A005789 -id:A005789 - cp's OEIS frontend

A274969 Number of walks in the first quadrant starting and ending at (0,0) consisting of 3n steps taken from {E=(1, 0), D=(-1, 1), S=(0, -1)}, no S step occurring before the final E step.

1, 1, 4, 21, 121, 728, 4488, 28101, 177859, 1134705, 7283640, 46981740, 304253964, 1976886616, 12880883408, 84130964709, 550649378199, 3610705776755, 23714554702020, 155979407872365, 1027269675638745, 6773476758296220, 44709685668953760, 295402076512228140, 1953492865541875476

Offset: 0

David Bevan, Jul 13 2016

Number of pushall stack words of length 3n. These consist of n 'ρ' letters, n 'λ' letters and n 'μ' letters, such that the count of 'λ' letters never exceeds the count of 'ρ' letters, the count of 'μ' letters never exceeds the count of 'λ' letters, and all the 'ρ' letters occur before any of the 'μ' letters.
A permutation of length n is 2-stack pushall sortable if and only if it can be sorted by a sequence of 3n operations represented by a pushall stack word of length 3n, where ρ corresponds to pushing to the 1st (Right) stack, λ corresponds to popping from the 1st stack and pushing to the 2nd (Left) stack, and μ corresponds to popping from the 2nd stack.
There is an obvious bijection between pushall stack words of length 3n using the letters 'ρ', 'λ', and 'μ', and pushall stack words of length 3n in which 'ρ' and 'μ' are the same symbol. In this way, a(n) is the number of words consisting of n 'λ' letters and 2n 'μ' letters, such that the count of 'λ' letters never exceeds the count of 'μ' letters in any prefix or suffix of the word. This allows a closed form (added below) based on two usages of "Andre's reflection method", in analogy with the Catalan numbers. - Janis Stipins, May 27 2019

			For n=2, the four walks are EEDDSS, EEDSDS, EDEDSS and EDESDS.

David Bevan, Table of n, a(n) for n = 0..300
Nicolas Borie and Justine Falque, Product-Coproduct Prographs and Triangulations of the Sphere, arXiv:2202.05757 [math.CO], 2022. Proceedings of the 34th Conference on Formal Power Series and Algebraic Combinatorics (Bangalore), Séminaire Lotharingien de Combinatoire XX (2022). See Proposition 1, p. 10.
Adeline Pierrot and Dominique Rossin, 2-stack pushall sortable permutations, arXiv:1303.4376 [cs.DM], 2013.

Walks in the first quadrant from (0,0) to (0,0) with steps from {E, D, S} A005789.
2-stack pushall sortable permutations A274970.
Cf. A259475.

CoefficientList[Module[{r=0},Do[r+=Coefficient[1-16z+64z^2+(21z-96z^2)f+(-4z+27z^2)f^2+(-4z^2+27z^3)f^3/.f->r,z,i]z^i,{i,0,30}];r],z]

N=O(z^35); f=1+N; while(f+N<>f=1+(5*z-32*z^2+(-4+27*z)*z*(1+z*f)*f^2)/(1-21*z+96*z^2), ); Vec(f+N) \\ Using that the g.f. is fixed point of T(f)=1+(5*z-32*z^2+(-4+27*z)*z*(1+z*f)*f^2)/(1-21*z+96*z^2). - M. F. Hasler, Jul 13 2016

a(n) = binomial(3*n,n) - 2*binomial(3*n,n-1) + binomial(3*n,n-2); \\ Janis Stipins, May 27 2019

The o.g.f. f=f(z) is algebraic, satisfying the cubic equation (1-16*z+64*z^2) + (-1+21*z-96*z^2)*f + (-4*z+27*z^2)*f^2 + (-4*z^2+27*z^3)*f^3 = 0.
a(n) = A259475(n,n). - Alois P. Heinz, Nov 19 2018
a(n) = binomial(3*n,n) - 2*binomial(3*n,n-1) + binomial(3*n,n-2). - Janis Stipins, May 27 2019
G.f.: (2*(1 - 6*x)*cos(arccos(1 - (27*x)/2)/6)/sqrt(4 - 27*x) + 4*sqrt(3)*sqrt(x)*sin(arcsin(3*sqrt(3)*sqrt(x)/2)/3) - 1)/(3*x). - Stefano Spezia, Feb 19 2022

Data double-checked by M. F. Hasler, Jul 13 2016

A321976 7-dimensional Catalan numbers.

Original entry on oeis.org

1, 1, 429, 1385670, 13672405890, 278607172289160, 9490348077234178440, 475073684264389879228560, 32103104214166146088869942000, 2760171874087743799855959353857200, 289232890341906497299306268771988273600, 35764585916110766978895474668714467232388000

Offset: 0

Seiichi Manyama, Nov 23 2018

nonn

Number of n X 7 Young tableaux.

Seiichi Manyama, Table of n, a(n) for n = 0..177
Wikipedia, Hook length formula

Row 7 of A060854.

Cf. A000108, A005789, A005790, A005791, A321975.

List([0..15],n->24883200*Factorial(7*n)/Product([0..6],k->Factorial(n+k))); # Muniru A Asiru, Nov 25 2018

[24883200*Factorial(7*n) / (Factorial(n)*Factorial(n + 1)*Factorial(n + 2)*Factorial(n + 3)*Factorial(n + 4)*Factorial(n + 5)*Factorial(n + 6)): n in [0..15]]; // Vincenzo Librandi, Nov 24 2018

Table[24883200*(7*n)!/(n!*(n+1)!*(n+2)!*(n+3)!*(n+4)!*(n+5)!*(n+6)!),{n,0,15}] (* Vincenzo Librandi, Nov 24 2018 *)

{a(n) = 24883200*(7*n)!/(n!*(n+1)!*(n+2)!*(n+3)!*(n+4)!*(n+5)!*(n+6)!)}

a(n) = 0!*1!*...*6! * (7*n)! / ( n!*(n+1)!*...*(n+6)! ).

a(n) ~ 3110400 * 7^(7*n + 1/2) / (Pi^3 * n^24). - Vaclav Kotesovec, Nov 23 2018

A065058 Number of paths to T(n,n,n) with T(i,j,k)= 0 if j>i or k>j and T(i,j,k) = T(i-1,j,k) + T(i,j-1,k) + T(i,j,k-1) and T(i,j,0) = 1.

Original entry on oeis.org

1, 1, 3, 18, 162, 1851, 24661, 365613, 5863881, 99895425, 1785024645, 33156724734, 635961987570, 12531882072719, 252701147866029, 5198011293931270, 108793300411597194, 2312049376195527621, 49804793378882733343, 1085910951385068915212, 23934948368968158240960

Offset: 0

Wouter Meeussen, Nov 06 2001

nonn

Similar to the "3-dimensional Catalan numbers" of A005789, but with paths starting from anywhere on z=0, instead of only from [0,0,0].

			a(3) = 18 because [3,3,3] can be reached from [x,y,0] in the following ways (along nondecreasing paths): 5 [1,1,0] + 5 [2,1,0] + 3 [2,2,0] + 2 [3,1,0] + 2 [3,2,0] + [3,3,0].

Alois P. Heinz, Table of n, a(n) for n = 0..500

Cf. A005789.

T[0, 0, 0] := 1; T[x_, y_, z_] := 0 /; (x< y || y< z); T[u_, v_, 0] := 1; T[, 0, 0] := 1 T[x, y_, z_] := (T[x, y, z]= T[x-1, y, z]+T[x, y-1, z] +T[x, y, z-1]) /; (y<=x ||z<=y)

a(n) ~ 13 * 3^(3*n+7/2) / (2^11 * Pi * n^4). - Vaclav Kotesovec, Sep 10 2014

A161581 a(n) = (3n)!/(n!(n+1)!(n+2)!).

Original entry on oeis.org

21, 231, 3003, 43758, 692835, 11685817, 207157665, 3823000545, 72931087320, 1430571328200, 28734046963560, 589047962752980, 12292044987448215, 260543149635912165, 5599392250947235125, 121830987186399315825

Offset: 3

Alexander Adamchuk, Jun 14 2009

nonn

3-d analog of the Catalan numbers A000108.

Winston de Greef, Table of n, a(n) for n = 3..704
Eric Weisstein's World of Mathematics, Binomial Sums.
Eric Weisstein's World of Mathematics, Central Binomial Coefficient.
Eric Weisstein's World of Mathematics, Catalan Number.

Cf. A000108, A005789, A006480.

A161581 := proc(n) (3*n)!/n!/(n+1)!/(n+2)! ; end: seq(A161581(n),n=3..40) ; # R. J. Mathar, Jun 16 2009
a := proc (n) options operator, arrow: factorial(3*n)/(factorial(n)*factorial(n+1)*factorial(n+2)) end proc: seq(a(n), n = 3 .. 20); # Emeric Deutsch, Jun 14 2009

a(n) = A006480(n)/((n+1)^2*(n+2)).
a(n) ~ 3^(3*n + 1/2) / (2*Pi*n^4). - Vaclav Kotesovec, Feb 21 2023
a(n) = (1/2)*A005789(n) for n >= 3. - Peter Bala, Mar 01 2023
D-finite with recurrence (n+2)*(n+1)*a(n) -3*(3*n-1)*(3*n-2)*a(n-1)=0. - R. J. Mathar, Nov 22 2024

Repetitions of information contained in other sequences removed by R. J. Mathar, Jun 16 2009

More terms from Emeric Deutsch, Jun 14 2009

A321977 8-dimensional Catalan numbers.

Original entry on oeis.org

1, 1, 1430, 23371634, 1489877926680, 231471904322784840, 67867669180627125604080, 32103104214166146088869942000, 22081374992701950398847674830857600, 20535535214275361308250745082811167425600, 24486819823897171791550434989846505231774984000

Offset: 0

Seiichi Manyama, Nov 23 2018

nonn

Number of n X 8 Young tableaux.

Seiichi Manyama, Table of n, a(n) for n = 0..146
Wikipedia, Hook length formula

Row 8 of A060854.

Cf. A000108, A005789, A005790, A005791, A321975, A321976.

List([0..15],n->125411328000*Factorial(8*n)/Product([0..7],k->Factorial(n+k))); # Muniru A Asiru, Nov 25 2018

[125411328000*Factorial(8*n)/(Factorial(n)*Factorial(n + 1)*Factorial(n + 2)*Factorial(n + 3)*Factorial(n + 4)*Factorial(n + 5)*Factorial(n + 6)*Factorial(n + 7)): n in [0..15]]; // Vincenzo Librandi, Nov 24 2018

Table[125411328000 (8 n)! / (n! (n+1)! (n+2)! (n+3)! (n+4)! (n+5)! (n+6)! (n + 7)!), {n, 0, 15}] (* Vincenzo Librandi, Nov 24 2018 *)

a(n) = 0!*1!*...*7! * (8*n)! / ( n!*(n+1)!*...*(n+7)! ).

a(n) ~ 1913625 * 2^(24*n + 14) / (Pi^(7/2) * n^(63/2)). - Vaclav Kotesovec, Nov 23 2018

A005792 Positive numbers that are the sum of 2 squares or 3 times a square.

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 9, 10, 12, 13, 16, 17, 18, 20, 25, 26, 27, 29, 32, 34, 36, 37, 40, 41, 45, 48, 49, 50, 52, 53, 58, 61, 64, 65, 68, 72, 73, 74, 75, 80, 81, 82, 85, 89, 90, 97, 98, 100, 101, 104, 106, 108, 109, 113, 116, 117, 121, 122, 125, 128, 130, 136, 137, 144, 145

Offset: 1

N. J. A. Sloane

Equivalently, numbers of the form k^2, k^2+m^2, or 3*k^2, where k >= 1, m >= 1.

Theorem (Golomb; Snover et al.): A triangle can be partitioned into n pairwise congruent triangles iff n is of the form k^2, k^2+m^2, or 3*k^2.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
A. Soifer, How Does One Cut A Triangle?, Chapter 2, CEME, Colorado Springs CO 1990.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Solomon W. Golomb, Replicating figures in the plane, The mathematical gazette 48.366 (1964): 403-412.
Murray Klamkin, Review of "How Does One Cut a Triangle?" by Alexander Soifer, Amer. Math. Monthly, October 1991, pp. 775-. [Annotated scanned copy of pages 775-777 only] See "Grand Problem 1".
S. Snover, Letter to N. J. A. Sloane, May 1991
S. L. Snover, C. Wavereis and J. K. Williams, Rep-tiling for triangles, Discrete Math. 91 (1991), no. 2, 193-200.
Index entries for sequences related to sums of squares

Union of positive terms of A000290, A000404, A033428.

Cf. A074764.

More terms from Larry Reeves (larryr(AT)acm.org), Mar 21 2001

Entry revised by N. J. A. Sloane, Nov 30 2016

A065078 Triangle read by rows: a(n,m) = T[n,m,m] where T[i,j,k] is the 3-dimensional pyramid defined by T[n,m,0]=1 and T[i,j,k]=0 if j>i or k>j and T[i,j,k]=T[i-1,j,k]+T[i,j-1,k]+T[i,j,k-1].

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 3, 10, 18, 1, 4, 22, 79, 162, 1, 5, 40, 220, 831, 1851, 1, 6, 65, 492, 2681, 10488, 24661, 1, 7, 98, 962, 6883, 37367, 149743, 365613, 1, 8, 140, 1715, 15318, 105731, 573051, 2336243, 5863881, 1, 9, 192, 2856, 30840, 258604, 1742770

Offset: 0

Wouter Meeussen, Nov 09 2001

Number of paths to T[n,m,m] counted from the bottom plane (or T[n,m,0]).

			[3,2,2] can be reached from 3*[1,1,0] + 3*[2,1,0] + 2*[2,2,0] + 1*[3,1,0] + 1*[3,2,0], so a(3,2) = 3 + 3 + 2 + 1 + 1 =10.
Triangle begins
1;
1, 1;
1, 2, 3;
1, 3, 10, 18;
1, 4, 22, 79, 162;

Wouter Meeussen, Further comments on this sequence

Last number in each row is A065058.

Cf. A005789, A065057.

T[0, 0, 0] := 1; T[x_, y_, z_] := 0 /; (x< y || y< z); T[u_, v_, 0] := 1; T[, 0, 0] := 1; T[x, y_, z_] := (T[x, y, z]= T[x-1, y, z]+T[x, y-1, z] +T[x, y, z-1]) /; (y<=x ||z<=y); Table[T[x, y, y], {x, 0, 10}, {y, 0, x}]

A087647 Triangle of 3-Narayana numbers, N(n,k), for n >= 1, 0 <= k <= 2n-2.

Original entry on oeis.org

1, 1, 3, 1, 1, 10, 20, 10, 1, 1, 22, 113, 190, 113, 22, 1, 1, 40, 400, 1456, 2212, 1456, 400, 40, 1, 1, 65, 1095, 7095, 20760, 29484, 20760, 7095, 1095, 65, 1, 1, 98, 2541, 26180, 127435, 320034, 433092, 320034, 127435, 26180, 2541, 98, 1, 1, 140, 5250, 79870

Offset: 1

Robert A. Sulanke (sulanke(AT)math.boisestate.edu), Sep 23 2003

N(n,k) counts ballot sequences for three candidates having length 3n, ending in a tie and having k instances of a vote for a weaker candidate being followed immediately by a vote for a stronger one.

Equivalently, N(n,k) counts the lattice paths P := p_1p_2...p_{3n} using the steps X := (1,0,0), Y := (0,1,0) and Z := (0,0,1), running from (0,0,0) to (n,n,n), lying in {(x,y,z) : 0<=x<=y<=z } and satisfying #{i : p_ip_{i+1} in {XY,XZ,YZ} } = k.

			1;
1,3,1;
1,10,20,10,1;
1,22,113,190,113,22,1;
1,40,400,1456,2212,1456,400,40,1;
1,65,1095,7095,20760,29484,20760,7095,1095,65,1;
1,98,2541,26180,127435,320034,433092,320034,127435,26180,2541,98,1

Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
R. A. Sulanke, Counting Lattice Paths by Narayana Polynomials Electronic J. Combinatorics 7, No. 1, R40, 1-9, 2000.
R. A. Sulanke, Generalizing Narayana and Schroeder Numbers to Higher Dimensions, Electron. J. Combin. 11 (2004), Research Paper 54, 20 pp.
R. A. Sulanke, Three-dimensional Narayana and Schröder numbers, Theoretical Computer Science, Volume 346, Issues 2-3, 28 November 2005, Pages 455-468.

Cf. A001263 (Narayana numbers), A005789 (= Sum[N(n, k), {k, 0, 2n-2}], 3-dimensional Catalan numbers), A056939 (antichains in the poset 3*m*n).

seq( seq( add(2*(-1)^(k-j)*binomial(3*n+1, k-j)* binomial(n+j,n)*binomial(n+j+1,n)*binomial(n+j+2,n)/(n+1)^2/(n+2), j = 0 .. k), k = 0 .. 2*n-2), n = 1 ..7 );

For 0<=k<=2n-2, N(n, k) := Sum[2*(-1)^(k-j)*C(3*n+1, k-j)*C(n+j, n)*C(n+j+1, n)*C(n+j+2, n)/(n+1)^2/(n+2), {j, 0, k}] = Sum[(-1)^(k-j)*C(3*n+1, k-j)*a(n, j), {j, 0, k}] where a(m, n) is an entry in the triangle of A056939.

Recurrence: If N_n(t) := Sum[t^k*N(n, k), {k, 0, 2n-2}] then (3n-4)(n+2)(n+1)^2 N_n(t) = (3n-2)(n+1)( 4(1+t+t^2) - 5(1+7t+t^2)n +3(1+7t+t^2)n^2 ) N_{n-1}(t) - (n-2)( -12 +29n -30n^2 +9n^3)(1-t)^4 N_{n-2}(t) +(3n-1)(n-2)(n-3)(n-4) (1-t)^6 N_{n-3}(t).

A185148 Number of rectangular arrangements of [1,3n] in 3 increasing sequences of size n and n monotonic sequences of size 3.

Original entry on oeis.org

1, 6, 53, 587, 7572, 109027, 1705249, 28440320, 499208817, 9134237407, 172976239886, 3371587949969, 67351686970929, 1374179898145980, 28557595591148315, 603118526483125869, 12920388129877471030, 280324904918707937001, 6151595155000424589327, 136384555249451824930126

Offset: 1

Olivier Gérard, Feb 15 2011

nonn

a(n) counts a subset of A025035(n).
a(n) counts a more general set than A005789(n).
a(n) is also the number of (3*n-1)-step walks on 3-dimensional cubic lattice from (1,0,0) to (n,n,n) with steps in {(1,0,0), (0,1,0), (0,0,1)} such that for each point (x,y,z) we have x<=y<=z or x>=y>=z. - Alois P. Heinz, Feb 29 2012

			For n = 2 the a(2) = 6 arrangements are:
+---+  +---+  +---+  +---+  +---+  +---+
|1 4|  |1 6|  |1 3|  |1 3|  |1 2|  |1 2|
|2 5|  |2 5|  |2 5|  |2 4|  |3 5|  |3 4|
|3 6|  |3 4|  |4 6|  |5 6|  |4 6|  |5 6|
+---+  +---+  +---+  +---+  +---+  +---+
Only the second of these arrangements is not counted by A005789(2).

Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 1..700 (terms 0..200 from Alois P. Heinz)

Cf. A025035, A005789.

Column k=3 of A208615. - Alois P. Heinz, Feb 29 2012

b:= proc(x, y, z) option remember;
      `if`(x=z, `if`(x=0, 1, 2*b(x-1, y, z)), `if`(x>0, b(x-1, y, z), 0)+
      `if`(y>x, b(x, y-1, z), 0)+ `if`(z>y, b(x, y, z-1), 0))
    end:
a:= n-> b(n-1, n$2):
seq(a(n), n=1..30);  # Alois P. Heinz, Feb 29 2012

b[x_, y_, z_] := b[x, y, z] = If[x == z, If[x == 0, 1, 2*b[x - 1, y, z]], If[x > 0, b[x - 1, y, z], 0] + If[y > x, b[x, y - 1, z], 0] + If[z > y, b[x, y, z - 1], 0]];
a[n_] := b[n - 1, n, n];
Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Nov 12 2017, after Alois P. Heinz *)

a(n) ~ c * 27^n / n^4, where c = 0.608287207375... . - Vaclav Kotesovec, Sep 03 2014, updated Sep 07 2016

More terms and example from Alois P. Heinz, Feb 22 2011

Extended beyond a(8) by Alois P. Heinz, Feb 22 2012

A241958 Duplicate of A217800.

Original entry on oeis.org

1, 2, 12, 110, 1274, 17136, 255816, 4124406, 70549050, 1264752060, 23555382240, 452806924752, 8939481277552, 180551099694400, 3719061442253520, 77933728043586630, 1658001861319441050, 35749633305661575300, 780123576993991461000, 17208112644166765652100, 383292388823513983713900

Offset: 0

Karol A. Penson, Aug 10 2014

dead

This is a duplicate of A217800 or of A007724. - Alois P. Heinz, Aug 22 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..200
K. Gorska and K. A. Penson, Multidimensional Catalan and related numbers as Hausdorff moments, arXiv preprint arXiv:1304.6008, 2013

[Factorial(3*n+3)/((4*(n+1)^2-1)*Factorial((n+1))^2*Factorial(n+ 2)): n in [0..20]]; // Vincenzo Librandi, Aug 30 2014

a := n -> (-1)^n*hypergeom([-2-2*n,-2*n,-2*n-1],[2, 3],1):
seq(round(evalf(a(n),32)),n=0..20); # Peter Luschny, Aug 29 2014

Table[(3 n + 3)!/((4 (n + 1)^2 - 1) ((n + 1)!)^2 (n + 2)!), {n, 0, 20}] (* Vincenzo Librandi, Aug 30 2014 *)
Table[(-1)^n HypergeometricPFQ[{-2 - 2 n, -2 n, -2 n - 1}, {2, 3},
1], {n, 0, 20}] (* Michael De Vlieger, Aug 22 2016 *)

a(n) = (3*n+3)!/((4*(n+1)^2-1)*((n+1)!)^2*(n+2)!); \\ Michel Marcus, Aug 10 2014

O.g.f.(in Maple notation): hypergeom([1/2, 1, 4/3, 5/3], [2, 5/2, 3], 27*z);
a(n) ~ (1/93312)*sqrt(3)*27^n*(314928*n^4-1644624*n^3+5545260*n^2 -15387660*n+38310503)/(Pi*n^8), for n -> infinity.
Representation of a(n) as the n-th power moment of a positive function on the segment [0,27]:
a(n) = int(x^n*W(x),x=0..27),n=0,1,2..., where
W(x) = 1/(Pi*sqrt(x))+sqrt(x)/Pi-(9/20)*sqrt(3)*2^(1/3)* hypergeom([-2/3, -1/6, 1/3], [2/3, 11/6], (1/27)*x)*x^(1/3)/ (sqrt(Pi)*Gamma(5/6)*Gamma(2/3))-(27/56)*2^(2/3)*Gamma(5/6) *Gamma(2/3)*hypergeom([-1/3, 1/6, 2/3], [4/3, 13/6], (1/27)*x)* x^(2/3)/Pi^(5/2).
W(x) for x->0 has the singularity 1/sqrt(x), W(27)=0.
This is the solution of the Hausdorff moment problem and is unique.
a(n) = (1/2)*(n+3)!/((4*(n+1)^2-1)*(n+1)!)*A005789(n), where A005789(n) are the three-dimensional Catalan numbers (see the Gorska and Penson link).
a(n) = A006480(n+1)/((2+n)*(1+2*n)*(3+2*n)). - Peter Luschny, Aug 15 2014
a(n) = (-1)^n*hypergeom([-2-2*n,-2*n,-2*n-1],[2,3],1). - Peter Luschny, Aug 29 2014
(2*n+3)*(n+2)*(n+1)*a(n) -3*(3*n+2)*(2*n-1)*(3*n+1)*a(n-1)=0. - R. J. Mathar, Jun 14 2016

cp's OEIS Frontend

A274969 Number of walks in the first quadrant starting and ending at (0,0) consisting of 3n steps taken from {E=(1, 0), D=(-1, 1), S=(0, -1)}, no S step occurring before the final E step.

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A321976 7-dimensional Catalan numbers.

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A065058 Number of paths to T(n,n,n) with T(i,j,k)= 0 if j>i or k>j and T(i,j,k) = T(i-1,j,k) + T(i,j-1,k) + T(i,j,k-1) and T(i,j,0) = 1.

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Mathematica

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A161581 a(n) = (3n)!/(n!(n+1)!(n+2)!).

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Maple

Formula

Extensions

A321977 8-dimensional Catalan numbers.

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GAP

Magma

Mathematica

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A005792 Positive numbers that are the sum of 2 squares or 3 times a square.

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A065078 Triangle read by rows: a(n,m) = T[n,m,m] where T[i,j,k] is the 3-dimensional pyramid defined by T[n,m,0]=1 and T[i,j,k]=0 if j>i or k>j and T[i,j,k]=T[i-1,j,k]+T[i,j-1,k]+T[i,j,k-1].

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