A000245 -id:A000245 - cp's OEIS frontend

A071724 a(n) = 3*binomial(2n, n-1)/(n+2), n > 0, with a(0)=1.

1, 1, 3, 9, 28, 90, 297, 1001, 3432, 11934, 41990, 149226, 534888, 1931540, 7020405, 25662825, 94287120, 347993910, 1289624490, 4796857230, 17902146600, 67016296620, 251577050010, 946844533674, 3572042254128, 13505406670700

Offset: 0

N. J. A. Sloane, Jun 06 2002

Number of standard tableaux of shape (n+1,n-1) (n>=1). - Emeric Deutsch, May 30 2004
From Gus Wiseman, Apr 12 2019: (Start)
Also the number of integer partitions (of any positive integer) such that n is the maximum number of unit steps East or South in the Young diagram starting from the upper-left square and ending in a boundary square in the lower-right quadrant. Also the number of integer partitions fitting in a triangular partition of length n but not of length n - 1. For example, the a(0) = 1 through a(4) = 9 partitions are:
  ()  (1)  (2)   (3)
           (11)  (22)
           (21)  (31)
                 (32)
                 (111)
                 (211)
                 (221)
                 (311)
                 (321)
(End)
The sequence (-1)^(n+1)*a(n), for n >= 1 and +1 for n = 0, is the so-called Z-sequence of the Riordan triangle A158909. For the notion of Z- and A-sequences for Riordan arrays see the W. Lang link under A006232 with details and references. - Wolfdieter Lang, Oct 22 2019

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Joerg Arndt, The a(4)=28 Young tableaux of shape [5, 3].
FindStat, St000384: The maximal part of the shifted composition of an integer partition.
Spencer J. Franks, Pamela E. Harris, Kimberly Harry, Jan Kretschmann, and Megan Vance, Counting Parking Sequences and Parking Assortments Through Permutations, arXiv:2301.10830 [math.CO], 2023.
Ricardo Gómez Aíza, Trees with flowers: A catalog of integer partition and integer composition trees with their asymptotic analysis, arXiv:2402.16111 [math.CO], 2024. See p. 21.
Zhicong Lin, David G.L. Wang, and Tongyuan Zhao, A decomposition of ballot permutations, pattern avoidance and Gessel walks, arXiv:2103.04599 [math.CO], 2021.
Murray Tannock, Equivalence classes of mesh patterns with a dominating pattern, MSc Thesis, Reykjavik Univ., May 2016.
Gus Wiseman, Young diagrams of all integer partitions fitting in a triangular partition of length n but not of length n - 1, n = 1...4.

Cf. A000245, A001622, A002421, A030237, A051924, A096771, A115720, A158909.
Cf. A325169, A325188, A325193, A325195, A325200.
Number of times n appears in A065770.
Column sums of A325189.
Row sums of A030237.

[1] cat [3*Binomial(2*n,n-1)/(n+2): n in [1..29]]; // Vincenzo Librandi, Jul 12 2017

A071724:= n-> 3*binomial(2*n, n-1)/(n+2); 1,seq(A071724(n), n=1..30); # G. C. Greubel, Mar 17 2021

Join[{1}, Table[3Binomial[2n, n-1]/(n+2), {n,1,30}]] (* Vincenzo Librandi, Jul 12 2017 *)
nn=7;
otbmax[ptn_]:=Max@@MapIndexed[#1+#2[[1]]-1&,Append[ptn,0]];
allip=Join@@Table[IntegerPartitions[n],{n,0,nn*(nn+1)/2}];
Table[Length[Select[allip,otbmax[#]==n&]],{n,0,nn}] (* Gus Wiseman, Apr 12 2019 *)

a(n)=if(n<1,n==0,3*(2*n)!/(n+2)!/(n-1)!)

[1]+[3*n*catalan_number(n)/(n+2) for n in (1..30)] # G. C. Greubel, Mar 17 2021

a(n) = A000245(n), n>0.
G.f.: (C(x)-1)*(1-x)/x = (1 + x^2 * C(x)^3)*C(x), where C(x) is g.f. for Catalan numbers, A000108.
G.f.: ((1-sqrt(1-4*x))/(2*x)-1)*(1-x)/x = A(x) satisfies x^2*A(x)^2 + (x-1)*(2*x-1)*A(x) + (x-1)^2 = 0.
G.f.: 1 + x*C(x)^3, where C(x) is g.f. for the Catalan numbers (A000108). Sequence without the first term is the 3-fold convolution of the Catalan sequence. - Emeric Deutsch, May 30 2004
a(n) is the n-th moment of the function defined on the segment (0, 4) of x axis: a(n) = Integral_{x=0..4} x^n*(-x^(1/2)*cos(3*arcsin((1/2)*x^(1/2)))/Pi) dx, n=0, 1... . - Karol A. Penson, Sep 29 2004
D-finite with recurrence -(n+2)*(n-1)*a(n) + 2*n*(2*n-1)*a(n-1) = 0. - R. J. Mathar, Jul 10 2017
a(n) ~ c*2^(2*n)*n^(-3/2), where c = 3/sqrt(Pi). - Stefano Spezia, Sep 23 2022
From Amiram Eldar, Sep 29 2022: (Start)
Sum_{n>=0} 1/a(n) = 14*(Pi/(3*sqrt(3)) + 1)/9.
Sum_{n>=0} (-1)^n/a(n) = 18/25 - 164*log(phi)/(75*sqrt(5)), where phi is the golden ratio (A001622). (End)

A047072 Array A read by diagonals: A(h,k)=number of paths consisting of steps from (0,0) to (h,k) such that each step has length 1 directed up or right and no step touches the line y=x unless x=0 or x=h.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 3, 2, 2, 3, 1, 1, 4, 5, 4, 5, 4, 1, 1, 5, 9, 5, 5, 9, 5, 1, 1, 6, 14, 14, 10, 14, 14, 6, 1, 1, 7, 20, 28, 14, 14, 28, 20, 7, 1, 1, 8, 27, 48, 42, 28, 42, 48, 27, 8, 1, 1, 9, 35, 75, 90, 42, 42, 90, 75, 35, 9, 1

Offset: 0

Clark Kimberling

			Array, A(n, k), begins as:
  1, 1,  1,  1,  1,   1,   1,   1, ...;
  1, 2,  1,  2,  3,   4,   5,   6, ...;
  1, 1,  2,  2,  5,   9,  14,  20, ...;
  1, 2,  2,  4,  5,  14,  28,  48, ...;
  1, 3,  5,  5, 10,  14,  42,  90, ...;
  1, 4,  9, 14, 14,  28,  42, 132, ...;
  1, 5, 14, 28, 42,  42,  84, 132, ...;
  1, 6, 20, 48, 90, 132, 132, 264, ...;
Antidiagonals, T(n, k), begins as:
  1;
  1,  1;
  1,  2,  1;
  1,  1,  1,  1;
  1,  2,  2,  2,  1;
  1,  3,  2,  2,  3,  1;
  1,  4,  5,  4,  5,  4,  1;
  1,  5,  9,  5,  5,  9,  5,  1;
  1,  6, 14, 14, 10, 14, 14,  6,  1;

G. C. Greubel, Antidiagonals n = 0..50, flattened
R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6.

Cf. A000007, A000027, A000096, A002420.
Cf. A005586, A025174, A047079, A063886.
Cf. A047073, A047074, A047079.
The following are all versions of (essentially) the same Catalan triangle: A009766, A030237, A033184, A059365, A099039, A106566, A130020, A047072.
Diagonals give A000108, A000245, A002057, A000344, A003517, A000588, A003518, A003519, A001392, ...

b:= func< n | n eq 0 select 1 else 2*Catalan(n-1) >;
function A(n,k)
  if k eq n then return b(n);
  elif k gt n then return Binomial(n+k-1, n) - Binomial(n+k-1, n-1);
  else return Binomial(n+k-1, k) - Binomial(n+k-1, k-1);
  end if; return A;
end function;
// [[A(n,k): k in [0..12]]: n in [0..12]];
T:= func< n,k | A(n-k, k) >;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Oct 13 2022

A[, 0]= 1; A[0, ]= 1; A[h_, k_]:= A[h, k]= If[(k-1>h || k-1Jean-François Alcover, Mar 06 2019 *)

def A(n,k):
    if (k==n): return 2*catalan_number(n-1) + 2*int(n==0)
    elif (k>n): return binomial(n+k-1, n) - binomial(n+k-1, n-1)
    else: return binomial(n+k-1, k) - binomial(n+k-1, k-1)
def T(n,k): return A(n-k, k)
# [[A(n,k) for k in range(12)] for n in range(12)]
flatten([[T(n,k) for k in range(n+1)] for n in range(12)]) # G. C. Greubel, Oct 13 2022

A(n, n) = 2*[n=0] - A002420(n),
A(n, n+1) = 2*A000108(n-1), n >= 1.
From G. C. Greubel, Oct 13 2022: (Start)
T(n, n-1) = A000027(n-2) + 2*[n<3], n >= 1.
T(n, n-2) = A000096(n-4) + 2*[n<5], n >= 2.
T(n, n-3) = A005586(n-6) + 4*[n<7] - 2*[n=3], n >= 3.
T(2*n, n) = 2*A000108(n-1) + 3*[n=0].
T(2*n-1, n-1) = T(2*n+1, n+1) = A000180(n).
T(3*n, n) = A025174(n) + [n=0]
Sum_{k=0..n} T(n, k) = 2*A063886(n-2) + [n=0] - 2*[n=1]
Sum_{k=0..n} (-1)^k * T(n, k) = A000007(n).
Sum_{k=0..floor(n/2)} T(n, k) = A047079(n). (End)

A026009 Triangular array T read by rows: T(n,0) = 1 for n >= 0; T(1,1) = 1; and for n >= 2, T(n,k) = T(n-1,k-1) + T(n-1,k) for k = 1,2,...,[(n+1)/2]; T(n,n/2 + 1) = T(n-1,n/2) if n is even.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 4, 6, 3, 1, 5, 10, 9, 1, 6, 15, 19, 9, 1, 7, 21, 34, 28, 1, 8, 28, 55, 62, 28, 1, 9, 36, 83, 117, 90, 1, 10, 45, 119, 200, 207, 90, 1, 11, 55, 164, 319, 407, 297, 1, 12, 66, 219, 483, 726, 704, 297, 1, 13, 78, 285, 702, 1209, 1430, 1001, 1, 14, 91, 363, 987, 1911, 2639, 2431, 1001

Offset: 0

Clark Kimberling

			From _Jonathon Kirkpatrick_, Jul 01 2016: (Start)
Triangle begins:
  1;
  1,  1;
  1,  2,  1;
  1,  3,  3;
  1,  4,  6,   3;
  1,  5, 10,   9;
  1,  6, 15,  19,   9;
  1,  7, 21,  34,  28;
  1,  8, 28,  55,  62,   28;
  1,  9, 36,  83, 117,   90;
  1, 10, 45, 119, 200,  207,   90;
  1, 11, 55, 164, 319,  407,  297;
  1, 12, 66, 219, 483,  726,  704,  297;
  1, 13, 78, 285, 702, 1209, 1430, 1001;
  ... (End)

G. C. Greubel, Rows n = 0..100 of the triangle, flattened

Diagonals of this sequence: A000217, A000245, A026012, A026013, A026014, A026015, A026016, A026017, A026018, A026019, A026020, A026021.

Sums involving this sequence: A026010, A027287, A027288, A027289, A027290, A027291, A027292.

[1] cat [Binomial(n,k) - Binomial(n,k-3): k in [0..Floor((n+2)/2)], n in [1..15]]; // G. C. Greubel, Mar 18 2021

T[n_, k_]:= Binomial[n, k] - Binomial[n, k-3];
Join[{1}, Table[T[n, k], {n,14}, {k,0,Floor[(n+2)/2]}]//Flatten] (* G. C. Greubel, Mar 18 2021 *)

[1]+flatten([[binomial(n,k) - binomial(n,k-3) for k in (0..(n+2)//2)] for n in (1..15)]) # G. C. Greubel, Mar 18 2021

T(n, k) = binomial(n, k) - binomial(n, k-3). - Darko Marinov (marinov(AT)lcs.mit.edu), May 17 2001

Sum_{k=0..floor((n+2)/2)} T(n, k) = A026010(n). - G. C. Greubel, Mar 18 2021

A130020 Triangle T(n,k), 0<=k<=n, read by rows given by [1,0,0,0,0,0,0,...] DELTA [0,1,1,1,1,1,1,...] where DELTA is the operator defined in A084938 .

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 5, 5, 0, 1, 4, 9, 14, 14, 0, 1, 5, 14, 28, 42, 42, 0, 1, 6, 20, 48, 90, 132, 132, 0, 1, 7, 27, 75, 165, 297, 429, 429, 0, 1, 8, 35, 110, 275, 572, 1001, 1430, 1430, 0, 1, 9, 44, 154, 429, 1001, 2002, 3432, 4862, 4862, 0

Offset: 0

Philippe Deléham, Jun 16 2007

Reflected version of A106566.

			Triangle begins:
  1;
  1, 0;
  1, 1,  0;
  1, 2,  2,   0;
  1, 3,  5,   5,   0;
  1, 4,  9,  14,  14,    0;
  1, 5, 14,  28,  42,   42,    0;
  1, 6, 20,  48,  90,  132,  132,    0;
  1, 7, 27,  75, 165,  297,  429,  429,    0;
  1, 8, 35, 110, 275,  572, 1001, 1430, 1430,    0;
  1, 9, 44, 154, 429, 1001, 2002, 3432, 4862, 4862,  0;
  ...

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Francesca Aicardi, Catalan triangle and tied arc diagrams, arXiv:2011.14628 [math.CO], 2020.

The following are all versions of (essentially) the same Catalan triangle: A009766, A030237, A033184, A047072, A059365, A099039, A106566, this sequence.
Diagonals give A000108, A000245, A002057, A000344, A003517, A000588, A003518, A003519, A001392, ...
Cf. A000108 (Catalan numbers), A106566 (row reversal), A210736.

A130020:= func< n,k | n eq 0 select 1 else (n-k)*Binomial(n+k-1, k)/n >;
[A130020(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 14 2022

T[n_, k_]:= (n-k)Binomial[n+k-1, k]/n; T[0, 0] = 1;
Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* Jean-François Alcover, Jun 14 2019 *)

{T(n, k) = if( k<0 || k>=n, n==0 && k==0, binomial(n+k, n) * (n-k)/(n+k))}; /* Michael Somos, Oct 01 2022 */

@CachedFunction
def A130020(n, k):
    if n==k: return add((-1)^j*binomial(n, j) for j in (0..n))
    return add(A130020(n-1, j) for j in (0..k))
for n in (0..10) :
    [A130020(n, k) for k in (0..n)]  # Peter Luschny, Nov 14 2012

T(n, k) = A106566(n, n-k).
Sum_{k=0..n} T(n,k) = A000108(n).
T(n, k) = (n-k)*binomial(n+k-1, k)/n with T(0, 0) = 1. - Jean-François Alcover, Jun 14 2019
Sum_{k=0..floor(n/2)} T(n-k, k) = A210736(n). - G. C. Greubel, Jun 14 2022
G.f.: Sum_{n>=0, k>=0} T(n, k)*x^k*z^n = 1/(1 - z*c(x*z)) where c(z) = g.f. of A000108.

A099039 Riordan array (1,c(-x)), where c(x) = g.f. of Catalan numbers.

Original entry on oeis.org

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

Offset: 0

Paul Barry, Sep 23 2004

Row sums are generalized Catalan numbers A064310. Diagonal sums are 0^n+(-1)^n*A030238(n-2). Inverse is A026729, as number triangle. Columns have g.f. (xc(-x))^k=((sqrt(1+4x)-1)/2)^k.

Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, ... ] DELTA [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ... ] where DELTA is the operator defined in A084938. - Philippe Deléham, May 31 2005

			Rows begin {1}, {0,1}, {0,-1,1}, {0,2,-2,1}, {0,-5,5,-3,1}, ...
Triangle begins
  1;
  0,    1;
  0,   -1,    1;
  0,    2,   -2,   1;
  0,   -5,    5,  -3,    1;
  0,   14,  -14,   9,   -4,   1;
  0,  -42,   42, -28,   14,  -5,  1;
  0,  132, -132,  90,  -48,  20, -6,  1;
  0, -429,  429, -297, 165, -75, 27, -7, 1;
Production matrix is
  0,  1,
  0, -1,  1,
  0,  1, -1,  1,
  0, -1,  1, -1,  1,
  0,  1, -1,  1, -1,  1,
  0, -1,  1, -1,  1, -1,  1,
  0,  1, -1,  1, -1,  1, -1,  1,
  0, -1,  1, -1,  1, -1,  1, -1,  1,
  0,  1, -1,  1, -1,  1, -1,  1, -1,  1

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened
George Beck and Karl Dilcher, A Matrix Related to Stern Polynomials and the Prouhet-Thue-Morse Sequence, arXiv:2106.10400 [math.CO], 2021. See (2.10) p. 6.
F. R. Bernhart, Catalan, Motzkin and Riordan numbers, Discr. Math., 204 (1999), 73-112.
D. Callan, A recursive bijective approach to counting permutations containing 3-letter patterns, arXiv:math/0211380 [math.CO], 2002.
E. Deutsch, Dyck path enumeration, Discrete Math., 204, 1999, 167-202.
R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6.
A. Robertson, D. Saracino and D. Zeilberger, Refined restricted permutations, arXiv:math/0203033 [math.CO], 2002.
L. W. Shapiro, S. Getu, Wen-Jin Woan and L. C. Woodson, The Riordan Group, Discrete Appl. Maths. 34 (1991) 229-239.

The three triangles A059365, A106566 and A099039 are the same except for signs and the leading term.
Cf. A033184, A000108.
Cf. A106566 (unsigned version), A059365
The following are all versions of (essentially) the same Catalan triangle: A009766, A030237, A033184, A059365, A099039, A106566, A130020, A047072.
Diagonals give A000108 A000245 A002057 A000344 A003517 A000588 A003518 A003519 A001392, ...

T[n_, k_]:= If[n == 0 && k == 0, 1, If[n == 0 && k > 0, 0, (-1)^(n + k)*Binomial[2*n - k - 1, n - k]*k/n]];  Table[T[n, k], {n, 0, 15}, {k, 0, n}] // Flatten (* G. C. Greubel, Dec 31 2017 *)

{T(n,k) = if(n == 0 && k == 0, 1, if(n == 0 && k > 0, 0, (-1)^(n + k)*binomial(2*n - k - 1, n - k)*k/n))};
for(n=0,15, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Dec 31 2017

T(n, k) = (-1)^(n+k)*binomial(2*n-k-1, n-k)*k/n for 0 <= k <= n with n > 0; T(0, 0) = 1; T(0, k) = 0 if k > 0. - Philippe Deléham, May 31 2005

A007001 Trajectory of 1 under the morphism 1 -> 12, 2 -> 123, 3 -> 1234, etc.

Original entry on oeis.org

1, 2, 1, 2, 3, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 1, 2, 3, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, 1, 2, 1, 2, 3, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 1, 2, 3, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, 1, 2, 1, 2, 3, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 1, 2, 3, 1, 2

Offset: 1

N. J. A. Sloane, Mira Bernstein, James Propp

Records in this sequence occur at positions: 1, 2, 5, 14, 42, 132, 429, 1430, ... (which appear to be the Catalan numbers A000108). - Robert G. Wilson v, May 07 2005
The records do occur at Catalan numbers. Of the first C(n) numbers, the number that are equal to k is A033184(n,k), with the one n last. - Franklin T. Adams-Watters, Mar 29 2009
Let (T(1) < T(2) < ... < T(A000108(m))) denote the sequence of Young tableaux of shape (2^m) ordered lexicographically with respect to their columns, and let f(T(i), T(j)) denote the first label of disagreement among T(i) and T(j). Then, empirically, if we take away the zeros from (f(T(1), T(A000108(m) - i + 1)) - f(T(A000108(m) - i), T(A000108(m) - i + 1)), i=1..A000108(m)-1), we obtain the first A000108(m - 1) - 1 terms in this sequence. This is illustrated in the below example. - John M. Campbell, Sep 07 2018
The average of the first k terms tends to 3 as k tends to infinity. - Andrew Slattery, Jan 19 2021

			From _John M. Campbell_, Sep 07 2018: (Start)
Letting m = 5, as above let (T(1) < T(2) < ... < T(42)) denote the lexicographic sequence of Young tableaux of shape (2, 2, 2, 2, 2). In this case, the sequence (f(T(1), T(43 - i)) - f(T(42 - i), T(43 - i)), i=1..41) is equal to (0, 1, 0, 0, 2, 0, 1, 0, 0, 2, 0, 0, 0, 3, 0, 1, 0, 0, 2, 0, 1, 0, 0, 2, 0, 0, 0, 3, 0, 1, 0, 0, 2, 0, 0, 0, 3, 0, 0, 0, 0). Removing the zeroes from this tuple, we obtain (1, 2, 1, 2, 3, 1, 2, 1, 2, 3, 1, 2, 3), which gives us the first 13 = A000108(m - 1) - 1 terms in this sequence. For example, the first term in the preceding tuple is 0 since T(1) and T(42) are respectively
   [ 5 10] [ 9 10]
   [ 4 9 ] [ 7 8 ]
   [ 3 8 ] [ 5 6 ]
   [ 2 7 ] [ 3 4 ]
   [ 1 6 ] [ 1 2 ]
and T(41) is equal to
   [ 9 10]
   [ 7 8 ]
   [ 5 6 ]
   [ 2 4 ]
   [ 1 3 ]
so that the first letter of disagreement between T(1) and T(42) is 2, and that between T(41) and T(42) is also 2. (End)

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
J. West, Generating trees and forbidden subsequences, Proc. 6th FPSAC [ Conference on Formal Power Series and Algebraic Combinatorics ] (1994), pp. 441-450 (see p. 443).

T. D. Noe, Table of n, a(n) for n = 1..4862 (8 iterations)
C. Banderier, A. Denise, P. Flajolet, M. Bousquet-Mélou et al., Generating Functions for Generating Trees, Discrete Mathematics 246(1-3), March 2002, pp. 29-55.
Antti Karttunen, Notes concerning A080237-tree and related sequences.
S. Lehr, J. Shallit and J. Tromp, On the vector space of the automatic reals, Theoret. Comput. Sci. 163 (1996), no. 1-2, 193-210.
James Propp and N. J. A. Sloane, Email, March 1994
Index entries for sequences that are fixed points of mappings

Cf. A000245, A085182. a(n)=A076050(n)-1. Partial sums: A080336. Positions of ones: A085197. The first occurrence of each n is at A000108(n). See A085180.

Nest[ Flatten[ # /. a_Integer -> Range[a + 1]] &, {1}, 6] (* Robert G. Wilson v, Jan 24 2006 *)

a(n)=local(v,w); if(n<1,0,v=[1]; while(#v

From n > 1 onward a(n) = A080237(A081291(n-1)). - Antti Karttunen, Jul 31 2003

More terms from Larry Reeves (larryr(AT)acm.org), Sep 22 2000

A325188 Regular triangle read by rows where T(n,k) is the number of integer partitions of n with origin-to-boundary graph-distance equal to k.

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 0, 2, 1, 0, 0, 2, 3, 0, 0, 0, 2, 5, 0, 0, 0, 0, 2, 8, 1, 0, 0, 0, 0, 2, 9, 4, 0, 0, 0, 0, 0, 2, 12, 8, 0, 0, 0, 0, 0, 0, 2, 13, 15, 0, 0, 0, 0, 0, 0, 0, 2, 16, 23, 1, 0, 0, 0, 0, 0, 0, 0, 2, 17, 32, 5, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Gus Wiseman, Apr 11 2019

The origin-to-boundary graph-distance of a Young diagram is the minimum number of unit steps right or down from the upper-left square to a nonsquare in the lower-right quadrant. It is also the side-length of the maximum triangular partition contained inside the diagram.

			Triangle begins:
  1
  0  1
  0  2  0
  0  2  1  0
  0  2  3  0  0
  0  2  5  0  0  0
  0  2  8  1  0  0  0
  0  2  9  4  0  0  0  0
  0  2 12  8  0  0  0  0  0
  0  2 13 15  0  0  0  0  0  0
  0  2 16 23  1  0  0  0  0  0  0
  0  2 17 32  5  0  0  0  0  0  0  0
  0  2 20 43 12  0  0  0  0  0  0  0  0
  0  2 21 54 24  0  0  0  0  0  0  0  0  0
  0  2 24 67 42  0  0  0  0  0  0  0  0  0  0
  0  2 25 82 66  1  0  0  0  0  0  0  0  0  0  0

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
N. Guru Sharan, Rook decomposition of the Partition function, arXiv:2507.20260 [math.CO], 2025. See p. 4.
N. Guru Sharan and Armin Straub, Partitions with Durfee triangles of fixed size, arXiv:2507.19047 [math.CO], 2025. See p. 10.
Eric Weisstein's World of Mathematics, Graph Distance.

Row sums are A000041.
Columns k=1-4 give: A130130, A325168, A382682, A384562.
Cf. A000245, A065770, A096771, A115994, A325169, A325183, A325187, A325189, A325191, A325195, A325200, A368986.

otb[ptn_]:=Min@@MapIndexed[#1+#2[[1]]-1&,Append[ptn,0]];
Table[Length[Select[IntegerPartitions[n],otb[#]==k&]],{n,0,15},{k,0,n}]

row(n)={my(r=vector(n+1)); forpart(p=n, my(w=#p); for(i=1, #p, w=min(w,#p-i+p[i])); r[w+1]++); r} \\ Andrew Howroyd, Jan 12 2024

Sum_{k=1..n} k*T(n,k) = A368986(n).

A234467 a(n) = 9binomial(8n + 9,n)/(8*n + 9).

Original entry on oeis.org

1, 9, 108, 1488, 22230, 350244, 5729724, 96395616, 1657248417, 28987537150, 514215324216, 9229030737264, 167283594343320, 3057857090083908, 56305821384711720, 1043424549990820800, 19445145508444588200, 364191559218548917713, 6851518654436447733980

Offset: 0

Tim Fulford, Dec 26 2013

Fuss-Catalan sequence is a(n,p,r) = r*binomial(n*p + r,n)/(n*p + r); this is the case p = 8, r = 9.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
J-C. Aval, Multivariate Fuss-Catalan Numbers, arXiv:0711.0906 [math.CO], 2007.
J-C. Aval, Multivariate Fuss-Catalan Numbers, Discrete Math., 308 (2008), 4660-4669.
Thomas A. Dowling, Catalan Numbers Chapter 7
Wojciech Mlotkowski, Fuss-Catalan Numbers in Noncommutative Probability, Docum. Mathm. 15: 939-955.
Wikipedia, Fuss-Catalan number

Cf. A007556, A234461, A234462, A234463, A234464, A234465, A234466.

Cf. A000108, A000245 (k = 3), A006629 (k = 4), A196678 (k = 5), A233668 (k = 6), A233743 (k = 7), A233835 (k = 8), A232265 (k = 10), A229963 (k = 11).

[9*Binomial(8*n+9, n)/(8*n+9): n in [0..30]]; // Vincenzo Librandi, Dec 26 2013

Table[9 Binomial[8 n + 9, n]/(8 n + 9), {n, 0, 40}] (* Vincenzo Librandi, Dec 26 2013 *)

PARI
```
a(n) = 9*binomial(8*n+9,n)/(8*n+9);
```

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(8/9))^9+x*O(x^n)); polcoeff(B, n)}

G.f. satisfies: A(x) = {1 + x*A(x)^(p/r)}^r, where p = 8, r = 9.
From Peter Bala, Oct 16 2015: (Start)
O.g.f.: (1/x) * series reversion (x*C(-x)^9), where C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f. for the Catalan numbers A000108. See cross-references for other Fuss-Catalan sequences with o.g.f. 1/x * series reversion (x*C(-x)^k), k = 3 through 11.
A(x)^(1/9) is the o.g.f. for A007556. (End)
D-finite with recurrence +7*n*(7*n+3)*(7*n+4)*(7*n+5)*(7*n+6)*(7*n+8)*(7*n+9)*a(n)-128*(2*n+1)*(4*n+1)*(4*n+3)*(8*n+1)*(8*n+3)*(8*n+5)*(8*n+7)*a(n-1) = 0. - R. J. Mathar, Feb 09 2020
E.g.f.: F([9/8, 5/4, 11/8, 3/2, 13/8, 7/4, 15/8], [1, 10/7, 11/7, 12/7, 13/7, 15/7, 16/7], 16777216*x/823543), where F is the generalized hypergeometric function. - Stefano Spezia, Feb 09 2020

A196678 a(n) = 5binomial(4n+5,n)/(4*n+5).

Original entry on oeis.org

1, 5, 30, 200, 1425, 10626, 81900, 647280, 5217300, 42724825, 354465254, 2973052680, 25168220350, 214762810500, 1845308367000, 15951899986272, 138638564739180, 1210677947695620, 10617706139119000, 93477423115076000

Offset: 0

Karol A. Penson, Oct 05 2011

This is a sequence  of power moments of the following signed function defined on the segment (0,256/27), in Maple notation:
-(1/2)*sqrt(2)*x^(1/4)*hypergeom([-5/12, -1/12, 5/4], [1/2, 3/4], (27/256)*x)/Pi+(5/4)*sqrt(x)*hypergeom([-1/6, 1/6, 3/2], [3/4, 5/4], (27/256)*x)/Pi-(15/64)*sqrt(2)*x^(3/4)*hypergeom([1/12, 5/12, 7/4], [5/4, 3/2], (27/256)*x)/Pi. This function is not positive on (0,256/27).
The two parameter Fuss-Catalan sequence is A(n,p,r) := r*binomial(n*p + r, n)/(n*p + r). This sequence is A(n,4,5). - Peter Bala, Oct 16 2015

C. H. Pah, M. R. Wahiddin, Combinatorial Interpretation of Raney Numbers and Tree Enumerations, Open Journal of Discrete Mathematics, 2015, 5, 1-9; http://www.scirp.org/journal/ojdm; http://dx.doi.org/10.4236/ojdm.2015.51001

Vincenzo Librandi, Table of n, a(n) for n = 0..110
C. B. Pah and M. Saburov, Single Polygon Counting on Cayley Tree of Order 4: Generalized Catalan Numbers, Middle-East Journal of Scientific Research 13 (Mathematical Applications in Engineering): 01-05, 2013, ISSN 1990-9233.
Karol A. Penson and Karol Zyczkowski, Product of Ginibre matrices : Fuss-Catalan and Raney distribution, arXiv version
Wikipedia, Fuss-Catalan number
Karol Zyczkowski, Karol A. Penson, Ion Nechita and Benoit Collins, Generating random density matrices, J. Math Phys. 52, 062201 (2011). arXiv version.

Cf. A000108, A002293, A000245 (k = 3), A006629 (k = 4), A233668 (k = 6), A233743 (k = 7), A233835 (k = 8), A234467 (k = 9), A232265 (k = 10), A229963 (k = 11).

[5*Binomial(4*n+5,n)/(4*n+5): n in [0..30]]; // Vincenzo Librandi, Oct 07 2011

O.g.f.: hypergeom([5/4, 3/2, 7/4], [7/3, 8/3], (256 z)/27)
E.g.f.: hypergeom([5/4, 3/2, 7/4], [1, 7/3, 8/3], (256 z)/27)
From _Peter Bala, Oct 16 2015: (Start)
O.g.f. A(x) = 1/x * series reversion (x*C(-x)^5), where C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f. for the Catalan numbers A000108. See cross-references for other Fuss-Catalan sequences with o.g.f. 1/x * series reversion (x*C(-x)^k), k = 3 through 11.
A(x)^(1/5) is the o.g.f. for A002293. (End)
D-finite with recurrence 3*n*(3*n+5)*(3*n+4)*a(n) -8*(4*n+1)*(2*n+1)*(4*n+3)*a(n-1)=0. - R. J. Mathar, Aug 01 2022

Offset changed from 1 to 0 and extended by Vincenzo Librandi, Oct 07 2011

A208355 Right edge of the triangle in A208101.

Original entry on oeis.org

1, 1, 1, 2, 2, 5, 5, 14, 14, 42, 42, 132, 132, 429, 429, 1430, 1430, 4862, 4862, 16796, 16796, 58786, 58786, 208012, 208012, 742900, 742900, 2674440, 2674440, 9694845, 9694845, 35357670, 35357670, 129644790, 129644790, 477638700, 477638700, 1767263190

Offset: 0

Reinhard Zumkeller, Mar 04 2012

nonn

Number of achiral polyominoes composed of n+1 triangular cells of the hyperbolic regular tiling with Schläfli symbol {3,oo}. A stereographic projection of this tiling on the Poincaré disk can be obtained via the Christensson link. An achiral polyomino is identical to its reflection. - Robert A. Russell, Jan 20 2024

			a(0)=1; a(1)=1; a(2)=1; a(3)=2. - _Robert A. Russell_, Jan 19 2024
____      ________
\  /  /\  \  /\  /  /\     /\
 \/  /__\  \/__\/  /__\   /__\____
     \  /         /\  /\  \  /\  /
      \/         /__\/__\  \/__\/

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Andrei Asinowski, Cyril Banderier, and Valerie Roitner, Generating functions for lattice paths with several forbidden patterns, (2019).
Malin Christensson, Make hyperbolic tilings of images, web page, 2019.
D. Levin, L. Pudwell, M. Riehl, and A. Sandberg, Pattern Avoidance on k-ary Heaps, Slides of Talk, 2014.
Zhicong Lin, David G. L. Wang, and Tongyuan Zhao, A decomposition of ballot permutations, pattern avoidance and Gessel walks, arXiv:2103.04599 [math.CO], 2021.

Cf. A099363, A208101.

Polyominoes: A001683(n+2) (oriented), A000207 (unoriented). A369314 (chiral), A000108 (rooted), A047749 ({4,oo}).

a208355 n = a208101 n n
a208355_list = map last a208101_tabl

[Ceiling(Catalan(n div 2)): n in [1..40]]; // Vincenzo Librandi, Feb 18 2014

A208355_list := proc(len) local D, b, h, R, i, k;
    D := [seq(0, j=0..len+2)]; D[1] := 1; b := true; h := 2; R := NULL;
    for i from 1 to 2*len do
        if b then
            for k from h by -1 to 2 do D[k] := D[k] - D[k-1] od;
            h := h + 1; R := R, abs(D[2]);
        else
            for k from 1 by 1 to h do D[k] := D[k] + D[k+1] od;
        fi;
        b := not b:
    od;
    return R
end:
A208355_list(38); # Peter Luschny, Dec 19 2017

T[, 0] = 1; T[n, 1] := n; T[n_, n_] := T[n - 1, n - 2]; T[n_, k_] /; 1 < k < n := T[n, k] = T[n - 1, k] + T[n - 1, k - 2];
a[n_] := T[n, n];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 03 2018, from A208101 *)
Table[If[EvenQ[n], Binomial[n,n/2]/(n/2+1), Binomial[n+1,(n+1)/2]/((n+3)/2)], {n,0,40}] (* Robert A. Russell, Jan 19 2024 *)

a(n) = A000108(floor((n+1)/2)), where A000108 = Catalan numbers.
a(n) = A208101(n,n).
a(n) = abs(A099363(n)).
Conjecture: -(n+3)*(n-2)*a(n) - 4*a(n-1) + 4*(n-1)^2*a(n-2) = 0. - R. J. Mathar, Aug 04 2015
From Robert A. Russell, Jan 19 2024: (Start)
a(2m) = C(2m,m)/(m+1); a(2m-1) = a(2m); a(n+2)/a(n) ~ 4.
a(n-1) = 2*A000207(n) - A001683(n+2) = A001683(n+2) - 2*A369314(n) = A000207(n) - A369314(n). (End)
G.f.: (G(z^2)+z*G(z^2)-1)/z, where G(z)=1+z*G(z)^2, the generating function for A000108. - Robert A. Russell, Jan 26 2024
G.f.: ((((1+z)*(1-sqrt(1-4*z^2)))/(2*z^2))-1)/z. - Robert A. Russell, Jan 28 2024
From Peter Bala, Feb 05 2024: (Start)
G.f.: 1/(1 + 2*x) * c(x/(1 + 2*x))^3, where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108.
a(n) = Sum_{k = 0..n} (-2)^(n-k)*binomial(n, k)*A000245(k+1).
a(n) = (-2)^n * hypergeom([-n, 3/2, 2], [1, 4], 2). (End)

cp's OEIS Frontend

A071724 a(n) = 3*binomial(2n, n-1)/(n+2), n > 0, with a(0)=1.

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A047072 Array A read by diagonals: A(h,k)=number of paths consisting of steps from (0,0) to (h,k) such that each step has length 1 directed up or right and no step touches the line y=x unless x=0 or x=h.

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A026009 Triangular array T read by rows: T(n,0) = 1 for n >= 0; T(1,1) = 1; and for n >= 2, T(n,k) = T(n-1,k-1) + T(n-1,k) for k = 1,2,...,[(n+1)/2]; T(n,n/2 + 1) = T(n-1,n/2) if n is even.

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A130020 Triangle T(n,k), 0<=k<=n, read by rows given by [1,0,0,0,0,0,0,...] DELTA [0,1,1,1,1,1,1,...] where DELTA is the operator defined in A084938 .

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A099039 Riordan array (1,c(-x)), where c(x) = g.f. of Catalan numbers.

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A007001 Trajectory of 1 under the morphism 1 -> 12, 2 -> 123, 3 -> 1234, etc.

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A325188 Regular triangle read by rows where T(n,k) is the number of integer partitions of n with origin-to-boundary graph-distance equal to k.

A234467 a(n) = 9binomial(8n + 9,n)/(8*n + 9).

A196678 a(n) = 5binomial(4n+5,n)/(4*n+5).