A071721 -id:A071721 - cp's OEIS frontend

A068875 Expansion of (1 + xC)C, where C = (1 - (1 - 4x)^(1/2))/(2x) is the g.f. for Catalan numbers, A000108.

1, 2, 4, 10, 28, 84, 264, 858, 2860, 9724, 33592, 117572, 416024, 1485800, 5348880, 19389690, 70715340, 259289580, 955277400, 3534526380, 13128240840, 48932534040, 182965127280, 686119227300, 2579808294648, 9723892802904, 36734706144304, 139067101832008

Offset: 0

N. J. A. Sloane, Jun 06 2002

A Catalan transform of A040000 under the mapping g(x) -> g(x*c(x)), where c(x) is the g.f. of A000108. Sequence A040000 can be retrieved using the mapping g(x) -> g(x*(1-x)). A040000(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k) * (-1)^k * a(n-k). a(n) and A040000 may be described as a Catalan pair. - Paul Barry, Nov 14 2004
a(n) is the number of Dyck (n+1)-paths all of whose nonterminal descents to ground level are of odd length. For example, a(2) counts UUUDDD, UUDUDD, UDUUDD, UDUDUD. - David Callan, Jul 25 2005
From Gary W. Adamson, Jul 11 2011: (Start)
a(n) is the sum of the top row terms in M^n, where M is the following infinite square production matrix:
  1, 1, 0, 0, 0, 0, ...
  0, 1, 1, 0, 0, 0, ...
  1, 1, 1, 1, 0, 0, ...
  0, 1, 1, 1, 1, 0, ...
  1, 1, 1, 1, 1, 1, ...
  ...
For example, the top row of M^3 = (2, 4, 3, 1) with sum = 10 = a(3). (End)
For n >= 1, a(n) is the number of binary trees with n+1 internal node in which one of the subtrees of the root is empty. Cf. A002057. [Sedgewick and Flajolet] - Geoffrey Critzer, Jan 05 2013
Empirical: a(n) is the number of entries of absolute value 1 that appear among all partitions in the canonical basis of the Temperley-Lieb algebra of order n. - John M. Campbell, Oct 17 2017
For n >= 1, a(n) is the number of Dyck paths of size n+2, whose corresponding unit interval graph has P3-hull number equal to 2. This result is due to Alrik Sandberg. - Per W. Alexandersson, Jan 09 2024

			G.f. = 1 + 2*x + 4*x^2 + 10*x^3 + 28*x^4 + 84*x^5 + 264*x^6 + 858*x^7 + ...
For example, the canonical basis of the Temperley-Lieb algebra of order 3 is {{{-3, 1}, {-2, -1}, {2, 3}}, {{-3, 3}, {-2, 2}, {-1, 1}}, {{-3, 3}, {-2, -1}, {1, 2}}, {{-3, -2}, {-1, 1}, {2, 3}}, {{-3, -2}, {-1, 3}, {1, 2}}}, and we see that the total number of entries of absolute value 1 that appear among the partitions in this basis is a(3) = 10.

R. Sedgewick and P. Flajolet, Analysis of Algorithms, Addison Wesley, 1996, p. 225.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
Paul Barry, Riordan Pseudo-Involutions, Continued Fractions and Somos 4 Sequences, arXiv:1807.05794 [math.CO], 2018.
Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group, arXiv:2112.11595 [math.CO], 2021.
S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin and E. K. Lloyd, Enumeration of polyene hydrocarbons: a complete mathematical solution, J. Chem. Inf. Comput. Sci., 35 (1995), 743-751. [Annotated scanned copy]
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.
Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.

A002420 and A262543 are essentially the same sequence as this.

Cf. A000108, A000984, A002057, A040000, A068875, A071721.

[1] cat [2*Binomial( 2*n, n)/(n+1): n in [1..30]]; // Vincenzo Librandi, Oct 17 2017

Z:=(1-sqrt(1-4*x))/2/x: G:=(2-(1+x)*Z)/Z: Gser:=series(-G, x=0, 30): (1,seq(coeff(Gser, x, n), n=2..26)); # Zerinvary Lajos, Dec 23 2006
Z:=-(1-z-sqrt(1-z))/sqrt(1-z): Zser:=series(Z, z=0, 32): (1,seq(coeff(Zser*4^n, z, n), n=2..26)); # Zerinvary Lajos, Jan 01 2007
A068875List := proc(m) local A, P, n; A := [1, 2]; P := [2];
for n from 1 to m - 2 do P := ListTools:-PartialSums([op(P), P[-1]]);
A := [op(A), P[-1]] od; A end: A068875List(26); # Peter Luschny, Mar 24 2022

nn=30;t=(1-(1-4x )^(1/2))/(2x);Prepend[Table[Coefficient[Series[1+x (y t -y+1)^2,{x,0,nn}],x ^n y],{n,2,nn}],1]  (* Geoffrey Critzer, Jan 05 2013 *)
a[ n_] := If[ n < 1, Boole[ n == 0], 2 Binomial[ 2 n, n]/(n + 1)]; (* Michael Somos, Jun 18 2014 *)
a[ n_] := SeriesCoefficient[ -1 + 4 / (1 + Sqrt[ 1 - 4 x]), {x, 0, n}]; (* Michael Somos, Jun 18 2014 *)
Table[If[n==0, 1, 2 CatalanNumber[n]], {n,0,25}] (* Peter Luschny, Feb 27 2017 *)

{a(n) = if( n<1, n==0, 2 * binomial( 2*n, n) / (n + 1))}; /* Michael Somos, Aug 17 2005 */

{a(n) = if( n<0, 0, polcoeff( -1 + 4 / (1 + sqrt(1 - 4*x + x * O(x^n))), n))}; /* Michael Somos, Aug 17 2005 */

Apart from initial term, twice Catalan numbers.
G.f.: (1 - x - sqrt(1 - 4*x)) / x. - Michael Somos, Apr 13 2012
From Paul Barry, Nov 14 2004: (Start)
G.f.: (1 + x*c(x))/(1 - x*c(x)), where c(x) is the g.f. of A000108.
a(n) = C(n)*(2-0^n), where C(n) = A000108(n).
a(n) = Sum_{j=0..n} Sum_{k=0..n} binomial(2*n, n-k) *((2*k + 1)/(n + k + 1)) * binomial(k, j) * (-1)^(j-k) * (2 - 0^j). (End)
Assuming offset 1, then series reversion of g.f. A(x) is -A(-x). - Michael Somos, Aug 17 2005
Assuming offset 2, then A(x) satisfies A(x - x^2) = x^2 - x^4 and so A(x) = C(x)^2 - C(x)^4, A(A(x)) = C(x)^4 - C(x)^8, A(A(A(x))) = C(x)^8 - C(x)^16, etc., where C(x) = (1-sqrt(1-4*x))/2 = x + x^2 + 2*x^3 + 5*x^4 + 14*x^5 + ... . - Paul D. Hanna, May 16 2008
Apart from initial term, INVERTi transform of A000984(n+1) = binomial(2*n+2,n+1), also, for n >= 1, a(n) = (1/Pi)*Integral_{x=0..4} x^(n-1)*sqrt(x*(4 - x)). - Groux Roland, Mar 15 2011
D-finite with recurrence (n+2)*a(n) - 2*(2*n+1)*a(n-1) = 0, n > 1. - R. J. Mathar, Nov 14 2011
For n > 0, a(n) = C(2*n+2, n+1) mod 4*C(2*n, n - 1). - Robert G. Wilson v, May 02 2012
For n > 0, a(n) = 2^(2*n + 1) * Gamma(n + 1/2)/(sqrt(Pi) * (n + 1)!). - Vaclav Kotesovec, Sep 16 2013
G.f.: 1 + 2*x/(Q(0) - x), where Q(k) = 2*x + (k + 1)/(2*k + 1) - 2*x*(k + 1)/(2*k + 1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Dec 03 2013
G.f.: 3 - 4*x - 2*S(0), where S(k) = 2*k + 1 - x*(2*k + 3)/(1 - x*(2*k + 1)/S(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Dec 23 2013
0 = a(n)*(16*a(n+1) - 10*a(n+2)) + a(n+1)*(2*a(n+1) + a(n+2)) for all n in Z. - Michael Somos, Jun 18 2014
If A(x)^t = 1 + 2*t*x + Sum_{n >= 2} t*P(n,t)*x^n, then we conjecture that all the zeros of the polynomial P(n,t) lie on the vertical line Re(t) = -n/2 in the complex plane. - Peter Bala, Oct 05 2015
a(n+1) = a(n) + (1/2)*(Sum_{k=0..n} a(k)*a(n-k)) if n > 0. - Michael Somos, Apr 22 2022
b(n) = a(n+1) - a(n) for all n in Z if b(0) = 2, a(-1) = -1, a(0) = 0, a(-1) = 3, a(-2) = -1 where b = A071721. - Michael Somos, Apr 23 2022

A002421 Expansion of (1-4*x)^(3/2) in powers of x.

Original entry on oeis.org

1, -6, 6, 4, 6, 12, 28, 72, 198, 572, 1716, 5304, 16796, 54264, 178296, 594320, 2005830, 6843420, 23571780, 81880920, 286583220, 1009864680, 3580429320, 12765008880, 45741281820, 164668614552, 595340375688, 2160865067312, 7871722745208, 28772503827312

Offset: 0

N. J. A. Sloane

Terms that are not divisible by 12 have indices in A019469. - Ralf Stephan, Aug 26 2004
From Ralf Steiner, Apr 06 2017: (Start)
By analytic continuation to the entire complex plane there exist regularized values for divergent sums such as:
Sum_{k>=0} a(k)^2/8^k = 2F1(-3/2,-3/2,1,2).
Sum_{k>=0} a(k) / 2^k = -i. (End)

			G.f. = 1 - 6*x + 6*x^2 + 4*x^3 + 6*x^4 + 12*x^5 + 28*x^6 + 72*x^7 + 198*x^8 + 572*x^9 + ...

A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 55.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
T. N. Thiele, Interpolationsrechnung. Teubner, Leipzig, 1909, p. 164.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, Notes on A984 and A2420-A2424

Cf. A007054, A004001, A002420, A002422, A002423, A002424.

Cf. A000257, A001622, A071721, A071724, A085687.

Concatenation([1], List([1..40], n-> 12*Factorial(2*n-4) /( Factorial(n)*Factorial(n-2)) )) # G. C. Greubel, Jul 03 2019

[1,-6] cat [12*Catalan(n-2)/n: n in [2..30]]; // Vincenzo Librandi, Jun 11 2012

A002421 := n -> 3*4^(n-1)*GAMMA(-3/2+n)/(sqrt(Pi)*GAMMA(1+n)):
seq(A002421(n), n=0..29); # Peter Luschny, Dec 14 2015

CoefficientList[Series[(1-4x)^(3/2),{x,0,40}],x] (* Vincenzo Librandi, Jun 11 2012 *)
a[n_]:= Binomial[ 3/2, n] (-4)^n; (* Michael Somos, Dec 04 2013 *)
a[n_]:= SeriesCoefficient[(1-4x)^(3/2), {x, 0, n}]; (* Michael Somos, Dec 04 2013 *)

{a(n) = binomial( 3/2, n) * (-4)^n}; /* Michael Somos, Dec 04 2013 */

{a(n) = if( n<0, 0, polcoeff( (1 - 4*x + x * O(x^n))^(3/2), n))}; /* Michael Somos, Dec 04 2013 */

((1-4*x)^(3/2)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Jul 03 2019

a(n) = Sum_{m=0..n} binomial(n, m)*K_m(4), where K_m(x) = K_m(n, 2, x) is a Krawtchouk polynomial. - Alexander Barg (abarg(AT)research.bell-labs.com)
a(n) ~ (3/4)*Pi^(-1/2)*n^(-5/2)*2^(2*n)*(1 + 15/8*n^-1 + ...). - Joe Keane (jgk(AT)jgk.org), Nov 22 2001
From Ralf Stephan, Mar 11 2004: (Start)
a(n) = 12*(2*n-4)! /(n!*(n-2)!), n > 1.
a(n) = 12*Cat(n-2)/n = 2(Cat(n-1) - 4*Cat(n-2)), in terms of Catalan numbers (A000108).
Terms that are not divisible by 12 have indices in A019469. (End)
Let rho(x)=(1/Pi)*(x*(4-x))^(3/2), then for n >= 4, a(n) = Integral_{x=0..4} (x^(n-4) *rho(x)) dx. - Groux Roland, Mar 16 2011
G.f.: (1-4*x)^(3/2) = 1 - 6*x + 12*x^2/(G(0) + 2*x); G(k) = (4*x+1)*k-2*x+2-2*x*(k+2)*(2*k+1)/G(k+1); for -1/4 <= x < 1/4, otherwise G(0) = 2*x; (continued fraction). - Sergei N. Gladkovskii, Dec 05 2011
G.f.: 1/G(0) where G(k) = 1 + 4*x*(2*k+1)/(1 - 1/(1 + (2*k+2)/G(k+1))); (continued fraction, 3-step). - Sergei N. Gladkovskii, Nov 18 2012
G.f.: G(0)/2, where G(k) = 2 + 2*x*(2*k-3)*G(k+1)/(k+1). - Sergei N. Gladkovskii, Jun 06 2013 [Edited by Michael Somos, Dec 04 2013]
0 = a(n+2) * (a(n+1) - 14*a(n)) + a(n+1) * (6*a(n+1) + 16*a(n)) for all n in Z. - Michael Somos, Dec 04 2013
A232546(n) = 3^n * a(n). - Michael Somos, Dec 04 2013
G.f.: hypergeometric1F0(-3/2;;4*x). - R. J. Mathar, Aug 09 2015
a(n) = 3*4^(n-1)*Gamma(-3/2+n)/(sqrt(Pi)*Gamma(1+n)). - Peter Luschny, Dec 14 2015
From Ralf Steiner, Apr 06 2017: (Start)
Sum_{k>=0} a(k)/4^k = 0.
Sum_{k>=0} a(k)^2/16^k = 32/(3*Pi).
Sum_{k>=0} a(k)^2*(k/8)/16^k = 1/Pi.
Sum_{k>=0} a(k)^2*(-k/24+1/8)/16^k = 1/Pi.
Sum_{k>=0} a(k-1)^2*(k-1/4)/16^k = 1/Pi.
Sum_{k>=0} a(k-1)^2*(2k-2)/16^k = 1/Pi.(End)
D-finite with recurrence: n*a(n) +2*(-2*n+5)*a(n-1)=0. - R. J. Mathar, Feb 20 2020
From Amiram Eldar, Mar 22 2022: (Start)
Sum_{n>=0} 1/a(n) = 4/3 + 10*Pi/(81*sqrt(3)).
Sum_{n>=0} (-1)^n/a(n) = 92/75 - 4*sqrt(5)*log(phi)/125, where phi is the golden ratio (A001622). (End)

A071722 Expansion of (1+x^2C^2)C^3, where C = (1-(1-4x)^(1/2))/(2x) is g.f. for Catalan numbers, A000108.

Original entry on oeis.org

1, 3, 10, 33, 110, 372, 1276, 4433, 15574, 55250, 197676, 712538, 2585292, 9434830, 34610400, 127553745, 472055910, 1753616370, 6536826780, 24443315550, 91664179620, 344655239760, 1299052403688, 4907335827258, 18576824685820

Offset: 0

N. J. A. Sloane, Jun 06 2002

nonn

gf=(1+x^2*C^2)*C^m: A000782 (m=1), A071721 (m=2), this sequence (m=3), A071723 (m=4).

Cf. A000108.

a := n -> (17*n^2 + 13*n + 6)*binomial(2*n, n)/((n + 1)*(n + 2)*(n + 3)): seq(a(n), n = 0..24);  # Peter Luschny, Dec 01 2024

a(n):=(3*binomial(2*n+2,n)+5*binomial(2*n,n+2))/(n+3); makelist(a(n),n,0,50);
/* Tani Akinari, Dec 01 2024 */

Conjecture: 2*(n+3)*a(n) + 4*(-3*n-4)*a(n-1) + (17*n-9)*a(n-2) + 2*(-2*n+5)*a(n-3) = 0. - R. J. Mathar, Aug 25 2013

a(n) = (3*binomial(2*n+2, n) + 5*binomial(2*n, n+2))/(n + 3). - Tani Akinari, Dec 01 2024

A071723 Expansion of (1+x^2C^2)C^4, where C = (1-(1-4x)^(1/2))/(2x) is g.f. for Catalan numbers, A000108.

Original entry on oeis.org

1, 4, 15, 54, 192, 682, 2431, 8710, 31382, 113696, 414086, 1515516, 5571750, 20569590, 76228095, 283481670, 1057628550, 3957577800, 14849601090, 55859886420, 210622646520, 795898303668, 3013646759910, 11432740177564, 43448822603452, 165396657221152

Offset: 0

N. J. A. Sloane, Jun 06 2002

nonn

gf=(1+x^2*C^2)*C^m: A000782 (m=1), A071721 (m=2), A071722 (m=3), this sequence (m=4).

Cf. A000108.

a := n -> (2*(2*n + 1)*(11*n^2 + 17*n + 12)*binomial(2*n, n))/((n + 1)*(n + 2)*(n + 3)*(n + 4)): seq(a(n), n = 0..25); # Peter Luschny, Dec 01 2024

a(n):=sum((k+1)*(k^2+k+1)*binomial(2*n-k,n),k,0,n)/(n+1); /* Vladimir Kruchinin, Sep 28 2011 */

a(n):=(4*binomial(2*n+3,n)+6*binomial(2*n+1,n+3))/(n+4); /* Tani Akinari, Dec 01 2024 */

a(n) = (Sum_{k=0..n} (k+1)*(k^2+k+1)*binomial(2*n-k,n))/(n+1). - Vladimir Kruchinin, Sep 28 2011
a(n) = (4*binomial(2*n+3,n)+6*binomial(2*n+1,n+3))/(n+4). - Tani Akinari, Dec 01 2024
D-finite with recurrence 2*(n+4)*a(n) +2*(-7*n-16)*a(n-1) +3*(9*n+4)*a(n-2) +6*(-2*n+3)*a(n-3)=0. - R. J. Mathar, Jul 13 2025

A319251 Triangle read by rows: T(n,k) is the number of permutations pi of [n] with k descents such that s(pi) avoids the patterns 132, 231, 312, and 321, where s denotes West's stack-sorting map.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 8, 8, 1, 1, 13, 28, 13, 1, 1, 19, 70, 70, 19, 1, 1, 26, 145, 250, 145, 26, 1, 1, 34, 266, 700, 700, 266, 34, 1, 1, 43, 448, 1666, 2548, 1666, 448, 43, 1, 1, 53, 708, 3528, 7644, 7644, 3528, 708, 53, 1

Offset: 1

Colin Defant, Sep 15 2018

Rows are symmetric, so T(n,k) = T(n,n-1-k).

T(n,k) is the number of permutations of [n] that avoid the patterns 1342, 2341, 3142, 3241, 3412, and 3421 and have k descents.

			Triangle begins:
  1,
  1, 1,
  1, 4, 1,
  1, 8, 8, 1,
  1, 13, 28, 13, 1,
  1, 19, 70, 70, 19, 1
  ...

Colin Defant, Stack-sorting preimages of permutation classes, arXiv:1809.03123 [math.CO], 2018.

Row sums give A071721. Cf. A001263, A319252.

Flatten[Table[Table[(1/n) Binomial[n, m + 1] Binomial[n, m] + Sum[Sum[(1/(n - i - 1)) Binomial[n - i - 1, j] Binomial[n - i - 1, j - 1] (1/i) Binomial[i, m - j + 1] Binomial[i, m - j], {j, 1, m}], {i, 1, n - 2}], {m, 0, n - 1}], {n, 1, 10}]]

T(n,k) = N(n,k+1) + Sum_{i=1..n-2} Sum_{j=1..m} N(n-i-1,j) * N(i,k-j+1), where N(i,j) = (1/i) * binomial(i,j) * binomial(i,j-1) are the Narayana numbers given in A001263.
From Vladimir Kruchinin, Nov 16 2020: (Start)
G.f.: A001263(x,y) + x*A001263(x,y)^2.
T(n,k) = N(n,k) +2*C(n-1,k-2)*C(n-1,k)/(n-1). (End)

A319252 Triangle read by rows: T(n,k) is the number of permutations pi of [n] with k+1 valleys such that s(pi) avoids the patterns 132, 231, 312, and 321, where s denotes West's stack-sorting map (0 <= k <= floor((n-1)/2)).

Original entry on oeis.org

1, 2, 4, 2, 8, 10, 16, 36, 4, 32, 112, 36, 64, 320, 200, 10, 128, 864, 880, 130, 256, 2240, 3360, 980, 28, 512, 5632, 11648, 5600, 476, 1024, 13824, 37632, 26880, 4536, 84, 2048, 33280, 115200, 114240, 31920, 1764

Offset: 1

Colin Defant, Sep 15 2018

T(n,k) is the number of permutations of [n] that avoid the patterns 1342, 2341, 3142, 3241, 3412, and 3421 and have k+1 valleys.

			Triangle begins:
   1;
   2;
   4,   2;
   8,  10;
  16,  36,  4;
  32, 112, 36;
  ...

C. Defant, Stack-sorting preimages of permutation classes, arXiv:1809.03123 [math.CO], 2018.

Row sums give A071721. Cf. A091894, A319251.

Flatten[Table[Table[(2^(n - 2 (m + 1) + 1)) Binomial[n - 1, 2 m] CatalanNumber[m] + Sum[Sum[(2^((n - i - 1) - 2 j + 1)) Binomial[n - i - 2, 2 j - 2] CatalanNumber[j - 1] (2^(i - 2 (m - j + 1) + 1)) Binomial[i - 1, 2 (m - j + 1) - 2] CatalanNumber[m - j], {j, 1, m}], {i, 1, n - 2}], {m, 0, Floor[(n - 1)/2]}], {n, 1, 12}]]

T(n,k) = V(n,k+1) + Sum_{i=1..n-2} Sum_{j=1..m} V(n-i-1,j) * V(i,k-j+1), where V(i,j) = 2^{i-2j+1} * (1/j) * binomial(i-1,2j-2) * binomial(2j-2,j-1) are the numbers given in A091894.

A163946 Triangle read by rows, A033184 * A091768 (diagonalized as an infinite lower triangular matrix).

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 5, 5, 6, 6, 14, 14, 18, 24, 22, 42, 42, 56, 84, 110, 92, 132, 132, 180, 288, 440, 552, 426, 429, 429, 594, 990, 1650, 2484, 2982, 2150, 1430, 1430, 2002, 3432, 6050, 10120, 14910, 17200, 11708, 4862, 4862, 6864, 12012, 22022, 39468, 65604, 94600, 105372, 68282

Offset: 0

Gary W. Adamson, Aug 06 2009

As an eigentriangle, equals A033184 * the diagonalized version of its eigensequence. (the eigensequence of triangle A033184 = A091768).
Right border = A091768, left border = Catalan sequence A000108.
Sum of n-th row terms = rightmost term of next row.

			First few rows of the triangle =
  1;
  1, 1;
  2, 2, 2;
  5, 5, 6, 6;
  14, 14, 18, 24, 22;
  42, 42, 56, 84, 110, 92;
  132, 132, 180, 288, 440, 552, 426;
  429, 429, 594, 990, 1650, 2484, 2982, 2150;
  1430, 1430, 2002, 3432, 6050, 10120, 14910, 17200, 11708;
  4862, 4862, 6864, 12012, 22022, 39468, 65604, 94600, 105372, 68282;
  ...
Row 3 = (5, 5, 6, 6) = (5, 5, 3, 1) * (1, 1, 2, 6); where (5, 5, 3, 1) = row 3 of triangle A033184 and (1, 1, 2, 6) = the first 3 terms of A091768 prefaced with a 1.

Cf. A033184, A091768, A000108, A071721.

Triangle read by rows, A033184 * A091768 (diagonalized such that the right border = (1, 1, 2, 6, 22, 92, 426, 2150,...) i.e. A091768 prefaced with a 1; with the rest zeros).

A114503 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n for which height of first peak + height of last peak = k (n>=1; 2<=k<=2n).

Original entry on oeis.org

1, 1, 0, 1, 1, 2, 1, 0, 1, 2, 4, 4, 2, 1, 0, 1, 5, 10, 11, 8, 4, 2, 1, 0, 1, 14, 28, 32, 26, 16, 8, 4, 2, 1, 0, 1, 42, 84, 98, 84, 57, 32, 16, 8, 4, 2, 1, 0, 1, 132, 264, 312, 276, 198, 120, 64, 32, 16, 8, 4, 2, 1, 0, 1, 429, 858, 1023, 924, 687, 438, 247, 128, 64, 32, 16, 8, 4, 2, 1, 0, 1

Offset: 1

Emeric Deutsch, Dec 02 2005

Row n has 2n-1 terms. Column 2 yields the Catalan numbers (A000108). T(n,3)=2T(n,2) (n>=3). Sum(kT(n,k),k=2..2n)=2[Catalan(n+1)-Catalan(n)] (A071721). The trivariate g.f., with z marking semilength, t marking height of the first peak and s marking height of the last peak, is G = (1-tzC-szC+tsz^2*C^2+tsz^2*C)/[(1-tzC)(1-szC)(1-tsz)]-1.

			T(5,6)=4 because we have UUDUUUDDDD, UUUDUDUDDD, UUUDDUUDDD and UUUUDDDUDD, where U=(1,1), D=(1,-1).
Triangle starts:
  1;
  1,0,1;
  1,2,1,0,1;
  2,4,4,2,1,0,1;
  5,10,11,8,4,2,1,0,1;
  ...

Krishna Menon and Anurag Singh, Grassmannian permutations avoiding identity, arXiv:2212.13794 [math.CO], 2022.

Cf. A000108, A071721.

C:=(1-sqrt(1-4*z))/2/z: g:=(1-2*t*z*C+t^2*z^2*C^2+t^2*z^2*C)/(1-t*z*C)^2/(1-t^2*z)-1: gser:=simplify(series(g,z=0,12)): for n from 1 to 10 do P[n]:=coeff(gser,z^n) od: for n from 1 to 10 do seq(coeff(P[n],t^j),j=2..2*n) od; # yields sequence in triangular form

G.f.: (1-2tzC+t^2*z^2*C^2+t^2*z^2*C)/[(1-tzC)^2*(1-t^2*z)]-1, where C=[1-sqrt(1-4z)]/(2z) is the Catalan function.

cp's OEIS Frontend

A068875 Expansion of (1 + x*C)*C, where C = (1 - (1 - 4*x)^(1/2))/(2*x) is the g.f. for Catalan numbers, A000108.

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A002421 Expansion of (1-4*x)^(3/2) in powers of x.

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A071722 Expansion of (1+x^2*C^2)*C^3, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108.

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A071723 Expansion of (1+x^2*C^2)*C^4, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108.

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A319251 Triangle read by rows: T(n,k) is the number of permutations pi of [n] with k descents such that s(pi) avoids the patterns 132, 231, 312, and 321, where s denotes West's stack-sorting map.

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A319252 Triangle read by rows: T(n,k) is the number of permutations pi of [n] with k+1 valleys such that s(pi) avoids the patterns 132, 231, 312, and 321, where s denotes West's stack-sorting map (0 <= k <= floor((n-1)/2)).

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A163946 Triangle read by rows, A033184 * A091768 (diagonalized as an infinite lower triangular matrix).

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A068875 Expansion of (1 + xC)C, where C = (1 - (1 - 4x)^(1/2))/(2x) is the g.f. for Catalan numbers, A000108.

A071722 Expansion of (1+x^2C^2)C^3, where C = (1-(1-4x)^(1/2))/(2x) is g.f. for Catalan numbers, A000108.

A071723 Expansion of (1+x^2C^2)C^4, where C = (1-(1-4x)^(1/2))/(2x) is g.f. for Catalan numbers, A000108.