A106566 -id:A106566 - cp's OEIS frontend

A111301 Triangle read by rows: T(n,k) is the number of Dyck n-paths containing k even-length descents to ground level.

1, 1, 1, 1, 2, 3, 5, 8, 1, 14, 23, 5, 42, 70, 19, 1, 132, 222, 68, 7, 429, 726, 240, 34, 1, 1430, 2431, 847, 145, 9, 4862, 8294, 3003, 583, 53, 1, 16796, 28730, 10712, 2275, 262, 11, 58786, 100776, 38454, 8736, 1183, 76, 1, 208012, 357238, 138890, 33252, 5068

Offset: 0

David Callan, Nov 02 2005

Column k is the sum of columns 2k and 2k+1 of A106566.

			Table begins
k: ..0....1....2....3....
n
0 |..1
1 |..1
2 |..1....1
3 |..2....3
4 |..5....8....1
5 |.14...23....5
6 |.42...70...19....1
7 |132..222...68....7
a(3,1)=3 because the Dyck 3-paths containing one even-length descent to ground level are UUDUDD, UDUUDD, UUDDUD.

G. C. Greubel, Table of n, a(n) for the first 76 rows, flattened
David Callan, The 136th manifestation of C_n , arXiv:math/0511010 [math.CO], 2005.

Row sums are the Catalan numbers A000108.

A143949 considers odd-length descents to the ground level. - Emeric Deutsch, Oct 05 2008

TableForm[Table[k/(n-k)Binomial[2n-2k, n]+(2k+1)/(2n-2k-1)Binomial[2n-2k-1, n], {n, 10}, {k, 0, n/2}]]
Join[{1}, Table[k/(n - k) Binomial[2 n - 2 k, n] + (2 k + 1)/(2 n - 2 k - 1) Binomial[2 n - 2 k - 1, n], {n, 25}, {k, 0, n/2}] // Flatten] (* G. C. Greubel, Jul 28 2017 *)

See Mathematica line.
From Emeric Deutsch, Oct 05 2008: (Start)
G.f.=G(s,z)=1/[1-z(1+szC)/(1-z^2*C^2)], where C=[1-sqrt(1-4z)]/(2z) is the Catalan function.
The trivariate g.f. H(t,s,z), where t (s) marks odd-length (even-length) descents to ground level and z marks semilength, is H=1/[1-z(t+szC)/(1-z^2*C^2)], where C=[1-sqrt(1-4z)]/(2z) is the Catalan function. (End)

A108747 Triangle read by rows: T(n,k) is the number of Grand Dyck paths of semilength n and having k returns to the x-axis.

Original entry on oeis.org

2, 2, 4, 4, 8, 8, 10, 20, 24, 16, 28, 56, 72, 64, 32, 84, 168, 224, 224, 160, 64, 264, 528, 720, 768, 640, 384, 128, 858, 1716, 2376, 2640, 2400, 1728, 896, 256, 2860, 5720, 8008, 9152, 8800, 7040, 4480, 2048, 512, 9724, 19448, 27456, 32032, 32032, 27456, 19712, 11264, 4608, 1024

Offset: 1

Emeric Deutsch, Jun 23 2005

A Grand Dyck path of semilength n is a path in the half-plane x >= 0, starting at (0,0), ending at (2n,0) and consisting of steps u=(1,1) and d=(1,-1).
Triangle T(n,k), 1 <= k <= n, read by rows, given by [0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...] DELTA [2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Jun 29 2005
T(n,k) is also equal to the number of grand Dyck paths of semilength n having k symmetric vertices. A symmetric vertex is a vertex in the first half of the path (not including the midpoint) that is a mirror image of a vertex in the second half, when with respect to the reflection along the vertical line through the midpoint of the path. - Sergi Elizalde, Feb 12 2021

			T(2,2)=4 because we have u(d)u(d), u(d)d(u), d(u)d(u) and d(u)u(d) (return steps to x-axis shown between parentheses).
Triangle begins:
   2;
   2,  4;
   4,  8,  8;
  10, 20, 24, 16;
  28, 56, 72, 64, 32;

Sergi Elizalde, The degree of symmetry of lattice paths, arXiv:2002.12874 [math.CO], 2020.
Sergi Elizalde, Measuring symmetry in lattice paths and partitions, Sem. Lothar. Combin. 84B.26, 12 pp (2020).

Cf. A000984 (row sums), A000108.

T:= (n,k)-> 2^k*k*binomial(2*n-k,n)/(2*n-k): for n from 1 to 10 do seq(T(n,k),k=1..n) od; # yields sequence in triangular form

nn=10;c=(1-(1-4x)^(1/2))/(2x);f[list_]:=Select[list,#>0&];Map[f,Drop[CoefficientList[Series[1/(1-2y x c),{x,0,nn}],{x,y}],1]]//Flatten  (* Geoffrey Critzer, Jan 30 2012 *)

T(n,1) = 2*A000108(n-1).
T(n,n) = 2^n.
T(n,k) = k * 2^k * binomial(2*n-k,n)/(2*n-k) (1 <= k <= n).
G.f.: 1/(1-2*t*z*C), where C = (1-sqrt(1-4*z))/(2*z) is the Catalan function.
T(n,k) = 2^k * A106566(n,k). - Philippe Deléham, Jun 29 2005

A127631 Square of Riordan array (1, x*c(x)) where c(x) is the g.f. of A000108.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 6, 4, 1, 0, 21, 16, 6, 1, 0, 80, 66, 30, 8, 1, 0, 322, 280, 143, 48, 10, 1, 0, 1348, 1216, 672, 260, 70, 12, 1, 0, 5814, 5385, 3150, 1344, 425, 96, 14, 1, 0, 25674, 24244, 14799, 6784, 2400, 646, 126, 16, 1

Offset: 0

Paul Barry, Jan 20 2007

Square of A106566. Row sums are A127632.

			Triangle begins
  1;
  0,      1;
  0,      2,      1;
  0,      6,      4,     1;
  0,     21,     16,     6,     1;
  0,     80,     66,    30,     8,     1;
  0,    322,    280,   143,    48,    10,    1;
  0,   1348,   1216,   672,   260,    70,   12,   1;
  0,   5814,   5385,  3150,  1344,   425,   96,  14,   1;
  0,  25674,  24244, 14799,  6784,  2400,  646, 126,  16,  1;
  0, 115566, 110704, 69828, 33814, 13002, 3960, 931, 160, 18, 1;

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

Cf. A106566, A121988, A129442.

[[k eq n select 1 else (k/n)*(&+[Binomial(2*j+k-1,j)*Binomial(2*n -k-j-1, n-k-j): j in [0..n-k]]): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Apr 05 2019

T[n_, k_]:= If[k==n, 1, (k/n)*Sum[Binomial[2*j-k-1, j-k]*Binomial[2*n-j- 1, n-j], {j,k,n}]]; Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Apr 05 2019 *)

T(n,k):=if k=n then 1 else if n=0 then 0 else (k*sum((binomial(-k+2*i-1,i-k))*(binomial(2*n-i-1,n-i)),i,k,n))/n; /* Vladimir Kruchinin, Apr 05 2019 */

{T(n,k) = if(k==n, 1, (k/n)*sum(j=0,n-k, binomial(2*j+k-1, j)* binomial(2*n-k-j-1, n-k-j)))}; \\ G. C. Greubel, Apr 05 2019

def T(n, k):
   if k == n: return 1
   return (k*sum(binomial(2*j+k-1, j)* binomial(2*n-k-j-1, n-k-j) for j in (0..n-k)))//n
[[T(n,k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Apr 05 2019

Riordan array (1, x*c(x)*c(x*c(x))), where c(x) is the g.f. of A000108.
T(n+1,1) = A129442(n) = A121988(n+1). - Philippe Deléham, Feb 27 2013
T(n,k) = (k/n)*Sum_{i=k..n} C(2*i-k-1,i-k)*C(2*n-i-1,n-i), T(n,n)=1. - Vladimir Kruchinin, Apr 05 2019

A146533 Catalan transform of A135092.

Original entry on oeis.org

1, 7, 21, 70, 245, 882, 3234, 12012, 45045, 170170, 646646, 2469012, 9464546, 36402100, 140408100, 542911320, 2103781365, 8167621770, 31762973550, 123708423300, 482462850870, 1883902560540, 7364346373020, 28817007546600, 112866612890850, 442437122532132, 1735714865318364

Offset: 0

Philippe Deléham, Oct 31 2008

nonn

Cf. A088218, A000984, A029651, A100320, A029609, A144706.

a[n_]:=(7*Binomial[2n,n]-5*KroneckerDelta[n,0])/2; Array[a,27,0] (* Stefano Spezia, Feb 14 2025 *)

a(n) = Sum_{k=0..n} A039599(n,k)*A010687(k) = Sum_{k=0..n} A106566(n,k)*A135092(k).
a(n) = (7*C(2n,n) - 5*0^n)/2.
From Stefano Spezia, Feb 14 2025: (Start)
G.f.: (7/sqrt(1 - 4*x) - 5)/2.
E.g.f.: (7*exp(2*x)*BesselI(0, 2*x) - 5)/2. (End)

a(23)-a(26) from Stefano Spezia, Feb 14 2025

A168377 Riordan array (1/(1 + x), x*c(x)), where c(x) is the o.g.f. of Catalan numbers A000108.

Original entry on oeis.org

1, -1, 1, 1, 0, 1, -1, 2, 1, 1, 1, 3, 4, 2, 1, -1, 11, 10, 7, 3, 1, 1, 31, 32, 21, 11, 4, 1, -1, 101, 100, 69, 37, 16, 5, 1, 1, 328, 329, 228, 128, 59, 22, 6, 1, -1, 1102, 1101, 773, 444, 216, 88, 29, 7, 1, 1, 3760, 3761, 2659, 1558, 785, 341, 125, 37, 8, 1

Offset: 0

Philippe Deléham, Nov 24 2009

			Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
   1;
  -1,   1;
   1,   0,   1;
  -1,   2,   1,  1;
   1,   3,   4,  2,  1;
  -1,  11,  10,  7,  3,  1;
   1,  31,  32, 21, 11,  4, 1;
  -1, 101, 100, 69, 37, 16, 5, 1;
  ...
From _Philippe Deléham_, Sep 14 2014: (Start)
Production matrix begins:
  -1, 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
   ... (End)

Emeric Deutsch, Luca Ferrari, and Simone Rinaldi, Production matrices and Riordan arrays, arXiv:math/0702638 [math.CO], 2007.
Emeric Deutsch, Luca Ferrari, and Simone Rinaldi, Production matrices and Riordan arrays, Annals of Combinatorics, 13 (2009), 65-85.
L. W. Shapiro, S. Getu, W.-J. Woan, and L. C. Woodson, The Riordan group, Discrete Applied Mathematics, 34(1-3) (1991), 229-239.
Wikipedia, Riordan array.

Cf. A000012, A000108, A000124, A023443, A032357, A033297, A033999, A091491, A096470, A106566, A127540.

A000108(n) = binomial(2*n, n)/(n+1);
A032357(n) = sum(k=0, n, (-1)^(n-k)*A000108(k));
T(n, k) = if ((k==0), (-1)^n, if ((n<0) || (k<0), 0, if (k==1, A032357(n-1), if (n > k-1, T(n, k-1) - T(n-1, k-2), 0))));
for(n=0, 10, for (k=0, n, print1(T(n, k), ", ")); print); \\ Petros Hadjicostas, Aug 08 2020

T(n,0) = (-1)^n and T(n,n) = 1.
Sum_{0 <= k <= n} T(n,k) = A032357(n).
From Petros Hadjicostas, Aug 08 2020: (Start)
T(n,k) = T(n,k-1) - T(n-1,k-2) for 2 <= k <= n with initial conditions T(n,0) = (-1)^n (n >= 0) and T(n,1) = A032357(n-1) (n >= 1).
T(n,2) = A033297(n).
T(n,n-1) = n - 2 for n >= 1.
|T(n,k)| = |A096470(n,n-k)| for 0 <= k <= n.
Bivariate o.g.f.: 1/((1 + x)*(1 - x*y*c(x))), where c(x) is the o.g.f. of A000108.
Bivariate o.g.f.: (1 - y + x*y*c(x))/((1 + x)*(1 - y + x*y^2)).
Bivariate o.g.f. of |T(n,k)|: (o.g.f. of T(n,k)) + 2*x/(1 - x^2). (End)

A236843 Triangle read by rows related to the Catalan transform of the Fibonacci numbers.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 5, 9, 4, 1, 14, 28, 14, 6, 1, 42, 90, 48, 27, 7, 1, 132, 297, 165, 110, 35, 9, 1, 429, 1001, 572, 429, 154, 54, 10, 1, 1430, 3432, 2002, 1638, 637, 273, 65, 12, 1, 4862, 11934, 7072, 6188, 2548, 1260, 350, 90, 13, 1, 16796, 41990, 25194, 23256, 9996, 5508, 1700, 544, 104, 15, 1

Offset: 0

Philippe Deléham, Feb 01 2014

Row sums are A109262(n+1).

			Triangle begins:
    1;
    1,   1;
    2,   3,   1;
    5,   9,   4,   1;
   14,  28,  14,   6,  1;
   42,  90,  48,  27,  7, 1;
  132, 297, 165, 110, 35, 9, 1;
Production matrix is:
  1...1
  1...2...1
  0...1...1...1
  0...1...1...2...1
  0...0...0...1...1...1
  0...0...0...1...1...2...1
  0...0...0...0...0...1...1...1
  0...0...0...0...0...1...1...2...1
  0...0...0...0...0...0...0...1...1...1
  0...0...0...0...0...0...0...1...1...2...1
  0...0...0...0...0...0...0...0...0...1...1...1
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Columns: A000108 (k=0), A000245 (k=1), A002057 (k=2), A003517 (k=3), A000588 (k=4), A001392 (k=5), A003519 (k=6), A090749 (k=7), A000590 (k=8).
Cf. A026674, A026726, A032766, A109262.
Cf. A000045, A039599, A106566.

F:=Factorial;
A236843:= func< n,k | (1/4)*(6*k+5-(-1)^k)*F(2*n-Floor(k/2))/(F(n-k)*F(n+Floor((k+1)/2)+1)) >;
[A236843(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 13 2022

T[n_, k_]:= (1/4)*(6*k+5-(-1)^k)*(2*n-Floor[k/2])!/((n-k)!*(n+Floor[(k+1)/2]+1)!);
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 13 2022 *)

T(n, k) = (1/4)*(6*k + 5 - (-1)^k)*(2*n - (k\2))!/((n-k)!*(n + (k+1)\2 + 1)!) \\ Andrew Howroyd, Jan 04 2023

F=factorial
def A236843(n,k): return (1/2)*(3*k+2+(k%2))*F(2*n-(k//2))/(F(n-k)*F(n+((k+1)//2)+1))
flatten([[A236843(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 13 2022

G.f. for the column k (with zeros omitted): C(x)^A032766(k+1) where C(x) is g.f. for Catalan numbers (A000108).
Sum_{k=0..n} T(n,k) = A109262(n+1).
Sum_{k=0..n} T(n+k,2k) = A026726(n).
Sum_{k=0..n} T(n+1+k,2k+1) = A026674(n+1).
T(n, k) = (1/4)*(6*k + 5 - (-1)^k)*(2*n - floor(k/2))!/((n-k)!*(n + floor((k+1)/2) + 1)!). - G. C. Greubel, Jun 13 2022

A307495 Expansion of Sum_{k>=0} k!((1 - sqrt(1 - 4x))/2)^k.

Original entry on oeis.org

1, 1, 3, 12, 57, 312, 1950, 13848, 111069, 998064, 9957186, 109305240, 1309637274, 17006109072, 237888664572, 3566114897520, 57030565449765, 969154436550240, 17439499379433690, 331268545604793240, 6624013560942038670, 139080391965533653200, 3059323407592802838180, 70355685298375014175440

Offset: 0

Ilya Gutkovskiy, Apr 10 2019

nonn

Catalan transform of A000142 (factorial numbers).
From Peter Bala, Jan 27 2020: (Start)
This sequence is the main diagonal of the lower triangular array formed by putting the sequence of factorial numbers in the first column (k = 0) of the array and then completing the triangle using the relation T(n,k) = T(n-1,k) + T(n,k-1) for k >= 1.
    1
    1    1
    2    3    3
    6    9   12   12
   24   33   45   57   57
  120  153  198  255  312  312
  ...
Alternatively, the sequence can be obtained by multiplying the sequence of factorial numbers by the array A106566.
(End)

P. Bala, A note on the Catalan transform of a sequence

Cf. A000108, A000142, A013999, A100100, A106566, A307496.

nmax = 23; CoefficientList[Series[Sum[k! ((1 - Sqrt[1 - 4 x])/2)^k, {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 23; CoefficientList[Series[1/(1 + ContinuedFractionK[-Floor[(k + 1)/2] (1 - Sqrt[1 - 4 x])/2, 1, {k, 1, nmax}]), {x, 0, nmax}], x]
Join[{1}, Table[1/n Sum[Binomial[2n - k - 1, n - k] k k!, {k, n}], {n, 23}]]

G.f.: 1 /(1 - x*c(x)/(1 - x*c(x)/(1 - 2*x*c(x)/(1 - 2*x*c(x)/(1 - 3*x*c(x)/(1 - 3*x*c(x)/(1 - ...))))))), a continued fraction, where c(x) = g.f. of Catalan numbers (A000108).
Sum_{n>=0} a(n)*(x*(1 - x))^n = g.f. of A000142.
a(n) = (1/n) * Sum_{k=1..n} binomial(2*n-k-1,n-k)*k*k! for n > 0.
a(n) ~ exp(1) * n!. - Vaclav Kotesovec, Aug 10 2019

A130655 Catalan transform of Catalan numbers C(n+1).

Original entry on oeis.org

1, 2, 7, 28, 119, 524, 2363, 10844, 50446, 237280, 1126437, 5389916, 25967972, 125868952, 613385075, 3003586196, 14771851093, 72936101780, 361419276386, 1796837068400, 8960207761500

Offset: 0

Philippe Deléham, Jun 21 2007

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A127632, A129442.

A130655 := proc(n)
    add(A106566(n,k)*A000108(k+1),k=0..n) ;
end proc: # R. J. Mathar, Mar 01 2015

CoefficientList[Series[2/(Sqrt[-1 + 2*Sqrt[1-4*x]] + Sqrt[1-4*x]),{x,0,20}],x] (* Vaclav Kotesovec, Jul 02 2015 *)

x='x+O('x^50); Vec(2/(sqrt(-1 + 2*sqrt(1-4*x)) + sqrt(1-4*x))) \\ G. C. Greubel, Mar 21 2017

a(n) = Sum_{k=0..n} A106566(n,k)*A000108(k+1).
Conjecture: 3*n*(n-2)*(n+2)*a(n) +4*(-10*n^3+21*n^2+7*n-15)*a(n-1) +16*(11*n^3-47*n^2+57*n-15)*a(n-2) -8*(2*n-5)*(4*n-9)*(4*n-7)*a(n-3)=0. - R. J. Mathar, Mar 01 2015
G.f.: (C(x*C(x))-1)/(x*C(x)), where C(x) is g.f. of Catalan numbers A000108. - Vladimir Kruchinin, Jul 02 2015
a(n) ~ 2^(4*n+3/2) / (sqrt(Pi) * n^(3/2) * 3^(n-1/2)). - Vaclav Kotesovec, Jul 02 2015

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A111301 Triangle read by rows: T(n,k) is the number of Dyck n-paths containing k even-length descents to ground level.

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A108747 Triangle read by rows: T(n,k) is the number of Grand Dyck paths of semilength n and having k returns to the x-axis.

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A127631 Square of Riordan array (1, x*c(x)) where c(x) is the g.f. of A000108.

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A146533 Catalan transform of A135092.

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A168377 Riordan array (1/(1 + x), x*c(x)), where c(x) is the o.g.f. of Catalan numbers A000108.

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A236843 Triangle read by rows related to the Catalan transform of the Fibonacci numbers.

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A307495 Expansion of Sum_{k>=0} k!*((1 - sqrt(1 - 4*x))/2)^k.

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A307495 Expansion of Sum_{k>=0} k!((1 - sqrt(1 - 4x))/2)^k.