A051924 -id:A051924 - cp's OEIS frontend

A051597 Rows of triangle formed using Pascal's rule except begin and end n-th row with n+1.

1, 2, 2, 3, 4, 3, 4, 7, 7, 4, 5, 11, 14, 11, 5, 6, 16, 25, 25, 16, 6, 7, 22, 41, 50, 41, 22, 7, 8, 29, 63, 91, 91, 63, 29, 8, 9, 37, 92, 154, 182, 154, 92, 37, 9, 10, 46, 129, 246, 336, 336, 246, 129, 46, 10, 11, 56, 175, 375, 582, 672, 582, 375, 175, 56, 11

Offset: 0

Asher Auel

Row sums give A033484(n).
The number of spotlight tilings of an (m+1) X (n+1) rectangle, read by antidiagonals. - Bridget Tenner, Nov 09 2007
T(n,k) = A134636(n,k) - A051601(n,k). - Reinhard Zumkeller, Nov 23 2012
T(n,k) = A209561(n+2,k+1), 0 <= k <= n. - Reinhard Zumkeller, Dec 26 2012
For a closed-form formula for arbitrary left and right borders of Pascal like triangle see A228196. - Boris Putievskiy, Aug 19 2013
For a closed-form formula for generalized Pascal's triangle see A228576. - Boris Putievskiy, Sep 09 2013

			Triangle begins as:
  1;
  2,  2;
  3,  4,  3;
  4,  7,  7,  4;
  5, 11, 14, 11, 5;

Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened
B. E. Tenner, Spotlight tiling, Ann. Combin. 14 (4) (2010) 553.
Index entries for triangles and arrays related to Pascal's triangle

Stripped variant of A072405, A122218.

Cf. A007318, A228196, A228576.

T:= function(n,k)
    if k<0 or k>n then return 0;
    elif k=0 or k=n then return n+1;
    else return T(n-1,k-1) + T(n-1,k);
    fi;
  end;
Flat(List([0..12], n-> List([0..n], k-> T(n,k) ))); # G. C. Greubel, Nov 18 2019

a051597 n k = a051597_tabl !! n !! k
a051597_row n = a051597_tabl !! n
a051597_tabl = iterate (\row -> zipWith (+) ([1] ++ row) (row ++ [1])) [1]
-- Reinhard Zumkeller, Nov 23 2012

function T(n,k)
    if k lt 0 or k gt n then return 0;
  elif k eq 0 or k eq n then return n+1;
  else return T(n-1,k-1) + T(n-1,k);
  end if;
  return T;
end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 18 2019

T:= proc(n, k) option remember;
      `if`(k<0 or k>n, 0,
      `if`(k=0 or k=n, n+1,
         T(n-1, k-1) + T(n-1, k) ))
    end:
seq(seq(T(n, k), k=0..n), n=0..14);  # Alois P. Heinz, May 27 2013

NestList[Append[ Prepend[Map[Apply[Plus, #] &, Partition[#, 2, 1]], #[[1]] + 1], #[[1]] + 1] &, {1}, 10] // Grid  (* Geoffrey Critzer, May 26 2013 *)
T[n_, k_] := T[n, k] = If[k<0 || k>n, 0, If[k==0 || k==n, n+1, T[n-1, k-1] + T[n-1, k]]]; Table[T[n, k], {n, 0, 14}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 09 2016, after Alois P. Heinz *)

T(n,k) = if(k<0 || k>n, 0, if(k==0 || k==n, n+1, T(n-1, k-1) + T(n-1, k) ));
for(n=0, 12, for(k=0, n, print1(T(n,k), ", "))) \\ G. C. Greubel, Nov 18 2019

@CachedFunction
def T(n, k):
    if (k<0 or k>n): return 0
    elif (k==0 or k==n): return n+1
    else: return T(n-1, k-1) + T(n-1, k)
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Nov 18 2019

T(2n,n) = A051924(n+1). - Philippe Deléham, Nov 26 2006
T(m,n) = binomial(m+n,m) - binomial(m+n-2,m-1) (correct up to offset and transformation of square indices to triangular indices). - Bridget Tenner, Nov 09 2007
T(0,n) = T(n,0) = n+1, T(n,k) = T(n-1,k) + T(n-1,k-1), 0 < k < n.
From Peter Bala, Feb 28 2013: (Start)
T(n,k) = binomial(n,k-1) + binomial(n,k) + binomial(n,k+1) for 0 <= k <= n.
O.g.f.: (1 - xt^2)/((1 - t)(1 - xt)(1 - (1+x)t)) = 1 + (2 + 2x)t + (3 + 4x + 3x^2)t^2 + ....
Row polynomials: ((1+x+x^2)*(1+x)^n - 1 - x^(n+2))/x. (End)

A325178 Difference between the length of the minimal square containing and the maximal square contained in the Young diagram of the integer partition with Heinz number n.

Original entry on oeis.org

0, 0, 1, 1, 2, 1, 3, 2, 0, 2, 4, 2, 5, 3, 1, 3, 6, 1, 7, 2, 2, 4, 8, 3, 1, 5, 1, 3, 9, 1, 10, 4, 3, 6, 2, 2, 11, 7, 4, 3, 12, 2, 13, 4, 1, 8, 14, 4, 2, 1, 5, 5, 15, 2, 3, 3, 6, 9, 16, 2, 17, 10, 2, 5, 4, 3, 18, 6, 7, 2, 19, 3, 20, 11, 1, 7, 3, 4, 21, 4, 2, 12

Offset: 1

Gus Wiseman, Apr 08 2019

nonn

The maximal square contained in the Young diagram of an integer partition is called its Durfee square, and its length is the rank of the partition.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The partition (3,3,2,1) has Heinz number 150 and diagram
  o o o
  o o o
  o o
  o
containing maximal square
  o o
  o o
and contained in minimal square
  o o o o
  o o o o
  o o o o
  o o o o
so a(150) = 4 - 2 = 2.

Richard P. Stanley, Enumerative Combinatorics, Volume 2, Cambridge University Press, 1999, p. 289.

Wikipedia, Durfee square.

Positions of zeros are A062457. Positions of 1's are A325179. Positions of 2's are A325180.

Cf. A001222, A046660, A051924, A056239, A061395, A093641, A096771, A115994, A243055, A257990, A263297, A325192, A325195.

durf[n_]:=Length[Select[Range[PrimeOmega[n]],Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]][[#]]>=#&]];
codurf[n_]:=If[n==1,0,Max[PrimeOmega[n],PrimePi[FactorInteger[n][[-1,1]]]]];
Table[codurf[n]-durf[n],{n,100}]

a(n) = A263297(n) - A257990(n).

A051960 a(n) = C(n)*(3n+2) where C(n) = Catalan numbers = A000108.

Original entry on oeis.org

2, 5, 16, 55, 196, 714, 2640, 9867, 37180, 140998, 537472, 2057510, 7904456, 30458900, 117675360, 455657715, 1767883500, 6871173870, 26747767200, 104268528210, 406975466040, 1590307356300, 6220814327520, 24357232569150, 95452906901976, 374369872911804

Offset: 0

Barry E. Williams, Jan 05 2000

If Y is a fixed 2-subset of a 2n-set X then a(n-1) is the number of n-subsets of X intersecting Y. - Milan Janjic, Oct 21 2007

a(n-1) is the number of vertices in the n-dimensional halohedron (or equivalently, n-cycle cubeahedron). - Vincent Pilaud, May 12 2020

Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

Robert Israel, Table of n, a(n) for n = 0..1600
Moa Apagodu and Doron Zeilberger, Using the "Freshman's Dream" to Prove Combinatorial Congruences, arXiv:1606.03351 [math.CO], 2016. Also Amer. Math. Monthly. 124 (2017), 597-608.
Satyan L. Devadoss, Timothy Heath, and Cid Vipismakul, Deformations of bordered Riemann surfaces and associahedral polytopes, arXiv:1002.1676 [math.AG], 2010.
S. L. Devadoss, T. Heath, and W. Vipismakul, Deformations of bordered surfaces and convex polytopes, Notices Amer. Math. Soc. 58 (2011), no. 4, 530-541.
S. B. Ekhad and M. Yang, Proofs of Linear Recurrences of Coefficients of Certain Algebraic Formal Power Series Conjectured in the On-Line Encyclopedia Of Integer Sequences, (2017).
Milan Janjic, Two Enumerative Functions.
Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019.

Cf. A000108 and A051924.
Cf. A024482 and A097613.
Half A028283.

[Catalan(n)*(3*n+2): n in [0..30]]; // Vincenzo Librandi, Oct 01 2015

a := n -> 4^n*(2+3*n)*GAMMA(1/2+n)/(sqrt(Pi)*GAMMA(2+n)):
seq(a(n), n=0..25); # Peter Luschny, Dec 14 2015

Table[CatalanNumber[n] (3n+2), {n,0,30}] (* Michael De Vlieger, Sep 30 2015 *)

a(n):=sum(binomial(n-k+1,k)*2^(n-2*k+1)*binomial(n,k),k,0,(n+1)/2); /* Vladimir Kruchinin, Sep 30 2015 */

a(n) = (3*n+2)*binomial(2*n, n)/(n+1);
vector(30, n, a(n-1)) \\ Altug Alkan, Sep 30 2015

(n+1)*a(n) - 2*(n+2)*a(n-1) - 4*(2*n-3)*a(n-2) = 0. - conjectured by R. J. Mathar, Oct 02 2014, verified by Robert Israel, Sep 30 2015
G.f.: (1 + 2*x)/(2*x*sqrt(1-4*x)) - 1/(2*x). - Vladimir Kruchinin, Sep 30 2015.
a(n) = Sum_{k=0..(n+1)/2} (binomial(n-k+1,k)*2^(n-2*k+1)*binomial(n,k)). - Vladimir Kruchinin, Sep 30 2015.
a(n) = 4^n*(2+3*n)*Gamma(n + 1/2)/(sqrt(Pi)*Gamma(n+2)). - Peter Luschny, Dec 14 2015
a(n - 1) = A051924(n) + A000108(n - 1). - F. Chapoton, Mar 05 2022
Sum_{n>=0} a(n)/8^n = 5*sqrt(2) - 4. - Amiram Eldar, May 06 2023
E.g.f.: exp(2*x)*(2*BesselI(0,2*x) + BesselI(1,2*x)). - Stefano Spezia, May 14 2025
a(n) = 2*binomial(2*n, n) + binomial(2*n, n-1) = 2*A000984(n) + A001791(n). - Peter Bala, Aug 23 2025

A330965 Array read by descending antidiagonals: A(n,k) = (1 + kn)C(n) where C(n) = Catalan numbers (A000108).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 6, 5, 1, 4, 10, 20, 14, 1, 5, 14, 35, 70, 42, 1, 6, 18, 50, 126, 252, 132, 1, 7, 22, 65, 182, 462, 924, 429, 1, 8, 26, 80, 238, 672, 1716, 3432, 1430, 1, 9, 30, 95, 294, 882, 2508, 6435, 12870, 4862, 1, 10, 34, 110, 350, 1092, 3300, 9438, 24310, 48620, 16796

Offset: 0

Andrew Howroyd, Jan 04 2020

			Array begins:
====================================================
n\k |   0    1    2    3     4     5     6     7
----+-----------------------------------------------
  0 |   1    1    1    1     1     1     1     1 ...
  1 |   1    2    3    4     5     6     7     8 ...
  2 |   2    6   10   14    18    22    26    30 ...
  3 |   5   20   35   50    65    80    95   110 ...
  4 |  14   70  126  182   238   294   350   406 ...
  5 |  42  252  462  672   882  1092  1302  1512 ...
  6 | 132  924 1716 2508  3300  4092  4884  5676 ...
  7 | 429 3432 6435 9438 12441 15444 18447 21450 ...
  ...

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

Andrew Howroyd, Table of n, a(n) for n = 0..1325

Columns k=0..12 are A000108, A000984, A001700, A051924, A051944, A051945, A050476, A050477, A050478, A050479, A050489, A050490, A050491.

A330965[n_, k_] := CatalanNumber[n]*(k*n + 1);
Table[A330965[k, n - k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Aug 24 2025 *)

T(n, k)={(1 + k*n)*binomial(2*n,n)/(n+1)}

A(n,k) = (1 + k*n)*binomial(2*n,n)/(n+1).
A(n,k) = 2*(k*n+1)*(2*n-1)*A(n-1,k)/((n+1)*(k*n-k+1)) for n > 0.
G.f. of column k: (k - 1 - (2*k-4)*x - (k-1)*sqrt(1 - 4*x))/(2*x*sqrt(1 - 4*x)).

A024482 a(n) = (1/2)*(binomial(2n, n) - binomial(2n-2, n-1)).

Original entry on oeis.org

2, 7, 25, 91, 336, 1254, 4719, 17875, 68068, 260338, 999362, 3848222, 14858000, 57500460, 222981435, 866262915, 3370764540, 13135064250, 51250632510, 200205672810, 782920544640, 3064665881940, 12007086477750, 47081501377326, 184753963255176, 725510446350004

Offset: 2

Clark Kimberling

nonn

Apparently the number of sawtooth patterns in all Dyck paths of semilength n, ([0,1],2,7,25,...). A sawtooth pattern is of the form (UD)^k, k >= 1. More generally, the number of sawtooth patterns of length > t in all Dyck paths with semilength (n+t), t >= 0. - David Scambler, Apr 23 2013

			The path udUududD has two sawtooth patterns, shown in lower case.

Stefano Spezia, Table of n, a(n) for n = 2..1600
Toufik Mansour and I. L. Ramirez, Enumerations of polyominoes determined by Fuss-Catalan words, Australas. J. Combin. 81 (3) (2021) 447-457, table 2.

Cf. A000108, A097613, A225015.

[(3*n-2)*Catalan(n-1)/2: n in [2..40]]; // G. C. Greubel, Apr 03 2024

Z:=(1-z-sqrt(1-4*z))/sqrt(1-4*z)/2: Zser:=series(Z, z=0, 32): seq(coeff(Zser, z, n), n=2..25); # Zerinvary Lajos, Jan 16 2007

Table[(Binomial[2n,n]-Binomial[2n-2,n-1])/2,{n,2,30}] (* Harvey P. Dale, Mar 04 2011 *)

[(3*n-2)*catalan_number(n-1)/2 for n in range(2,41)] # G. C. Greubel, Apr 03 2024

a(n) = A051924(n)/2. - Zerinvary Lajos, Jan 16 2007
From R. J. Mathar, Nov 09 2018: (Start)
D-finite with recurrence n*a(n) - (5*n-4)*a(n-1) + 2*(2*n-5)*a(n-2) = 0.
n*(3*n-5)*a(n) - 2*(3*n-2)*(2*n-3)*a(n-1) = 0. (End)
a(n) ~ 3*2^(2*n-3)/sqrt(n*Pi). - Stefano Spezia, May 09 2023
From G. C. Greubel, Apr 03 2024: (Start)
a(n) = (3*n-2)*A000108(n-1)/2.
G.f.: ((1-x)*sqrt(1-4*x) - (1+x)*(1-4*x))/(2*(1-4*x)).
E.g.f.: (1/2)*( -1 - x + exp(2*x)*( (1-x)*BesselI(0, 2*x) + x*BesselI(1, 2*x) ) ). (End)

A325227 Regular triangle read by rows where T(n,k) is the number of integer partitions of n such that the lesser of the maximum part and the number of parts is k.

Original entry on oeis.org

0, 0, 1, 0, 2, 0, 0, 2, 1, 0, 0, 2, 3, 0, 0, 0, 2, 4, 1, 0, 0, 0, 2, 6, 3, 0, 0, 0, 0, 2, 6, 6, 1, 0, 0, 0, 0, 2, 8, 9, 3, 0, 0, 0, 0, 0, 2, 8, 13, 6, 1, 0, 0, 0, 0, 0, 2, 10, 16, 11, 3, 0, 0, 0, 0, 0, 0, 2, 10, 20, 17, 6, 1, 0, 0, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Apr 12 2019

			Triangle begins:
  1
  2  0
  2  1  0
  2  3  0  0
  2  4  1  0  0
  2  6  3  0  0  0
  2  6  6  1  0  0  0
  2  8  9  3  0  0  0  0
  2  8 13  6  1  0  0  0  0
  2 10 16 11  3  0  0  0  0  0
  2 10 20 17  6  1  0  0  0  0  0
  2 12 24 25 11  3  0  0  0  0  0  0
  2 12 28 33 19  6  1  0  0  0  0  0  0
  2 14 32 44 29 11  3  0  0  0  0  0  0  0
  2 14 38 53 43 19  6  1  0  0  0  0  0  0  0
Row n = 9 counts the following partitions:
  (9)          (54)        (333)      (4221)    (51111)
  (111111111)  (63)        (432)      (4311)
               (72)        (441)      (5211)
               (81)        (522)      (6111)
               (22221)     (531)      (42111)
               (222111)    (621)      (411111)
               (2211111)   (711)
               (21111111)  (3222)
                           (3321)
                           (32211)
                           (33111)
                           (321111)
                           (3111111)

FindStat, St000533: The maximal number of non-attacking rooks on a Ferrers shape

Column k = 2 is A265283. Column k = 3 is A325228.

Cf. A051924, A096771, A115720, A263297, A325188, A325189, A325192, A325193, A325194, A325224, A325225, A325229, A325232.

Table[Length[Select[IntegerPartitions[n],Min[Length[#],Max[#]]==k&]],{n,15},{k,n}]

A325232 Number of integer partitions (of any nonnegative integer) whose sum minus the lesser of their maximum part and their number of parts is n.

Original entry on oeis.org

2, 3, 6, 10, 18, 27, 44, 64, 97, 138, 200, 276, 390, 528, 724, 968, 1301, 1712, 2266, 2946, 3842, 4947, 6372, 8122, 10362, 13094, 16544, 20754, 26010, 32392, 40308, 49876, 61648, 75845, 93178, 114006, 139308, 169586, 206158, 249814, 302267, 364664, 439330

Offset: 0

Gus Wiseman, Apr 13 2019

nonn

			The a(0) = 1 through a(4) = 18 partitions:
  ()   (2)   (3)    (4)     (5)
  (1)  (11)  (22)   (32)    (33)
       (21)  (31)   (41)    (42)
             (111)  (221)   (51)
             (211)  (321)   (222)
             (311)  (411)   (322)
                    (1111)  (331)
                    (2111)  (421)
                    (3111)  (511)
                    (4111)  (2211)
                            (3211)
                            (4211)
                            (5111)
                            (11111)
                            (21111)
                            (31111)
                            (41111)
                            (51111)

Giovanni Resta, Table of n, a(n) for n = 0..75
FindStat, St000533: The maximal number of non-attacking rooks on a Ferrers shape

Number of times n appears in A325224.

Cf. A051924, A257541, A263297, A325193, A325194, A325224, A325225, A325227.

nn=30;
mindif[ptn_]:=If[ptn=={},0,Total[ptn]-Min[Length[ptn],Max[ptn]]];
allip=Array[IntegerPartitions,2*nn+2,0,Join];
Table[Length[Select[allip,mindif[#]==n&]],{n,0,nn}]

For n > 0, a(n) = Sum_{k > 0} A325227(n + k, k).

More terms from Giovanni Resta, Apr 15 2019

A051945 a(n) = C(n)(5n+1) where C(n) = Catalan numbers (A000108).

Original entry on oeis.org

1, 6, 22, 80, 294, 1092, 4092, 15444, 58630, 223652, 856596, 3292016, 12688732, 49031400, 189885240, 736808220, 2863971270, 11149451940, 43465121700, 169657266240, 662976162420, 2593424304120, 10154564564040, 39794915183400, 156078401826204, 612605246582952

Offset: 0

Barry E. Williams, Dec 20 1999

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

Andrew Howroyd, Table of n, a(n) for n = 0..200

Column k=5 of A330965.

Cf. A016813, A000108, A051924.

[Catalan(n)*(5*n+1):n in [0..27] ]; // Marius A. Burtea, Jan 05 2020

R:=PowerSeriesRing(Rationals(),29); (Coefficients(R!((2-3*x-2*Sqrt(1-4*x))/(x*Sqrt(1-4*x))))); // Marius A. Burtea, Jan 05 2020

Table[CatalanNumber[n](5n+1),{n,0,30}] (* Harvey P. Dale, Jul 27 2020 *)

a(n) = (5*n+1)*binomial(2*n, n)/(n+1)  \\ Michel Marcus, Jul 12 2013

(n+1)*(5n-4)*a(n) - 2*(5n+1)(2n-1)*a(n-1) = 0. - R. J. Mathar, Jul 09 2012
G.f.: (2 - 3*x - 2*sqrt(1 - 4*x))/(x*sqrt(1 - 4*x)). - Ilya Gutkovskiy, Jun 13 2017
From Peter Bala, Aug 23 2025: (Start)
a(n) = binomial(2*n, n) + 4*binomial(2*n, n-1) = A000984(n) + 4*A001791(n).
a(n) ~ 4^n * 5/sqrt(Pi*n). (End)
E.g.f.: exp(2*x)*((1 + 5*x)*BesselI(0, 2*x) - BesselI(1, 2*x) - 5*x*BesselI(2, 2*x)). - Stefano Spezia, Aug 29 2025

Terms a(21) and beyond from Andrew Howroyd, Jan 04 2020

A325193 Number of integer partitions whose sum plus co-rank is n, where co-rank is maximum of length and largest part.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 3, 2, 5, 5, 8, 8, 14, 14, 22, 24, 35, 39, 54, 62, 84, 97, 127, 148, 192, 224, 284, 334, 418, 492, 610, 716, 880, 1035, 1259, 1480, 1790, 2100, 2522, 2958, 3533, 4135, 4916, 5742, 6798, 7928, 9344, 10878, 12778, 14846, 17378, 20156, 23520

Offset: 0

Gus Wiseman, Apr 12 2019

nonn

			The a(4) = 2 through a(12) = 14 partitions:
  (2)   (21)  (3)    (31)   (4)     (33)    (5)      (43)     (6)
  (11)        (22)   (211)  (32)    (41)    (42)     (51)     (44)
              (111)         (221)   (222)   (322)    (332)    (52)
                            (311)   (321)   (331)    (421)    (333)
                            (1111)  (2111)  (411)    (2221)   (422)
                                            (2211)   (3211)   (431)
                                            (3111)   (4111)   (511)
                                            (11111)  (21111)  (2222)
                                                              (3221)
                                                              (3311)
                                                              (4211)
                                                              (22111)
                                                              (31111)
                                                              (111111)

FindStat, St000784: The maximum of the length and the largest part of the integer partition

Row sums of A325194.

Cf. A051924, A065770, A071724, A096771, A115720, A252464, A257990, A263297, A325178, A325189, A325192.

Table[Sum[Length[Select[IntegerPartitions[k],Max[Length[#],Max[#]]==n-k&]],{k,0,n}],{n,0,30}]

A129869 Number of positive clusters of type D_n.

Original entry on oeis.org

-1, 1, 5, 20, 77, 294, 1122, 4290, 16445, 63206, 243542, 940576, 3640210, 14115100, 54826020, 213286590, 830905245, 3241119750, 12657425550, 49483369320, 193641552390, 758454277620, 2973183318300, 11664026864100, 45791597230002, 179892016853724

Offset: 1

F. Chapoton, May 24 2007

This is also the number of antichains in the poset of positive-but-not-simple roots of type D_n.
If Y is a fixed 2-subset of a (2n+1)-set X then a(n+1) is the number of (n+2)-subsets of X intersecting Y. - Milan Janjic, Oct 28 2007
Define an array by m(1,k)=k, m(n,1) = n*(n-1) + 1, and m(n,k) = m(n,k-1) + m(n-1,k) otherwise. This yields m(n,1) = A002061(n) and on the diagonal, m(n,n) = a(n+1). - J. M. Bergot, Mar 30 2013

			a(3) = 5 because the type D3 is the same as type A3 and there are 5 positive clusters among the 14 clusters in type A3.

G. C. Greubel, Table of n, a(n) for n = 1..1000
JL Baril, S Kirgizov, The pure descent statistic on permutations, Preprint, 2016, See Cor. 6.
Paul Barry, On a Central Transform of Integer Sequences, arXiv:2004.04577 [math.CO], 2020.
F. Chapoton and L. Manivel, Triangulations and Severi varieties, arXiv:1109.6490 [math.AG], 2011.
S. Fomin and A. Zelevinsky, Y-systems and generalized associahedra, Ann. of Math. (2) 158 (2003), no. 3, 977-1018.
Milan Janjic, Two Enumerative Functions
M. A. A. Obaid, S. K. Nauman, W. M. Fakieh, C. M. Ringel, The numbers of support-tilting modules for a Dynkin algebra, 2014 and J. Int. Seq. 18 (2015) 15.10.6 .

Cf. A051924.

[(3*n-4)/n * Binomial(2*n-3,n-1) : n in [1..30]]; // Wesley Ivan Hurt, Jan 24 2017

a := n -> (1/8)*4^n*GAMMA(-1/2+n)*(3*n-4)/(sqrt(Pi)*GAMMA(1+n)) - 0^(n-1)/2;
seq(a(n), n=1..26); # Peter Luschny, Dec 14 2015

Table[((3*n-4)/n)*Binomial[2n-3,n-1],{n,30}] (* Harvey P. Dale, May 23 2012 *)

(3*n-4)/n*binomial(2*n-3,n-1) $n=1..22;

[(3*n-4)/n*binomial(2*n-3,n-1) for n in range(1,20)]

a(n) = (3*n-4)/n * C(2*n-3,n-1).
Starting with "1" = the Narayana transform (A001263) of [1, 4, 7, 10, 13, 16, ...]. - Gary W. Adamson, Jul 29 2001
G.f.: x^2*(sqrt(1-4*x)*(2*x+1)-4*x+1)/(sqrt(1-4*x)*(4*x^2-5*x+1) +12*x^2-7*x+1)-x. - Vladimir Kruchinin, Sep 27 2011
2*n*a(n) +(-13*n+14)*a(n-1) +10*(2*n-5)*a(n-2)=0. - R. J. Mathar, Apr 11 2013
a(n) = (1/8)*4^n*Gamma(n-1/2)*(3*n-4)/(sqrt(Pi)*Gamma(1+n)) - 0^(n-1)/2. - Peter Luschny, Dec 14 2015

cp's OEIS Frontend

A051597 Rows of triangle formed using Pascal's rule except begin and end n-th row with n+1.

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A325178 Difference between the length of the minimal square containing and the maximal square contained in the Young diagram of the integer partition with Heinz number n.

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A051960 a(n) = C(n)*(3n+2) where C(n) = Catalan numbers = A000108.

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A330965 Array read by descending antidiagonals: A(n,k) = (1 + k*n)*C(n) where C(n) = Catalan numbers (A000108).

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A024482 a(n) = (1/2)*(binomial(2n, n) - binomial(2n-2, n-1)).

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A325227 Regular triangle read by rows where T(n,k) is the number of integer partitions of n such that the lesser of the maximum part and the number of parts is k.

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A330965 Array read by descending antidiagonals: A(n,k) = (1 + kn)C(n) where C(n) = Catalan numbers (A000108).

A051945 a(n) = C(n)(5n+1) where C(n) = Catalan numbers (A000108).