A033184 -id:A033184 - cp's OEIS frontend

A054445 Triangle read by rows giving partial row sums of triangle A033184(n,m), n >= m >= 1 (Catalan triangle).

1, 2, 1, 5, 3, 1, 14, 9, 4, 1, 42, 28, 14, 5, 1, 132, 90, 48, 20, 6, 1, 429, 297, 165, 75, 27, 7, 1, 1430, 1001, 572, 275, 110, 35, 8, 1, 4862, 3432, 2002, 1001, 429, 154, 44, 9, 1, 16796, 11934, 7072, 3640, 1638, 637, 208, 54, 10, 1, 58786, 41990, 25194, 13260

Offset: 0

Wolfdieter Lang, Apr 27 2000 and May 08 2000

In the language of the Shapiro et al. reference (given in A053121) such a lower triangular (ordinary) convolution array, considered as a matrix, belongs to the Riordan-group. The g.f. for the row polynomials p(n,x) (increasing powers of x) is (c(z)^2)/(1-x*z*c(z)) with c(z) = g.f. A000108 (Catalan numbers).

This coincides with the lower triangular Catalan convolution matrix A033184 with first row and first column deleted: a(n,m)= A033184(n+2,m+2), n >= m >= 0, a(n,m) := 0 if n

The Catalan convolution matrix R(n,m) = A033184(n+1,m+1), n >= m >= 0, is the only Riordan-type matrix with R(0,0)=1 whose partial row sums (prs) matrix satisfies (prs(R))(n,m)= R(n+1,m+1), n >= m >= 0.

Riordan array (c(x)^2,x*c(x)) where c(x)is the g.f. of A000108. - Philippe Deléham, Nov 11 2009

			Triangle starts:
    1;
    2,  1;
    5,  3,  1;
   14,  9,  4,  1;
   42, 28, 14,  5,  1;
  132, 90, 48, 20,  6,  1;
  ...
Fourth row polynomial (n=3): p(3,x)= 14 + 9*x + 4*x^2 + x^3.
Top row of M^3 = [14, 9, 4, 1, 0, 0, 0, ...].

Cf. A033184, A000108. Row sums: a(n+1, 1).

T[n_, k_] := SeriesCoefficient[((2-2*x)*y)/(2*y+x*Sqrt[1-4*y]-x), {x, 0, n}, {y, 0, k}]; Table[T[n-k+2, k], {n, 0, 10}, {k, n+1, 1, -1}] // Flatten (* Jean-François Alcover, Apr 13 2015, after Vladimir Kruchinin *)
T[ n_, k_] := (k + 1) Binomial[2 n - k, n] / (n + 1); (* Michael Somos, Oct 01 2018 *)

tabl(nn) = {
  default(seriesprecision, nn+1);
  my( gf = ((2-2*x)*y)/(2*y+x*sqrt(1-4*y)-x) + O(x^nn) );
  for (n=0, nn-1,  my( P = polcoeff(gf, n, x) );
    for (k=0, nn-1, print1(polcoeff(P, k, y), ", "); );
    print(); );
} \\ Michel Marcus, Apr 13 2015

T(n, m) = Sum_{k=m..n} A033184(n+1, k+1), (partial row sums in columns m).

Column m recursion: a(n, m)= sum(a(j-1, m)*A033184(n-j+1, 1), j=m..n) + A033184(n+1, m+1) if n >= m >= 0, a(n, m) := 0 if n

G.f. for column m: (c(x)^2)*(x*c(x))^m, m >= 0, with c(x) = g.f. A000108.

From Gary W. Adamson, Jan 19 2012: (Start)

n-th row of the triangle = top row of M^n, where M is the following infinite square production matrix:

2, 1, 0, 0, 0, ...

1, 1, 1, 0, 0, ...

1, 1, 1, 1, 0, ...

1, 1, 1, 1, 1, ...

...

(End)

G.f.: (((2-2*x)*y)/(2*y+x*sqrt(1-4*y)-x)-1)/(x*y). - Vladimir Kruchinin, Apr 13 2015

T(n, m) = (m+1) * binomial(2*n - m, n) / (n+1) if n>=m>=1. - Michael Somos, Oct 01 2018

A172380 Eigentriangle of Catalan triangle A033184.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 3, 1, 0, 5, 10, 6, 1, 0, 14, 36, 31, 10, 1, 0, 42, 137, 156, 75, 15, 1, 0, 132, 544, 787, 510, 155, 21, 1, 0, 429, 2235, 4017, 3331, 1380, 287, 28, 1, 0, 1430, 9445, 20809, 21405, 11411, 3255, 490, 36, 1, 0, 4862, 40876, 109486, 136921, 90665

Offset: 0

Paul Barry, Feb 01 2010

Row sums are A091768.
Production matrix of inverse is matrix with general term (-1)^(n-k+1)C(k,n-k+1).
Diagonal sums are A172382. Product of A033184 and A172380 is the matrix A172381.

			Triangle begins
  1;
  0,    1;
  0,    1,    1;
  0,    2,    3,     1;
  0,    5,   10,     6,     1;
  0,   14,   36,    31,    10,     1;
  0,   42,  137,   156,    75,    15,    1;
  0,  132,  544,   787,   510,   155,   21,   1;
  0,  429, 2235,  4017,  3331,  1380,  287,  28,  1;
  0, 1430, 9445, 20809, 21405, 11411, 3255, 490, 36, 1;
Production matrix of inverse is
  0,  1;
  0, -1,  1;
  0,  0, -2,  1;
  0,  0,  1, -3,  1;
  0,  0,  0,  3, -4,   1;
  0,  0,  0, -1,  6,  -5,   1;
  0,  0,  0,  0, -4,  10,  -6,   1;
  0,  0,  0,  0,  1, -10,  15,  -7,   1;
  0,  0,  0,  0,  0,   5, -20,  21,  -8,  1;
  0,  0,  0,  0,  0,  -1,  15, -35,  28, -9,   1;
  0,  0,  0,  0,  0,   0,  -6,  35, -56, 36, -10, 1;

Paul Barry, Invariant number triangles, eigentriangles and Somos-4 sequences, arXiv preprint arXiv:1107.5490 [math.CO], 2011.

A228334 Triangle read by rows: the X-transformation of the Catalan triangle A033184.

Original entry on oeis.org

1, 0, 1, 0, 3, 1, 0, 14, 10, 1, 0, 84, 90, 21, 1, 0, 594, 825, 308, 36, 1, 0, 4719, 7865, 4004, 780, 55, 1, 0, 40898, 78078, 49686, 13650, 1650, 78, 1, 0, 379236, 804440, 606424, 214200, 37400, 3094, 105, 1, 0, 3711916, 8565960, 7379904, 3162816, 724812, 88179, 5320, 136, 1

Offset: 0

N. J. A. Sloane, Aug 26 2013

			Triangle begins:
  1;
  0,   1;
  0,   3,   1;
  0,  14,  10,   1;
  0,  84,  90,  21,   1;
  0, 594, 825, 308,  36,   1;
  ...

Fangfang Cai, Qing-Hu Hou, Yidong Sun, Arthur L.B. Yang, Combinatorial identities related to 2×2 submatrices of recursive matrices, arXiv:1808.05736 [math.CO], 2018.
Yidong Sun and Fei Ma, Minors of a Class of Riordan Arrays Related to Weighted Partial Motzkin Paths, arXiv preprint arXiv:1305.2015 [math.CO], 2013.
Yidong Sun and Fei Ma, Four transformations on the Catalan triangle, arXiv preprint arXiv:1305.2017 [math.CO], 2013.
Yidong Sun and Fei Ma, Some new binomial sums related to the Catalan triangle, Electronic Journal of Combinatorics 21(1) (2014), #P1.33

Cf. A033184, A179898, A228335, A228336, A228337.

nn = 9;
c[n_, k_] := Binomial[2n-k, n] (k+1)/(n+1);
a[0, 0] = 1;
a[n_, k_] := Table[c[n+k+i-1, 2k+j-1], {i, 1, 2}, {j, 1, 2}] // Det;
Table[a[n, k], {n, 0, nn}, {k, 0, n}] // Flatten (* Jean-François Alcover, Aug 12 2018 *)

C(n, k) = (k<=n)*binomial(2*n-k, n)*(k+1)/(n+1);
aX(nn) = {for (n = 0, nn, for (k = 0, n, print1(matdet(matrix(2, 2, i, j, C(n+k+i-1, 2*k+j-1))), ", ");); print(););} \\ Michel Marcus, Feb 13 2014

More terms from Michel Marcus, Feb 13 2014

A-number for Catalan triangle changed by Michel Marcus, Feb 13 2014

A228335 Triangle read by rows: the Y-transformation of the Catalan triangle A033184.

Original entry on oeis.org

1, 1, 1, 3, 6, 1, 14, 40, 15, 1, 84, 300, 175, 28, 1, 594, 2475, 1925, 504, 45, 1, 4719, 22022, 21021, 7644, 1155, 66, 1, 40898, 208208, 231868, 107016, 23100, 2288, 91, 1, 379236, 2068560, 2598960, 1439424, 403920, 58344, 4095, 120, 1, 3711916, 21414900, 29651400, 18976896, 6523308, 1247103, 129675, 6800, 153, 1

Offset: 0

N. J. A. Sloane, Aug 26 2013

From Roger Ford, Feb 08 2020: (Start)
For 2n-step octant walks, the numbers for the x axis ending locations are equal to numbers in row n of the example triangle.
Example: For n = 2, all 4-step octant walks starting at (0,0) and ending on the x axis are as follows:
  EWEW EEWW   EEEW EWEE EEWE   EEEE
  ENSW        EENS ENSE ENES
  (0,0)       (2,0)            (4,0) - x axis ending location
  3           6                1     - number of walks
  These numbers match row 2 of the example triangle. (End)

			Triangle begins:
    1;
    1,    1;
    3,    6,    1;
   14,   40,   15,    1;
   84,  300,  175,   28,    1;
  594, 2475, 1925,  504,   45,    1;
  ...

Yidong Sun and Fei Ma, Four transformations on the Catalan triangle, arXiv preprint arXiv:1305.2017 [math.CO], 2013.
Yidong Sun and Fei Ma, Some new binomial sums related to the Catalan triangle, Electronic Journal of Combinatorics 21(1) (2014), #P1.33.

Cf. A033184, A228334, A228336, A228337.

nn = 9;
c[n_, k_] := Binomial[2n-k, n] (k+1)/(n+1);
a[0, 0] = 1;
a[n_, k_] := Table[c[n+k+i, 2k+j], {i, 2}, {j, 2}] // Det;
Table[a[n, k], {n, 0, nn}, {k, 0, n}] // Flatten (* Jean-François Alcover, Aug 12 2018 *)

C(n, k) = (k<=n)*binomial(2*n-k, n)*(k+1)/(n+1);
T(n, k) = matdet(matrix(2, 2, i, j, C(n+k+i, 2*k+j))); \\ Michel Marcus, Feb 13 2014

More terms from Michel Marcus, Feb 13 2014

A-number for Catalan triangle changed by Michel Marcus, Feb 13 2014

A228336 Triangle read by rows: the Z-transformation of the Catalan triangle A033184.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 4, 6, 3, 1, 10, 15, 12, 4, 1, 25, 45, 36, 20, 5, 1, 70, 126, 126, 70, 30, 6, 1, 196, 392, 392, 280, 120, 42, 7, 1, 588, 1176, 1344, 960, 540, 189, 56, 8, 1, 1764, 3780, 4320, 3600, 2025, 945, 280, 72, 9, 1, 5544, 11880, 14850, 12375, 8250, 3850, 1540, 396, 90, 10, 1

Offset: 0

N. J. A. Sloane, Aug 26 2013

			Triangle begins:
   1;
   1,   1;
   2,   2,   1;
   4,   6,   3,  1;
  10,  15,  12,  4,  1;
  25,  45,  36, 20,  5, 1;
  70, 126, 126, 70, 30, 6, 1;
  ...

Yidong Sun and Fei Ma, Four transformations on the Catalan triangle, arXiv preprint arXiv:1305.2017 [math.CO], 2013.
Yidong Sun and Fei Ma, Some new binomial sums related to the Catalan triangle, Electronic Journal of Combinatorics 21(1) (2014), #P1.33.

Cf. A033184, A228334, A228335, A228337.

c[n_, k_] := Boole[k <= n] Binomial[2n - k, n] (k + 1)/(n + 1);
T[n_, k_] := Module[{nn, kk}, If[OddQ[n], nn = (n + 1)/2, nn = n/2]; If[OddQ[k], kk = (k - 1)/2, kk = k/2]; If [OddQ[n], If[OddQ[k], c[nn + kk, 2kk + 1] c[nn + kk + 1, 2kk + 2], c[nn + kk, 2kk] c[nn + kk, 2kk + 1]], If[OddQ[k], c[nn + kk + 1, 2kk + 1] c[nn + kk + 1, 2kk + 2], c[nn + kk, 2kk] c[nn + kk + 1, 2kk + 1]]]];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Oct 04 2018, from PARI *)

C(n, k) = (k<=n)*binomial(2*n-k, n)*(k+1)/(n+1);
T(n, k) = {my(nn, kk); if (n % 2, nn = (n+1)/2, nn = n/2); if (k % 2, kk = (k-1)/2, kk = k/2); if ((n % 2), if (k % 2, C(nn+kk, 2*kk+1)*C(nn+kk+1, 2*kk+2), C(nn+kk, 2*kk)*C(nn+kk, 2*kk+1)), if (k % 2, C(nn+kk+1, 2*kk+1)*C(nn+kk+1, 2*kk+2), C(nn+kk, 2*kk)*C(nn+kk+1, 2*kk+1)));} \\ Michel Marcus, Feb 13 2014

More terms from Michel Marcus, Feb 13 2014

A-number for Catalan triangle changed by Michel Marcus, Feb 13 2014

A228337 Irregular triangle read by rows: the W-transformation of the Catalan triangle A033184.

Original entry on oeis.org

1, 2, 4, 1, 10, 4, 20, 21, 1, 56, 70, 6, 140, 238, 50, 1, 420, 792, 210, 8, 1176, 2604, 990, 91, 1, 3696, 8778, 3850, 462, 10, 11088, 29106, 15675, 2772, 144, 1, 36036, 99528, 59202, 12376, 858, 12, 113256, 335049, 228085, 60060, 6240, 209, 1

Offset: 0

N. J. A. Sloane, Aug 26 2013

			Triangle begins:
     1;
     2;
     4,    1;
    10,    4;
    20,   21,    1;
    56,   70,    6;
   140,  238,   50,    1;
   420,  792,  210,    8;
  1176, 2604,  990,   91,    1;
  ...

Yidong Sun and Fei Ma, Four transformations on the Catalan triangle, arXiv preprint arXiv:1305.2017 [math.CO], 2013.
Yidong Sun and Fei Ma, Some new binomial sums related to the Catalan triangle, Electronic Journal of Combinatorics 21(1) (2014), #P1.33.

Cf. A033184, A228334, A228335, A228336.

nn = 12;
c[n_, k_] := If[k <= n, Binomial[2n-k, n] (k+1)/(n+1), 0];
a[n_, k_] := Table[c[If[OddQ[n], (n-1)/2+k+2i-2, n/2+k+i-1], 2k+j-1], {i, 1, 2}, {j, 1, 2}] // Permanent;
Table[a[n, k], {n, 0, nn}, {k, 0, n/2}] // Flatten (* Jean-François Alcover, Aug 12 2018 *)

C(n, k) = (k<=n)*binomial(2*n-k, n)*(k+1)/(n+1);
matperm(M)=my(n=#M,t);sum(i=1,n!,t=numtoperm(n,i);prod(j=1,n,M[j,t[j]])); \\ from Rosetta code
W(n, k) = my(nn); if (n % 2, nn = (n-1)/2; matperm(matrix(2, 2, i, j, C(nn+k+2*i-2, 2*k+j-1))), nn = n/2; matperm(matrix(2, 2, i, j, C(nn+k+i-1, 2*k+j-1))));
aW(nn) = {for (n=0, nn, for (k=0, n\2, print1(W(n, k), ", ");); print(););} \\ Michel Marcus, Feb 13 2014

More terms from Michel Marcus, Feb 13 2014

A-number for Catalan triangle changed by Michel Marcus, Feb 13 2014

A116363 a(n) = dot product of row n in Catalan triangle A033184 with row n in Pascal's triangle.

Original entry on oeis.org

1, 2, 7, 30, 141, 698, 3571, 18686, 99385, 535122, 2908863, 15932766, 87809541, 486421770, 2706138987, 15110359038, 84637982961, 475381503266, 2676447372535, 15100548901790, 85357620588541, 483304834607322

Offset: 0

Paul D. Hanna, Feb 04 2006

nonn

			The dot product of Catalan row 4 and Pascal row 4 equals a(4) = [14,14,9,4,1]*[1,4,6,4,1] = 141
which is equivalent to obtaining the final term in these repeated partial sums of Pascal row 4:
  1,4, 6, 4, 1
   5,11,15,16
    16,31,47
     47,94
      141

G. C. Greubel, Table of n, a(n) for n = 0..1000
D. Drake, Bijections from Weighted Dyck Paths to Schröder Paths, J. Int. Seq. 13 (2010) # 10.9.2.

Cf. A033184.

List([0..30], n-> Sum([0..n], j-> Binomial(n,j)*Binomial(2*n-j+1, n-j)*(j+1)/(2*n-j+1))); # G. C. Greubel, May 12 2019

[(&+[Binomial(n,j)*Binomial(2*n-j+1, n-j)*(j+1)/(2*n-j+1): j in [0..n]]): n in [0..30]]; // G. C. Greubel, May 12 2019

Table[Sum[Binomial[n, j]*Binomial[2*n-j+1, n-j]*(j+1)/(2*n-j+1), {j,0,n} ], {n,0,30}] (* G. C. Greubel, May 12 2019 *)

a(n)=sum(k=0,n,binomial(n,k)*binomial(2*n-k+1,n-k)*(k+1)/(2*n-k+1))
for(n=0,30,print1(a(n),", "))

[sum(binomial(n,j)*binomial(2*n-j+1, n-j)*(j+1)/(2*n-j+1) for j in (0..n)) for n in (0..30)] # G. C. Greubel, May 12 2019

a(n) = Sum_{k=0..n} C(n,k)*C(2*n-k+1,n-k)*(k+1)/(2*n-k+1).
G.f. A(x) satisfies: d/dx[log(1 - 4*x*A(x))] = -4*(1-5*x)/(1-13*x+43*x^2-7*x^3).
O.g.f.: 2*(R+x)/(R*(R+x+1)), where R = sqrt(x^2+6*x+1). [Dan Drake, May 19 2010]
Conjecture: +(2*n+5)*(n+1)*a(n) +4*(-3*n^2-9*n+5)*a(n-1) +(2*n+7)*(n-1)*a(n-2)=0. - R. J. Mathar, Jun 22 2016

A116364 Row squared sums of Catalan triangle A033184.

Original entry on oeis.org

1, 2, 9, 60, 490, 4534, 45689, 489920, 5508000, 64276492, 773029466, 9531003552, 119990158054, 1537695160070, 20009930706137, 263883333450760, 3521003563829212, 47470845904561648, 645960472314074400

Offset: 0

Paul D. Hanna, Feb 04 2006

nonn

Number of 321-avoiding permutations in which the length of the longest increasing subsequence is n. Example: a(2)=9 because we have 12, 132, 312, 213, 231, 3142, 3412, 2143 and 2413. Column sums of triangle in A126217 (n >= 1). - Emeric Deutsch, Sep 07 2007

			The dot product of Catalan row 4 with itself equals
  a(4) = [14,14,9,4,1]*[14,14,9,4,1] = 490
which is equivalent to obtaining the final term in these repeated partial sums of Catalan row 4:
  14,   14,    9,    4,    1
     28,   37,   41,   42
        65,  106,  148
          171,  319
             490

G. C. Greubel, Table of n, a(n) for n = 0..830

Cf. A033184, A116363.

Cf. A126217.

List([0..30], n-> Sum([0..n], j-> (Binomial(2*n-j+1, n-j)* (j+1)/(2*n-j+1))^2 )); # G. C. Greubel, May 12 2019

[(&+[(Binomial(2*n-j+1, n-j)*(j+1)/(2*n-j+1))^2: j in [0..n]]): n in [0..30]]; // G. C. Greubel, May 12 2019

a:=proc(k) options operator, arrow: sum((2*k-n+1)^2*binomial(n+1,k+1)^2/(n+1)^2,n=k..2*k) end proc: 1,seq(a(k),k=1..17); # Emeric Deutsch, Sep 07 2007

Table[Sum[(Binomial[2*n-j+1, n-j]*(j+1)/(2*n-j+1))^2, {j, 0, n}], {n, 0, 30}] (* G. C. Greubel, May 12 2019 *)

a(n)=sum(k=0,n,((k+1)*binomial(2*n-k+1,n-k)/(2*n-k+1))^2)

[sum(( binomial(2*n-j+1, n-j)*(j+1)/(2*n-j+1) )^2 for j in (0..n)) for n in (0..30)] # G. C. Greubel, May 12 2019

a(n) = Sum_{k=0..n} (C(2*n-k+1,n-k)*(k+1)/(2*n-k+1))^2.

A127742 Triangle read by rows with shape A000041 which refines the Catalan triangle A033184 using sequence A048996.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 5, 4, 1, 3, 1, 14, 10, 4, 6, 3, 4, 1, 42, 28, 10, 4, 15, 12, 1, 8, 6, 5, 1, 132, 84, 28, 20, 42, 30, 12, 6, 20, 24, 4, 10, 10, 6, 1, 429, 264, 84, 56, 25, 126, 84, 60, 15, 12, 56, 60, 24, 24, 1, 25, 40, 10, 12, 15, 7, 1, 1430, 858, 264, 168, 140, 396, 252, 168, 75

Offset: 1

Alford Arnold, Feb 23 2007

Contribution from R. J. Mathar, Jul 16 2010: (Start)
The entries count Dyck paths of length 2n which have a step/stride pattern between consecutive returns to the horizontal axes (after sorting) equivalent to the k-th partition of n.
Equivalent means that each distance between two x (where y=0) is divided by 2 prior to comparison.
Example: if the x-values are (0,4,8,10,16) with 2n=16, the strides are 4,4,2,6, equal to 2,2,1,3 after division by 2, and contribute to the 1^1,2^2,3^1 partition T(8,14). (End)

			The triangle begins
1
1 1
2 2 1
5 4 1 3 1
14 10 4 6 3 4 1
etc

More terms from R. J. Mathar, Jul 16 2010

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cp's OEIS Frontend

A174687 Central coefficients T(2n,n) of the Catalan triangle A033184.

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A054445 Triangle read by rows giving partial row sums of triangle A033184(n,m), n >= m >= 1 (Catalan triangle).

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A172380 Eigentriangle of Catalan triangle A033184.

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A228334 Triangle read by rows: the X-transformation of the Catalan triangle A033184.

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A228335 Triangle read by rows: the Y-transformation of the Catalan triangle A033184.

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A228336 Triangle read by rows: the Z-transformation of the Catalan triangle A033184.

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A228337 Irregular triangle read by rows: the W-transformation of the Catalan triangle A033184.

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A116363 a(n) = dot product of row n in Catalan triangle A033184 with row n in Pascal's triangle.

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