A(x,y) = f/(1-x^2*y*f^4), where f=(1-sqrt(1-4*x))/(2*x) is the Catalan g.f. (A000108).
Row sums equal A088218(n) = C(2*n-1,n).
T(n,0) = A000108(n) (the Catalan numbers).
T(n,1) = A000344(n).
T(n,2) = A001392(n).
Sum_{k=0..floor(n/2)} k*T(n,k) = A000346(n-2).
Eigenvector is defined by: A119244(n) = Sum_{k=0..[n\2]} T(n,k)*A119244(k).
...
T(n,k) = (4*k+1)/(n+2*k+1)*binomial(2*n,n+2*k). Compare with A158483. - Peter Bala, Mar 20 2009
T(n,k) = A039599(n, 2*k). - Johannes W. Meijer, Sep 04 2013
A002894(n) = Sum_{k=0..floor(n/2)} (binomial(2k,k)^2)*(4^(n-2*k))*T(n,k). - Bradley Klee, Feb 26 2018

A355173 The Fuss-Catalan triangle of order 1, read by rows. Related to binary trees.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 1, 3, 5, 0, 1, 4, 9, 14, 0, 1, 5, 14, 28, 42, 0, 1, 6, 20, 48, 90, 132, 0, 1, 7, 27, 75, 165, 297, 429, 0, 1, 8, 35, 110, 275, 572, 1001, 1430, 0, 1, 9, 44, 154, 429, 1001, 2002, 3432, 4862, 0, 1, 10, 54, 208, 637, 1638, 3640, 7072, 11934, 16796

Offset: 0

Peter Luschny, Jun 25 2022

The Fuss-Catalan triangle of order m is a regular, (0, 0)-based table recursively defined as follows: Set row(0) = [1] and row(1) = [0, 1]. Now assume row(n-1) already constructed and duplicate the last element of row(n-1). Next apply the cumulative sum m times to this list to get row(n). Here m = 1. (See the Python program for a reference implementation.)

This definition also includes the classical Fuss-Catalan numbers, since T(n, n) = A000108(n), or row 2 in A355262. For m = 2 see A355172 and for m = 3 A355174. More generally, for n >= 1 all Fuss-Catalan sequences (A355262(n, k), k >= 0) are the main diagonals of the Fuss-Catalan triangles of order n - 1.

			Table T(n, k) begins:
  [0] [1]
  [1] [0, 1]
  [2] [0, 1, 2]
  [3] [0, 1, 3,  5]
  [4] [0, 1, 4,  9,  14]
  [5] [0, 1, 5, 14,  28,  42]
  [6] [0, 1, 6, 20,  48,  90,  132]
  [7] [0, 1, 7, 27,  75, 165,  297, 429]
  [8] [0, 1, 8, 35, 110, 275,  572, 1001, 1430]
  [9] [0, 1, 9, 44, 154, 429, 1001, 2002, 3432, 4862]
Seen as an array reading the diagonals starting from the main diagonal:
  [0] 1, 1, 2,  5,  14,   42,  132,   429,  1430,   4862,   16796, ...  A000108
  [1] 0, 1, 3,  9,  28,   90,  297,  1001,  3432,  11934,   41990, ...  A000245
  [2] 0, 1, 4, 14,  48,  165,  572,  2002,  7072,  25194,   90440, ...  A099376
  [3] 0, 1, 5, 20,  75,  275, 1001,  3640, 13260,  48450,  177650, ...  A115144
  [4] 0, 1, 6, 27, 110,  429, 1638,  6188, 23256,  87210,  326876, ...  A115145
  [5] 0, 1, 7, 35, 154,  637, 2548,  9996, 38760, 149226,  572033, ...  A000588
  [6] 0, 1, 8, 44, 208,  910, 3808, 15504, 62016, 245157,  961400, ...  A115147
  [7] 0, 1, 9, 54, 273, 1260, 5508, 23256, 95931, 389367, 1562275, ...  A115148

A000108 (main diagonal), A000245 (subdiagonal), A002057 (diagonal 2), A000344 (diagonal 3), A000027 (column 2), A000096 (column 3), A071724 (row sums), A000958 (alternating row sums), A262394 (main diagonal of array).
Variants: A009766 (main variant), A030237, A130020.
Cf. A123110 (triangle of order 0), A355172 (triangle of order 2), A355174 (triangle of order 3), A355262 (Fuss-Catalan array).
Cf. A115144, A115145, A000588, A115147, A115148.

from functools import cache
from itertools import accumulate
@cache
def Trow(n: int) -> list[int]:
    if n == 0: return [1]
    if n == 1: return [0, 1]
    row = Trow(n - 1) + [Trow(n - 1)[n - 1]]
    return list(accumulate(row))
for n in range(11): print(Trow(n))

The general formula for the Fuss-Catalan triangles is, for m >= 0 and 0 <= k <= n:
FCT(n, k, m) = (m*(n - k) + m + 1)*(m*n + k - 1)!/((m*n + 1)!*(k - 1)!) for k > 0 and FCT(n, 0, m) = 0^n. The case considered here is T(n, k) = FCT(n, k, 1).
T(n, k) = (n - k + 2)*(n + k - 1)!/((n + 1)!*(k - 1)!) for k > 0; T(n, 0) = 0^n.
The g.f. of row n of the FC-triangle of order m is 0^n + (x - (m + 1)*x^2) / (1 - x)^(m*n + 2), thus:
T(n, k) = [x^k] (0^n + (x - 2*x^2)/(1 - x)^(n + 2)).

A360144 a(n) = Sum_{k=0..n} binomial(2n+3k,n-k).

Original entry on oeis.org

1, 3, 14, 69, 344, 1721, 8621, 43206, 216570, 1085574, 5441294, 27272044, 136679882, 684959516, 3432431414, 17199626276, 86182614207, 431824008713, 2163629549132, 10840520569183, 54313805146415, 272122594209738, 1363372115057995, 6830627007245263

Offset: 0

Seiichi Manyama, Jan 27 2023

nonn

Cf. A001700, A032443, A108080, A360143.

Cf. A000108, A000344, A000984.

A360144 := proc(n)
    add(binomial(2*n+3*k,n-k),k=0..n) ;
end proc:
seq(A360144(n),n=0..70) ; # R. J. Mathar, Mar 12 2023

a(n) = sum(k=0, n, binomial(2*n+3*k, n-k));

my(N=30, x='x+O('x^N)); Vec(1/(sqrt(1-4*x)*(1-x*(2/(1+sqrt(1-4*x)))^5)))

G.f.: 1 / ( sqrt(1-4*x) * (1 - x * c(x)^5) ), where c(x) is the g.f. of A000108.
D-finite with recurrence +n*(697*n-7543)*a(n) +(697*n^2+23641*n-3800)*a(n-1) +2*(-32006*n^2+199879*n-255053)*a(n-2) +(283953*n^2-2288641*n+4072186)*a(n-3) +2*(-186566*n^2+1774989*n-4013515)*a(n-4) +(146221*n^2-1648033*n+4472550)*a(n-5) +(38223*n^2-307771*n+532906)*a(n-6) -10*(1511*n-6875)*(2*n-13)*a(n-7)=0. - R. J. Mathar, Mar 12 2023
a(n) = binomial(2*n, n)*hypergeom([1, (1+2*n)/3, 2*(1+n)/3, 1+2*n/3, -n], [(1+n)/4, (2+n)/4, (3+n)/4, 1+n/4], -3^3/4^4). - Stefano Spezia, Jun 17 2025

A050145 T(n,k)=M(2n,n-1,k-1), 0<=k<=n, n >= 0, array M as in A050144.

Original entry on oeis.org

0, 1, 0, 2, 1, 1, 5, 4, 5, 1, 14, 14, 20, 7, 1, 42, 48, 75, 35, 9, 1, 132, 165, 275, 154, 54, 11, 1, 429, 572, 1001, 637, 273, 77, 13, 1, 1430, 2002, 3640, 2548, 1260, 440, 104, 15, 1, 4862, 7072, 13260, 9996, 5508, 2244, 663, 135, 17, 1

Offset: 0

Clark Kimberling

First 7 columns of T are A000108, A002057, A000344, A000588, A001392, A000589, A000590.

			Rows: {0}; {1,0}; {2,1,1}; ...

cp's OEIS Frontend

A052722 Expansion of e.g.f. (1 - 2*x - sqrt(1-4*x))^2 * (1 - sqrt(1-4*x))/8.

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A115144 Fifth convolution of A115140.

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A143955 Sum of the altitudes of the leftmost valleys of all Dyck paths of semilength n (if path has no valley, then this altitude is taken to be 0).

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A026029 Number of (s(0), s(1), ..., s(2n)) such that s(i) is a nonnegative integer and |s(i) - s(i-1)| = 1 for i = 1,2,...,n, s(0) = 3, s(2n) = 3. Also T(2n,n), where T is defined in A026022.

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A050155 Triangle T(n,k), k>=0 and n>=1, read by rows defined by: T(n,k) = (2k+3)*binomial(2n,n-k-1)/(n+k+2).

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A071718 Expansion of (1+x^2*C)*C^3, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108.

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A119245 Triangle, read by rows, defined by: T(n,k) = (4*k+1)*binomial(2*n+1, n-2*k)/(2*n+1) for n >= 2*k >= 0.

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A052722 Expansion of e.g.f. (1 - 2x - sqrt(1-4x))^2 * (1 - sqrt(1-4*x))/8.

A071718 Expansion of (1+x^2C)C^3, where C = (1-(1-4x)^(1/2))/(2x) is g.f. for Catalan numbers, A000108.

A119245 Triangle, read by rows, defined by: T(n,k) = (4k+1)binomial(2n+1, n-2k)/(2n+1) for n >= 2k >= 0.

A360144 a(n) = Sum_{k=0..n} binomial(2n+3k,n-k).