A025227 -id:A025227 - cp's OEIS frontend

A052709 Expansion of g.f. (1-sqrt(1-4x-4x^2))/(2*(1+x)).

0, 1, 1, 3, 9, 31, 113, 431, 1697, 6847, 28161, 117631, 497665, 2128127, 9183489, 39940863, 174897665, 770452479, 3411959809, 15181264895, 67833868289, 304256253951, 1369404661761, 6182858317823, 27995941060609, 127100310290431, 578433619525633, 2638370120138751

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

A simple context-free grammar.
Number of lattice paths from (0,0) to (2n-2,0) that stay (weakly) in the first quadrant and such that each step is either U=(1,1), D=(1,-1), or L=(3,1). Equivalently, underdiagonal lattice paths from (0,0) to (n-1,n-1) and such that each step is either (1,0), (0,1), or (2,1). E.g., a(4)=9 because in addition to the five Dyck paths from (0,0) to (6,0) [UDUDUD, UDUUDD, UUDDUD, UUDUDD, UUUDDD] we have LDUD, LUDD, ULDD and UDLD. - Emeric Deutsch, Dec 21 2003
Hankel transform of a(n+1) is A006125(n+1). - Paul Barry, Apr 01 2007
Also, a(n+1) is the number of walks from (0,0) to (n,0) using steps (1,1), (1,-1) and (0,-1). See the U(n,k) array in A071943, where A052709(n+1) = U(n,0). - N. J. A. Sloane, Mar 29 2013
Diagonal sums of triangle in A085880. - Philippe Deléham, Nov 15 2013
From Gus Wiseman, Jun 17 2021: (Start)
Conjecture: For n > 0, also the number of sequences of length n - 1 covering an initial interval of positive integers and avoiding three terms (..., x, ..., y, ..., z, ...) such that x <= y <= z. The version avoiding the strict pattern (1,2,3) is A226316. Sequences covering an initial interval are counted by A000670. The a(1) = 1 through a(4) = 9 sequences are:
  ()  (1)  (1,1)  (1,2,1)
           (1,2)  (1,3,2)
           (2,1)  (2,1,1)
                  (2,1,2)
                  (2,1,3)
                  (2,2,1)
                  (2,3,1)
                  (3,1,2)
                  (3,2,1)
(End)

N. J. A. Sloane, Table of n, a(n) for n = 0..499
Marilena Barnabei, Flavio Bonetti, Niccolò Castronuovo, and Matteo Silimbani, Ascending runs in permutations and valued Dyck paths, Ars Mathematica Contemporanea (2019) Vol. 16, No. 2, 445-463.
Paul Barry, Riordan arrays, generalized Narayana triangles, and series reversion, Linear Algebra and its Applications, 491 (2016) 343-385.
Paul Barry, On a transformation of Riordan moment sequences, arXiv:1802.03443 [math.CO], 2018.
Paul Barry, Characterizations of the Borel triangle and Borel polynomials, arXiv:2001.08799 [math.CO], 2020.
Daniel Birmajer, Juan B. Gil, David S. Kenepp, and Michael D. Weiner, Restricted generating trees for weak orderings, arXiv:2108.04302 [math.CO], 2021.
Daniel Birmajer, Juan B. Gil, Peter R. W. McNamara, and Michael D. Weiner, Enumeration of colored Dyck paths via partial Bell polynomials, arXiv:1602.03550 [math.CO], 2016.
Xiang-Ke Chang, X.-B. Hu, H. Lei, and Y.-N. Yeh, Combinatorial proofs of addition formulas, The Electronic Journal of Combinatorics, 23(1) (2016), #P1.8.
Brian Drake, Limits of areas under lattice paths, Discrete Math. 309 (2009), no. 12, 3936-3953.
M. Dziemianczuk, On Directed Lattice Paths With Additional Vertical Steps, arXiv preprint arXiv:1410.5747 [math.CO], 2014.
James East and Nicholas Ham, Lattice paths and submonoids of Z^2, arXiv:1811.05735 [math.CO], 2018.
L. Ferrari, E. Pergola, R. Pinzani, and S. Rinaldi, Jumping succession rules and their generating functions, Discrete Math., 271 (2003), 29-50.
Nancy S. S. Gu, Nelson Y. Li, and Toufik Mansour, 2-Binary trees: bijections and related issues, Discr. Math., 308 (2008), 1209-1221.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 664
J. P. S. Kung and A. de Mier, Catalan lattice paths with rook, bishop and spider steps, Journal of Combinatorial Theory, Series A 120 (2013) 379-389. - _N. J. A. Sloane_, Dec 27 2012
D. Merlini, D. G. Rogers, R. Sprugnoli, and M. C. Verri, On some alternative characterizations of Riordan arrays, Canad. J. Math., 49 (1997), 301-320.

Diagonal entries of A071943 and A071945.
Cf. A000108, A000670, A025227, A052709, A056986, A071943, A085880.
Cf. A102726, A117434, A158005, A226316, A333217, A335479.

[0] cat [(&+[Binomial(n,k+1)*Binomial(2*k,n-1): k in [0..n-1]])/n: n in [1..30]]; // G. C. Greubel, May 30 2022

spec := [S,{C=Prod(B,Z),S=Union(B,C,Z),B=Prod(S,S)},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);

InverseSeries[Series[(y-y^2)/(1+y^2), {y, 0, 24}], x] (* then A(x)= y(x) *) (* Len Smiley, Apr 12 2000 *)
CoefficientList[Series[(1 -Sqrt[1 -4x -4x^2])/(2(1+x)), {x, 0, 33}], x] (* Vincenzo Librandi, Feb 12 2016 *)

a(n)=polcoeff((1-sqrt(1-4*x*(1+x+O(x^n))))/2/(1+x),n)

[sum(binomial(k, n-k-1)*catalan_number(k) for k in (0..n-1)) for n in (0..30)] # G. C. Greubel, May 30 2022

a(n) + a(n-1) = A025227(n).
a(n) = Sum_{k=0..floor((n-1)/2)} (2*n-2-2*k)!/(k!*(n-k)!*(n-1-2*k)!). - Emeric Deutsch, Nov 14 2001
D-finite with recurrence: n*a(n) = (3*n-6)*a(n-1) + (8*n-18)*a(n-2) + (4*n-12)*a(n-3), n>2. a(1)=a(2)=1.
a(n) = b(1)*a(n-1) + b(2)*a(n-2) + ... + b(n-1)*a(1) for n>1 where b(n)=A025227(n).
G.f.: A(x) = x/(1-(1+x)*A(x)). - Paul D. Hanna, Aug 16 2002
G.f.: A(x) = x/(1-z/(1-z/(1-z/(...)))) where z=x+x^2 (continued fraction). - Paul D. Hanna, Aug 16 2002; revised by Joerg Arndt, Mar 18 2011
a(n+1) = Sum_{k=0..n} Catalan(k)*binomial(k, n-k). - Paul Barry, Feb 22 2005
From Paul Barry, Mar 14 2006: (Start)
G.f. is x*c(x*(1+x)) where c(x) is the g.f. of A000108.
Row sums of A117434. (End)
a(n+1) = (1/(2*Pi))*Integral_{x=2-2*sqrt(2)..2+2*sqrt(2)} x^n*(4+4x-x^2)/(2*(1+x)). - Paul Barry, Apr 01 2007
From Gary W. Adamson, Jul 22 2011: (Start)
For n>0, a(n) is the upper left term in M^(n-1), where M is an infinite square production matrix as follows:
  1, 1, 0, 0, 0, 0, ...
  2, 1, 1, 0, 0, 0, ...
  2, 2, 1, 1, 0, 0, ...
  2, 2, 2, 1, 1, 0, ...
  2, 2, 2, 2, 1, 1, ...
  ... (End)
G.f.: x*Q(0), where Q(k) = 1 + (4*k+1)*x*(1+x)/(k+1 - x*(1+x)*(2*k+2)*(4*k+3)/(2*x*(1+x)*(4*k+3) + (2*k+3)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 14 2013
a(n) ~ sqrt(2-sqrt(2))*2^(n-1/2)*(1+sqrt(2))^(n-1)/(n^(3/2)*sqrt(Pi)). - Vaclav Kotesovec, Jun 29 2013
a(n+1) = Sum_{k=0..floor(n/2)} A085880(n-k,k). - Philippe Deléham, Nov 15 2013

Better g.f. and recurrence from Michael Somos, Aug 03 2000

More terms from Larry Reeves (larryr(AT)acm.org), Oct 03 2000

A366266 G.f. A(x) satisfies A(x) = 1 + x + x*A(x)^3.

Original entry on oeis.org

1, 2, 6, 30, 170, 1050, 6846, 46374, 323154, 2301618, 16680246, 122607342, 911868282, 6849381194, 51885977838, 395941193718, 3040818657954, 23485437201762, 182297207394150, 1421357996034750, 11126867651367498, 87421958424703098, 689130671539597854

Offset: 0

Seiichi Manyama, Oct 06 2023

nonn

Cf. A025227, A366267, A366268.

Cf. A366221, A366364.

a(n) = sum(k=0, n, binomial(2*k+1, n-k)*binomial(3*k, k)/(2*k+1));

a(n) = Sum_{k=0..n} binomial(2*k+1,n-k) * binomial(3*k,k)/(2*k+1).
a(n) = A366221(n) + A366221(n-1).
G.f.: A(x) = 1/B(-x) where B(x) is the g.f. of A366364.

A366267 G.f. A(x) satisfies A(x) = 1 + x + x*A(x)^4.

Original entry on oeis.org

1, 2, 8, 56, 448, 3920, 36288, 349440, 3464448, 35125760, 362522624, 3795914240, 40224968704, 430579701760, 4648899846144, 50568103690240, 553632271155200, 6096025799852032, 67464070696927232, 750003531943903232, 8371814935842258944

Offset: 0

Seiichi Manyama, Oct 06 2023

nonn

Cf. A025227, A366266, A366268.

Cf. A366272, A366365.

nmax = 20; A[_] = 1;
Do[A[x_] = 1 + x + x*A[x]^4 + O[x]^(nmax+1) // Normal, {nmax+1}];
CoefficientList[A[x], x] (* Jean-François Alcover, Mar 03 2024 *)

a(n) = sum(k=0, n, binomial(3*k+1, n-k)*binomial(4*k, k)/(3*k+1));

a(n) = Sum_{k=0..n} binomial(3*k+1,n-k) * binomial(4*k,k)/(3*k+1).
a(n) = A366272(n) + A366272(n-1).
G.f.: A(x) = 1/B(-x) where B(x) is the g.f. of A366365.

A068764 Generalized Catalan numbers 2xA(x)^2 -A(x) +1 -x =0.

Original entry on oeis.org

1, 1, 4, 18, 88, 456, 2464, 13736, 78432, 456416, 2697088, 16141120, 97632000, 595912960, 3665728512, 22703097472, 141448381952, 885934151168, 5575020435456, 35230798994432, 223485795258368, 1422572226146304, 9083682419818496, 58169612565614592, 373486362257899520, 2403850703479816192

Offset: 0

Wolfdieter Lang, Mar 04 2002

a(n) = K(2,2; n)/2 with K(a,b; n) defined in a comment to A068763.

Hankel transform is A166232(n+1). - Paul Barry, Oct 09 2009

			G.f. = 1 + x + 4*x^2 + 18*x^3 + 88*x^4 + 456*x^5 + 2464*x^6 + 13736*x^7 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A025227, A025228, A025229, A025230.

Cf. A071356, A001003, A025235.

Table[SeriesCoefficient[(1-Sqrt[1-8*x*(1-x)])/(4*x),{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 13 2012 *)
Round@Table[4^(n-1) Hypergeometric2F1[(1-n)/2, 1-n/2, 2, 1/2] + KroneckerDelta[n]/Sqrt[2], {n, 0, 20}] (* Vladimir Reshetnikov, Nov 07 2015 *)
a[ n_] := If[ n < 1, Boole[n == 0], 4^(n - 1) Hypergeometric2F1[ (1 - n)/2, (2 - n)/2, 2, 1/2]]; (* Michael Somos, Nov 08 2015 *)

a(n):=sum(binomial(n-1,k-1)*1/k*sum(binomial(k,j)*binomial(k+j,j-1),j,1,k),k,1,n); /* Vladimir Kruchinin, Aug 11 2010 */

{a(n) = my(A); if( n<1, n==0, n--;  A = x * O(x^n); n! * simplify( polcoeff( exp(4*x + A) * besseli(1, 2*x * quadgen(8) + A), n)))}; /* Michael Somos, Mar 31 2007 */

x='x+O('x^66); Vec((1-sqrt(1-8*x*(1-x)))/(4*x)) \\ Joerg Arndt, May 06 2013

G.f.: (1-sqrt(1-8*x*(1-x)))/(4*x).
a(n+1) = 2*sum(a(k)*a(n-k), k=0..n), n>=1, a(0) = 1 = a(1).
a(n) = (2^n)*p(n, -1/2) with the row polynomials p(n, x) defined from array A068763.
E.g.f. (offset -1) is exp(4*x)*BesselI(1, 2*sqrt(2)*x)/(sqrt(2)*x). - Vladeta Jovovic, Mar 31 2004
The o.g.f. satisfies A(x) = 1 + x*(2*A(x)^2 - 1), A(0) = 1. - Wolfdieter Lang, Nov 13 2007
a(n) = subs(t=1,(d^(n-1)/dt^(n-1))(-1+2*t^2)^n)/n!, n >= 2, due to the Lagrange series for the given implicit o.g.f. equation. This formula holds also for n=1 if no differentiation is used. - Wolfdieter Lang, Nov 13 2007, Feb 22 2008
1/(1-x/(1-x-2x/(1-x/(1-x-2x/(1-x/(1-x-2x/(1-..... (continued fraction). - Paul Barry, Jan 29 2009
a(n) = A166229(n)/(2-0^n). - Paul Barry, Oct 09 2009
a(n) = sum(binomial(n-1,k-1)*1/k*sum(binomial(k,j)*binomial(k+j,j-1),j,1,k),k,1,n), n>0. - Vladimir Kruchinin, Aug 11 2010
D-finite with recurrence: (n+1)*a(n) = 4*(2*n-1)*a(n-1) - 8*(n-2)*a(n-2). - Vaclav Kotesovec, Oct 13 2012
a(n) ~ sqrt(1+sqrt(2))*(4+2*sqrt(2))^n/(2*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 13 2012
a(n) = 4^(n-1)*hypergeom([(1-n)/2,1-n/2], [2], 1/2) + 0^n/sqrt(2). - Vladimir Reshetnikov, Nov 07 2015
0 = a(n)*(+64*a(n+1) - 160*a(n+2) + 32*a(n+3)) + a(n+1)*(+32*a(n+1) + 48*a(n+2) - 20*a(n+3)) + a(n+2)*(+4*a(n+2) + a(n+3)) for all n>=0. - Michael Somos, Nov 08 2015
a(n) = (-1)^n * Sum_{k=0..n} (-2)^k * binomial(n,k) * binomial(2*k+1,n) / (2*k+1). - Seiichi Manyama, Jul 24 2023

A219534 G.f. satisfies A(x) = 1 + x*(A(x)^2 + A(x)^4).

Original entry on oeis.org

1, 2, 12, 100, 968, 10208, 113792, 1318832, 15732064, 191878592, 2381917824, 29995598208, 382257383168, 4920505410816, 63882881030656, 835554927932160, 10999486798112256, 145626782310460416, 1937772463214168064, 25901381584638605312, 347618773649248088064

Offset: 0

Paul D. Hanna, Nov 21 2012

nonn

			G.f.: A(x) = 1 + 2*x + 12*x^2 + 100*x^3 + 968*x^4 + 10208*x^5 +...
Related expansions:
A(x)^2 = 1 + 4*x + 28*x^2 + 248*x^3 + 2480*x^4 + 26688*x^5 +...
A(x)^4 = 1 + 8*x + 72*x^2 + 720*x^3 + 7728*x^4 + 87104*x^5 +...
The g.f. satisfies A(x) = G(x*A(x)^2) and G(x) = A(x/G(x)^2) where
G(x) = 1 + 2*x + 4*x^2 + 12*x^3 + 40*x^4 + 144*x^5 + 544*x^6 +...+ A025227(n+1)*x^n +...

G. C. Greubel, Table of n, a(n) for n = 0..850

Cf. A025227, A219535, A219536, A219537, A219538.

Cf. A006318, A364167.

nmax=20; aa=ConstantArray[0,nmax]; aa[[1]]=2; Do[AGF=1+Sum[aa[[n]]*x^n,{n,1,j-1}]+koef*x^j; sol=Solve[Coefficient[1+x*(AGF^2+AGF^4)-AGF,x,j]==0,koef][[1]];aa[[j]]=koef/.sol[[1]],{j,2,nmax}]; Flatten[{1,aa}] (* Vaclav Kotesovec, Sep 10 2013 *)

/* Formula A(x) = 1 + x*(A(x)^2 + A(x)^4): */
{a(n)=local(A=1);for(i=1,n,A=1+x*(A^2+A^4) +x*O(x^n));polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))

/* Formula using Series Reversion: */
{a(n)=local(A=1,G=(1-sqrt(1-4*x-4*x^2+x^3*O(x^n)))/(2*x));A=sqrt((1/x)*serreverse(x/G^2));polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))

Let G(x) = (1 - sqrt(1-4*x-4*x^2))/(2*x), then g.f. A(x) satisfies:
(1) A(x) = sqrt( (1/x)*Series_Reversion(x/G(x)^2) ),
(2) A(x) = G(x*A(x)^2) and G(x) = A(x/G(x)^2),
where x*G(x) is the g.f. of A025227.
Recurrence: 3*n*(3*n-1)*(3*n+1)*(131*n^3 - 666*n^2 + 1075*n - 558)*a(n) = 2*(26200*n^6 - 172500*n^5 + 431572*n^4 - 521613*n^3 + 316327*n^2 - 89058*n + 8640)*a(n-1) - 12*(n-2)*(1441*n^5 - 8767*n^4 + 19186*n^3 - 18172*n^2 + 6930*n - 810)*a(n-2) + 8*(n-3)*(n-2)*(2*n-5)*(131*n^3 - 273*n^2 + 136*n - 18)*a(n-3). - Vaclav Kotesovec, Sep 10 2013
a(n) ~ c*d^n/n^(3/2), where d = 2/81*(7217783 + 10611 * sqrt(786))^(1/3) + 74654/(81*(7217783 + 10611 * sqrt(786))^(1/3)) + 400/81 = 14.48001092254652246... is the root of the equation -16 + 132*d - 400*d^2 + 27*d^3 = 0 and c = 1/2358*sqrt(262)*sqrt((213070976 + 3034746 * sqrt(786))^(1/3) * ((213070976 + 3034746 * sqrt(786))^(2/3) + 336670 + 1310*(213070976 + 3034746 * sqrt(786))^(1/3)))/((213070976 + 3034746 * sqrt(786))^(1/3)*sqrt(Pi)) = 0.1929450901182412149... - Vaclav Kotesovec, Sep 10 2013
a(n) = (1/n) * Sum_{k=0..floor(n-1)/2} 2^(n-k) * binomial(n,k) * binomial(3*n-k,n-1-2*k) for n > 0. - Seiichi Manyama, Apr 01 2024
a(n) = Sum_{k=0..n} binomial(n,k) * binomial(2*n+2*k+1,n)/(2*n+2*k+1). - Seiichi Manyama, Apr 03 2024

A007971 INVERTi transform of central trinomial coefficients (A002426).

Original entry on oeis.org

0, 1, 2, 2, 4, 8, 18, 42, 102, 254, 646, 1670, 4376, 11596, 31022, 83670, 227268, 621144, 1706934, 4713558, 13072764, 36398568, 101704038, 285095118, 801526446, 2259520830, 6385455594, 18086805002, 51339636952, 146015545604

Offset: 0

David Dumas (dumas(AT)TCNJ.EDU)

nonn

Number of paths of a walk on the integers, allowing steps of size 0, +1, and -1, which return to the starting point for the first time at time n. [David P. Sanders (dps(AT)fciencias.unam.mx), May 04 2009]

			G.f. = x + 2*x^2 + 2*x^3 + 4*x^4 + 8*x^5 + 18*x^6 + 42*x^7 + 102*x^8 + 254*x^9 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group and Chebyshev polynomials, arXiv:2502.13673 [math.CO], 2025.

Cf. A001006, A002426, A005043.

Cf. A025227.

CoefficientList[Series[1-Sqrt[1-2x-3x^2],{x,0,40}],x] (* Harvey P. Dale, Dec 17 2012 *)
a[1]:=1;a[2]:=2;a[n_]:=a[n]=1/2 Sum[a[k] a[n-k],{k,1,n-1}];
Join[{0},Map[a,Range[24]]] (* Oliver Seipel, Nov 03 2024, after Schröder 1870 *)

x='x+O('x^50); concat([0], Vec(1 - sqrt(1 - 2*x - 3*x^2))) \\ G. C. Greubel, Feb 26 2017

A002426(n) = Sum_{i=1..n} a(i)*A002426(n-i), n>0. - Michael Somos, Jun 14 2000
G.f.: 1 - sqrt(1 - 2*x - 3*x^2). - Michael Somos, Jun 14 2000
a(0)=0, a(1)=1, a(2)=2, then a(n) = (1/2) *(a(1)*a(n-1)+a(2)*a(n-2)+....+a(n-1)*a(1)). - Benoit Cloitre, Oct 24 2003
a(n) = 2^(1-n)*Sum_{k=1..n} (binomial(k,n-k)*A000108(k-1)*3^(n-k)), n>0. - Vladimir Kruchinin, Feb 05 2011
G.f.: 1-sqrt(1-2*x-3*(x^2)) =  x/G(0) ; G(k) = 1-2*x/(1+x/(1+x/(1-2*x/(1-x/(2-x/G(k+1)))))) ; (continued fraction). - Sergei N. Gladkovskii, Dec 11 2011
a(n+2) = 2 * A001006(n). - Michael Somos, Jun 14 2000
For n>1, a(n) = 2 * (A005043(n-1) + A005043(n-2)). - Ralf Stephan, Jul 06 2003
0 = a(n) * (9*a(n+1) + 15*a(n+2) - 12*a(n+3)) + a(n+1) * (-3*a(n+1) + 10*a(n+2) - 5*a(n+3)) + a(n+2) * (a(n+2) + a(n+3)) for all n>0. - Michael Somos, Jan 25 2014
n*a(n) + (-2*n+3)*a(n-1) + *(-n+3)*a(n-2) = 0. - R. J. Mathar, Sep 06 2016

Name corrected by Michael Somos, Mar 23 2012

A366268 G.f. A(x) satisfies A(x) = 1 + x + x*A(x)^5.

Original entry on oeis.org

1, 2, 10, 90, 930, 10530, 126282, 1576410, 20268930, 266591490, 3569991370, 48509238810, 667157894050, 9269347395490, 129908752970890, 1834347364277530, 26071297610067970, 372683901080814850, 5354668071305293450, 77286026066830771930

Offset: 0

Seiichi Manyama, Oct 06 2023

nonn

Cf. A025227, A366266, A366267.

Cf. A366273, A366366.

a(n) = sum(k=0, n, binomial(4*k+1, n-k)*binomial(5*k, k)/(4*k+1));

a(n) = Sum_{k=0..n} binomial(4*k+1,n-k) * binomial(5*k,k)/(4*k+1).
a(n) = A366273(n) + A366273(n-1).
G.f.: A(x) = 1/B(-x) where B(x) is the g.f. of A366366.

A068763 Irregular triangle of the Fibonacci polynomials of A011973 multiplied diagonally by the Catalan numbers.

Original entry on oeis.org

1, 1, 1, 2, 2, 5, 6, 1, 14, 20, 6, 42, 70, 30, 2, 132, 252, 140, 20, 429, 924, 630, 140, 5, 1430, 3432, 2772, 840, 70, 4862, 12870, 12012, 4620, 630, 14, 16796, 48620, 51480, 24024, 4620, 252, 58786, 184756, 218790

Offset: 0

Wolfdieter Lang, Mar 04 2002

The row length sequence of this array is [1,2,2,3,3,4,4,5,5,...] = A008619(n+2), n>=0.
The row polynomials p(n,x) := Sum_{m=0..floor((n+1)/2)} a(n,m)*x^m produce, for x = (b-a^2)/a^2 (not 0), the two parameter family of sequences K(a,b; n) := (a^(n+1))*p(n,(b-a^2)/a^2) with g.f. K(a,b; x) := (1-sqrt(1-4*x*(a+x*(b-a^2))))/(2*x).
Some members are: K(1,1; n)=A000108(n) (Catalan), K(1,2; n)=A025227(n-1), K(2,1; n)=A025228(n-1), K(1,3; n)=A025229(n-1), K(3,1; n)=A025230(n-1). For a=b=2..10 the sequences K(a,a; n)/a are A068764-A068772.
The column sequences (without leading 0's) are: A000108 (Catalan), A000984 (central binomial), A002457, 2*A002802, 5*A020918, 14*A020920, 42*A020922, ...
a(n,m) is the number of ways to designate exactly m cherries over all binary trees with n internal nodes.  A cherry is an internal node whose descendants are both external nodes.  Cf. A091894 which gives the number of binary trees with m cherries. - Geoffrey Critzer, Jul 24 2020
This irregular triangle is essentially that of A011973 with its diagonals multiplied by the Catalan numbers of A000108. The diagonals of this triangle are then rows of the Pascal matrix A007318 multiplied by the Catalan numbers. - Tom Copeland, Dec 23 2023

			The irregular triangle begins:
   n\m    0     1     2     3    4   5
   0:     1
   1:     1     1
   2:     2     2
   3:     5     6     1
   4:    14    20     6
   5:    42    70    30     2
   6:   132   252   140    20
   7:   429   924   630   140    5
   8:  1430  3432  2772   840   70
   9:  4862 12870 12012  4620  630  14
  10: 16796 48620 51480 24024 4620 252
  ...
p(3,x) = 5 + 6*x + x^2.

P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 213.
Tad White, Quota Trees, arXiv:2401.01462 [math.CO], 2024. See p. 20.

Cf. A025227(n-1) (row sums).
Cf. A000007(n) (alternating row sums).
Cf. A000108, A001263, A007318, A011973, A033832, A034867, A133437 and A134264.

nn = 10; b[z_] := (1 - Sqrt[1 - 4 z])/(2 z);Map[Select[#, # > 0 &] &,
CoefficientList[Series[v b[v z] /. v -> (1 + u z ), {z, 0, nn}], {z, u}]] // Grid (* Geoffrey Critzer, Jul 24 2020 *)

a(n, m) = binomial(n+1-m, m)*C(n-m) if 0 <= m <= floor((n+1)/2), otherwise 0, with C(n) := A000108(n) (Catalan).
G.f. for column m=1, 2, ...: (x^(2*m-1))*C(m-1)/(1-4*x)^((2*m-1)/2); m=0: c(x), g.f. for A000108 (Catalan).
G.f. for row polynomials p(n, x): c(z) + x*z*c(x*(z^2)/(1-4*z))/sqrt(1-4*z) = (1-sqrt(1-4*z*(1+x*z)))/(2*z), where c(x) is the g.f. of A000108 (Catalan).
G.f. for triangle: (1 - sqrt(1 - 4*x (1 + y*x)))/(2*x). - Geoffrey Critzer, Jul 24 2020
The alternating row sums are {1,repeat 0} = {A000007(n)}{n>=0}. From the second comment p(n, x=-1) = K(1,0; n), with g.f. K(1,0; x) = 1. Or from the g.f. of the triangle with y = -1. Thanks to Aaron Li for asking for a proof. - _Wolfdieter Lang, Jan 21 2023
The series expansion of f(x) = (1 + 2sx - sqrt(1 + 4sx + 4d^2x^2))/(2x) at x = 0 is (s^2 - d^2) x + (2 d^2s - 2 s^3) x^2 + (d^4 - 6 d^2 s^2 + 5 s^4) x^3  + (-6 d^4 s + 20 d^2 s^3 - 14 s^5) x^4 + ..., containing the coefficients of this array. With s = (a+b)/2 and d = (a-b)/2, then f(x)/ab = g(x) = (1 + (a+b)x - sqrt((1+(a+b)x)^2 - 4abx^2))/(2abx) = x - (a + b) x^2 + (a^2 + 3 a b + b^2) x^3 - (a^3 + 6 a^2 b + 6 a b^2 + b^3) x^4 + ..., containing the Narayana polynomials of A001263, which can be simply transformed into A033282. The compositional inverse about the origin of g(x) is g^(-1)(x) = x/((1-ax)(1-bx)) = x/((1-(s+d)x)(1-(s-d)x)) = x + (a + b) x^2 + (a^2 + a b + b^2) x^3 + (a^3 + a^2 b + a b^2 + b^3) x^4 + ..., containing the complete homogeneous symmetric polynomials h_n(a,b) = (a^n - b^n)/(a-b), which are the polynomials of A034867 when expressed in s and d, e.g., ((s + d)^7 - (s - d)^7)/(2 d) = d^6 + 21 d^4 s^2 + 35 d^2 s^4 + 7 s^6. A133437 and A134264 for compositional inversion of o.g.f.s can be used to relate the sets of polynomials above. - Tom Copeland, Nov 28 2023

Title changed by Tom Copeland, Dec 23 2023

A068772 Generalized Catalan numbers 10xA(x)^2 -A(x) +1 -9*x =0.

Original entry on oeis.org

1, 1, 20, 410, 8600, 184200, 4020000, 89205000, 2008700000, 45816140000, 1056825200000, 24618524200000, 578457724000000, 13695679012000000, 326448619920000000, 7827776361090000000, 188701194087000000000

Offset: 0

Wolfdieter Lang, Mar 04 2002

This is the tenth member in the a-family of sequences K(a,a; n), a=1,2,3,...,n>=0, defined in a comment to the array A068763.

Fung Lam, Table of n, a(n) for n = 0..700

Cf. A000108, A068764-A068771, A025227-A025230.

a[0] = 1; a[1] = 1; a[n_] := (360 (2 - n) a[n - 2] + 20 (2 n - 1) a[n - 1])/(n + 1); Table[a[n], {n, 0, 20}] (* Wesley Ivan Hurt, Mar 04 2014 *)
CoefficientList[Series[(1-Sqrt[1-40*x*(1-9*x)])/(20*x), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 06 2014 *)

a(n) = (10^n) * p(n, -9/10) with the row polynomials p(n, x) defined from array A068763.
a(n+1) = 10*sum(a(k)*a(n-k), k=0..n), n>=1, a(0)=1=a(1).
G.f.: (1-sqrt(1-40*x*(1-9*x)))/(20*x).
Recurrence: (n+1)*a(n) = 360*(2-n)*a(n-2) + 20*(2*n-1)*a(n-1). - Fung Lam, Mar 05 2014
a(n) ~ sqrt(5+5*sqrt(10)) * (20+2*sqrt(10))^n / (10*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 06 2014

A068766 Generalized Catalan numbers 4xA(x)^2 -A(x)+1-3*x=0.

Original entry on oeis.org

1, 1, 8, 68, 608, 5664, 54528, 538944, 5441024, 55889408, 582348800, 6140864512, 65414742016, 702897995776, 7609805045760, 82929151328256, 908978855215104, 10014523823357952, 110840574196580352, 1231847926116384768

Offset: 0

Wolfdieter Lang, Mar 04 2002

a(n)=K(4,4; n)/4 with K(a,b; n) defined in a comment to A068763.

Fung Lam, Table of n, a(n) for n = 0..925

Cf. A000108, A068764-5, A068767-72, A025227-30.

a := n -> `if`(n=0,1,simplify(2^n*GegenbauerC(n-1, -n, -2))/(2*n)):
seq(a(n), n=0..19); # Peter Luschny, May 09 2016

CoefficientList[Series[(1-Sqrt[1-16*x*(1-3*x)])/(8*x), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 04 2014 *)

a(n)=(4^n)*p(n, -3/4) with the row polynomials p(n, x) defined from array A068763.
a(n+1)= 4*sum(a(k)*a(n-k), k=0..n), n>=1, a(0)=1=a(1).
G.f.: (1-sqrt(1-16*x*(1-3*x)))/(8*x).
Recurrence: (n+1)*a(n) = 48*(2-n)*a(n-2) + 8*(2*n-1)*a(n-1). - Fung Lam, Mar 04 2014
a(n) ~ sqrt(6) * 12^n / (4*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 04 2014
a(n) = 2^n*GegenbauerC(n-1, -n, -2)/(2*n) for n>=1. - Peter Luschny, May 09 2016

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A052709 Expansion of g.f. (1-sqrt(1-4x-4x^2))/(2*(1+x)).

A068764 Generalized Catalan numbers 2xA(x)^2 -A(x) +1 -x =0.

A068772 Generalized Catalan numbers 10xA(x)^2 -A(x) +1 -9*x =0.

A068766 Generalized Catalan numbers 4xA(x)^2 -A(x)+1-3*x=0.