A000108 -id:A000108 - cp's OEIS frontend

A126984 Expansion of 1/(1+2xc(x)), c(x) the g.f. of Catalan numbers A000108.

1, -2, 2, -4, 2, -12, -12, -72, -190, -700, -2308, -8120, -28364, -100856, -360792, -1301904, -4727358, -17268636, -63405012, -233885784, -866327748, -3220976616, -12016209192, -44966763504, -168750724428, -634935132312, -2394717424552, -9051945482032

Offset: 0

Philippe Deléham, Mar 21 2007

Hankel transform is (-2)^n.

Hankel transform omitting first term is (-2)^n omitting first term. Hankel transform omitting first two terms is 2*(-1)^n*A014480(n). - Michael Somos, May 16 2022

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

Cf. A000108, A014480, A039599, A268600, A268601.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 1/(2-Sqrt(1-4*x)) )); // G. C. Greubel, May 28 2019

c:=(1-sqrt(1-4*x))/2/x: ser:=series(1/(1+2*x*c),x=0,32): seq(coeff(ser,x,n),n=0..30); # Emeric Deutsch, Mar 24 2007

CoefficientList[Series[1/(2-Sqrt[1-4*x]), {x,0,30}], x] (* G. C. Greubel, May 28 2019 *)

my(x='x+O('x^30)); Vec(1/(2-sqrt(1-4*x))) \\ G. C. Greubel, May 28 2019

(1/(2-sqrt(1-4*x))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 28 2019

a(n) = Sum_{k=0..n} A039599(n,k)*(-3)^k.
G.f.: 1/(2 - sqrt(1-4*x)). - G. C. Greubel, May 28 2019
(-1)^n*a(n) = A268600(n) - A268601(n). - Michael Somos, May 16 2022
D-finite with recurrence 3*n*a(n) +2*(-4*n+9)*a(n-1) +8*(-2*n+3)*a(n-2)=0. - R. J. Mathar, Nov 22 2024
a(n) = Sum_{k = 0..n} A009766(n-1, k)*(-2)^(n-k) for n >= 1. - Peter Bala, Jun 18 2025

Corrected and extended by Emeric Deutsch, Mar 24 2007

A141223 Expansion of 1/(sqrt(1-4x)(1-3xc(x))), where c(x) is the g.f. of A000108.

Original entry on oeis.org

1, 5, 24, 113, 526, 2430, 11166, 51105, 233190, 1061510, 4822984, 21879786, 99135076, 448707992, 2029215114, 9170247393, 41416383366, 186957126702, 843575853984, 3804927658878, 17156636097156, 77339426905812, 348553445817084, 1570548863858778, 7075531788285276

Offset: 0

Paul Barry, Jun 14 2008

Binomial transform of A126932. Hankel transform is (-1)^n.

Row sums of the Riordan matrix (1/(1-4*x),(1-sqrt(1-4*x))/(2*sqrt(1-4*x))) (A188481). - Emanuele Munarini, Apr 01 2001

Michael De Vlieger, Table of n, a(n) for n = 0..1000
Paul Barry, On a Central Transform of Integer Sequences, arXiv:2004.04577 [math.CO], 2020.
Isaac DeJager, Madeleine Naquin, Frank Seidl, Colored Motzkin Paths of Higher Order, VERUM 2019.

Cf. A000108.
Cf. A000302, A000984, A026641.
Cf. A006256, A385605, A385632.

CoefficientList[Series[(3-12x+Sqrt[1-4x])/(4-34x+72x^2),{x,0,100}],x] (* Emanuele Munarini, Apr 01 2011 *)

makelist(sum(binomial(n+k,k)*3^(n-k),k,0,n),n,0,12); /* Emanuele Munarini, Apr 01 2011 */

a(n) = Sum_{k=0..n} C(2*n-k,n-k)*3^k.
From Emanuele Munarini, Apr 01 2011: (Start)
a(n) = [x^n] 1/((1-x)^(n+1) * (1-3*x)). [Corrected by Seiichi Manyama, Aug 03 2025]
a(n) = 3^(2*n+1)/2^(n+2) + (1/4)*Sum_{k=0..n} binomial(2*k,k)*(9/2)^(n-k).
D-finite with recurrence: 2*(n+2)*a(n+2) - (17*n+30)*a(n+1) + 18*(2*n+3)*a(n) = 0.
G.f.: (3-12*x+sqrt(1-4*x))/(4-34*x+72*x^2). (End)
G.f.: (1/(1-4*x)^(1/2)+3)/(4-18*x) = (2 + x/(Q(0)-2*x))/(2-9*x) where Q(k) = 2*(2*k+1)*x + (k+1) - 2*(k+1)*(2*k+3)*x/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Mar 18 2013
a(n) ~ 3^(2*n + 1) / 2^(n + 1). - Vaclav Kotesovec, Sep 15 2021
a(n) = Sum_{k=0..n} 2^(n-k) * binomial(2*n+1,k). - Seiichi Manyama, Aug 03 2025
a(n) = 3^(2*n+1)*2^(-n-1) - binomial(2*n+1, n)*(hypergeom([1, -1-n], [1+n], -1/2) - 1). - Stefano Spezia, Aug 05 2025
a(n) = Sum_{k=0..n} 3^k * (-2)^(n-k) * binomial(2*n+1,k) * binomial(2*n-k,n-k). - Seiichi Manyama, Aug 07 2025

A158832 Main diagonal in the array A158825 of coefficients of successive iterations of x*C(x), where C(x) is the Catalan function (A000108).

Original entry on oeis.org

1, 2, 12, 110, 1330, 19852, 351792, 7209036, 167607066, 4357308098, 125219900520, 3941126688798, 134808743674176, 4979127855477336, 197480359402576304, 8370550907396970684, 377599345119560766534, 18061714498169627460982

Offset: 1

Paul D. Hanna, Mar 28 2009

nonn

Triangle A158835 transforms A158831 into this sequence, where A158831 is the previous diagonal in A158825.

Triangle A158835 transforms this sequence into A158833, the next diagonal in A158825.

			Array of coefficients in the i-th iteration of x*Catalan(x):
(1),1,2,5,14,42,132,429,1430,4862,16796,58786,208012,...;
1,(2),6,21,80,322,1348,5814,25674,115566,528528,2449746,...;
1,3,(12),54,260,1310,6824,36478,199094,1105478,6227712,...;
1,4,20,(110),640,3870,24084,153306,993978,6544242,43652340,...;
1,5,30,195,(1330),9380,67844,500619,3755156,28558484,...;
1,6,42,315,2464,(19852),163576,1372196,11682348,100707972,...;
1,7,56,476,4200,38052,(351792),3305484,31478628,303208212,...;
1,8,72,684,6720,67620,693048,(7209036),75915708,807845676,...;
1,9,90,945,10230,113190,1273668,14528217,(167607066),...;
1,10,110,1265,14960,180510,2212188,27454218,344320262,(4357308098),...; ...
where terms in parenthesis form the initial terms of this sequence.

Paul D. Hanna, Table of n, a(n), n = 1..50.

Cf. A158825, A158831, A158833, A158834.

a[n_] := Module[{x, F, G}, F = InverseSeries[x - x^2 + O[x]^(n+2)]; G = x; For[i = 1, i <= n, i++, G = (F /. x -> G)]; Coefficient[G, x, n]];
Array[a, 18] (* Jean-François Alcover, Jul 13 2018, from PARI *)

{a(n)=local(F=serreverse(x-x^2+O(x^(n+2))),G=x); for(i=1,n,G=subst(F,x,G));polcoeff(G,n)}

A161629 E.g.f. satisfies: A(x) = exp( xCatalan(xA(x)) ), where Catalan(x) = (1-sqrt(1-4x))/(2x) is the g.f. of A000108.

Original entry on oeis.org

1, 1, 3, 25, 349, 6821, 171421, 5265625, 191160201, 8007548617, 380157603481, 20171371753061, 1182973489103869, 75984447924612397, 5305029326492409333, 400014338565211619761, 32396515980658185762961

Offset: 0

Paul D. Hanna, Jun 17 2009

nonn

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 25*x^3/3! + 349*x^4/4! + 6821*x^5/5! +...
log(A(x))/x = 1 + x*A(x) + 2*x^2*A(x)^2 + 5*x^3*A(x)^3 + 14*x^4*A(x)^4 +...+ A000108(n)*x^n*A(x)^n +...
log(A(x))/x = 1 + x + 6*x^2/2! + 63*x^3/3! + 988*x^4/4! + 20725*x^5/5! +...+ A213644(n)*x^n/n! +...
log(A(x)) = x + 2*x^2/2! + 18*x^3/3! + 252*x^4/4! + 4940*x^5/5! +...+ A213643(n)*x^n/n! +...
Ordinary Generating Function:
O.g.f.: 1 + x + 3*x^2 + 25*x^3 + 349*x^4 + 6821*x^5 + 171421*x^6 +...
O.g.f.: 1 + x/(1-x) + 2*x^2/(1-2*x)^3 + 6*2!*x^3/(1-3*x)^5 + 20*3!*x^4/(1-4*x)^7 + 70*4!*x^5/(1-5*x)^9 + 252*5!*x^6/(1-6*x)^11 +...+ (2*n-2)!/(n-1)!*x^n/(1-n*x)^(2*n-1) +...

Cf. A213643 (log), A214689, A000108.

Flatten[{1,Table[Sum[n!/k!*(n-k+1)^(k-1)*Binomial[2*n-k, n-k]*k/(2*n-k),{k,0,n}],{n,1,20}]}] (* Vaclav Kotesovec, Feb 26 2014 *)

{a(n,m=1)=if(n==0,1,sum(k=0,n,n!/k!*m*(m+n-k)^(k-1)*binomial(2*n-k,n-k)*k/(2*n-k)))}

{a(n,m=1)=local(A=1+x+x*O(x^n));for(i=1,n,A=exp((1-sqrt(1-4*x*A))/(2*A)));n!*polcoeff(A^m,n)}

/* O.g.f.: */
{a(n)=polcoeff(1+sum(m=1, n, (2*m-2)!/(m-1)!*x^m/(1-m*x+x*O(x^n))^(2*m-1)), n)}

E.g.f.: A(x) = exp(F(x)) where F(x) = x + F(x)^2*exp(F(x)) is the e.g.f. of A213643.
E.g.f.: A(x) = Sum_{n>=0} a(n)*x^n/n!, where
a(n) = Sum_{k=0..n} n! * (n-k+1)^(k-1)/k! * C(2*n-k,n-k)*k/(2*n-k).
Let A(x)^m = Sum_{n>=0} a(n,m)*x^n/n!, then
a(n,m) = Sum_{k=0..n} n! * m*(n-k+m)^(k-1)/k! * C(2*n-k,n-k)*k/(2*n-k).
...
O.g.f.: A(x) = 1 + Sum_{n>=1} (2*n-2)!/(n-1)! * x^n/(1 - n*x)^(2*n-1).
a(n) ~ n^(n-1) * sqrt((r*s^3*(1-6*r*s+8*r^2*s^2)) / (1 - (-2+8*r+r^2)*s + 4*r*(-4+4*r+r^2)*s^2 + 4*r^2*(8+r)*s^3)) / (exp(n) * r^n), where s = 1.370489293947401403417767032... is the root of the equation log(s)*(1-s*log(s)) + 2*(1+s) = (1+2*s) * sqrt((1+s)/s), and r = log(s)*(1-s*log(s)) = 0.179036084709909351719214... - Vaclav Kotesovec, Feb 26 2014

A355341 G.f.: A(x) = Sum_{n=-oo..+oo} x^(n(n+1)/2) C(x)^n, where C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers (A000108).

Original entry on oeis.org

2, 1, -2, 1, -3, 0, 1, -4, 2, 0, 1, -5, 5, 0, 0, 1, -6, 9, -2, 0, 0, 1, -7, 14, -7, 0, 0, 0, 1, -8, 20, -16, 2, 0, 0, 0, 1, -9, 27, -30, 9, 0, 0, 0, 0, 1, -10, 35, -50, 25, -2, 0, 0, 0, 0, 1, -11, 44, -77, 55, -11, 0, 0, 0, 0, 0, 1, -12, 54, -112, 105, -36, 2, 0, 0, 0, 0, 0, 1, -13, 65, -156, 182, -91, 13, 0, 0, 0, 0, 0, 0, 1, -14, 77, -210, 294, -196, 49, -2, 0, 0, 0, 0, 0, 0, 1, -15, 90, -275, 450, -378, 140, -15, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Paul D. Hanna, Jul 21 2022

sign

			G.f.: A(x) = 2 + x - 2*x^2 + x^3 - 3*x^4 + x^6 - 4*x^7 + 2*x^8 + x^10 - 5*x^11 + 5*x^12 + x^15 - 6*x^16 + 9*x^17 - 2*x^18 + x^21 - 7*x^22 + 14*x^23 - 7*x^24 + x^28 - 8*x^29 + 20*x^30 - 16*x^31 + 2*x^32 + x^36 - 9*x^37 + 27*x^38 - 30*x^39 + 9*x^40 + x^45 - 10*x^46 + 35*x^47 - 50*x^48 + 25*x^49 - 2*x^50 + ...
such that
A(x) = ... + x^6/C(x)^4 + x^3/C(x)^3 + x/C(x)^2 + 1/C(x) + 1 + x*C(x) + x^3*C(x)^2 + x^6*C(x)^3 + x^10*C(x)^4 + ...
where
C(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 + 429*x^7 + 1430*x^8 + 4862*x^9 + 16796*x^10 + ... + A000108(n)*x^n + ...
The coefficients of x^k in x^(n*(n+1)/2) * (C(x)^n + 1/C(x)^(n+1)) begin:
n = 0: [2, -1, -1, -2, -5, -14, -42, -132, -429, -1430, -4862, -16796, ...];
n = 1: [0,  2, -1,  1,  3,   9,  28,   90,  297,  1001,  3432,  11934, ...];
n = 2: [0,  0,  0,  2, -1,   5,  13,   39,  123,   401,  1340,   4565, ...];
n = 3: [0,  0,  0,  0,  0,   0,   2,   -1,   11,    28,    89,    293, ...];
n = 4: [0,  0,  0,  0,  0,   0,   0,    0,    0,     0,     2,     -1, ...]; ...
forming a table the column sums of which yield this sequence.
The g.f. may also be written as
A(x) = 2 + (-2*x + 1)*x + (-3*x + 1)*x^3 + (2*x^2 - 4*x + 1)*x^6 + (5*x^2 - 5*x + 1)*x^10 + (-2*x^3 + 9*x^2 - 6*x + 1)*x^15 + (-7*x^3 + 14*x^2 - 7*x + 1)*x^21 + (2*x^4 - 16*x^3 + 20*x^2 - 8*x + 1)*x^28 + (9*x^4 - 30*x^3 + 27*x^2 - 9*x + 1)*x^36 + (-2*x^5 + 25*x^4 - 50*x^3 + 35*x^2 - 10*x + 1)*x^45 + ...
compare to
(1 - 2*y*x)/(1-x + y*x^2) = 1 + (-2*y + 1)*x + (-3*y + 1)*x^2 + (2*y^2 - 4*y + 1)*x^3 + (5*y^2 - 5*y + 1)*x^4 + (-2*y^3 + 9*y^2 - 6*y + 1)*x^5 + (-7*y^3 + 14*y^2 - 7*y + 1)*x^6 + (2*y^4 - 16*y^3 + 20*y^2 - 8*y + 1)*x^7 + (9*y^4 - 30*y^3 + 27*y^2 - 9*y + 1)*x^8 + (-2*y^5 + 25*y^4 - 50*y^3 + 35*y^2 - 10*y + 1)*x^9 + ...
The terms of this sequence may be written as a triangle (see triangle A244422):
2,
1, -2,
1, -3, 0,
1, -4, 2, 0,
1, -5, 5, 0, 0,
1, -6, 9, -2, 0, 0,
1, -7, 14, -7, 0, 0, 0,
1, -8, 20, -16, 2, 0, 0, 0,
1, -9, 27, -30, 9, 0, 0, 0, 0,
1, -10, 35, -50, 25, -2, 0, 0, 0, 0,
1, -11, 44, -77, 55, -11, 0, 0, 0, 0, 0,
1, -12, 54, -112, 105, -36, 2, 0, 0, 0, 0, 0,
1, -13, 65, -156, 182, -91, 13, 0, 0, 0, 0, 0, 0,
1, -14, 77, -210, 294, -196, 49, -2, 0, 0, 0, 0, 0, 0,
1, -15, 90, -275, 450, -378, 140, -15, 0, 0, 0, 0, 0, 0, 0,
1, -16, 104, -352, 660, -672, 336, -64, 2, 0, 0, 0, 0, 0, 0, 0,
...

Paul D. Hanna, Table of n, a(n) for n = 0..1275

Cf. A244422, A355342, A355343.

{a(n) = my(A,C = serreverse(x-x^2 +x^2*O(x^n))/x);
A = sum(m=-n-1,n+1, x^(m*(m+1)/2) * C^m); polcoeff(A,n)}
for(n=0,70,print1(a(n),", "))

{a(n) = my(A,C = serreverse(x-x^2 +x^2*O(x^n))/x, M = sqrtint(2*n+9));
A = sum(m=0,M, x^(m*(m+1)/2) * (C^m + 1/C^(m+1))); polcoeff(A,n)}
for(n=0,70,print1(a(n),", "))

G.f. A(x) = Sum_{n>=0} a(n) * x^n is equal to the following expressions; here, C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers (A000108).
(1) A(x) = 1/C(x) * Product_{n>=1} (1 + x^n/C(x)) * (1 + x^(n-1)*C(x)) * (1-x^n), by the Jacobi triple product identity.
(2) A(x) = Sum_{n=-oo..+oo} x^(n*(n+1)/2) * C(x)^n.
(3) A(x) = Sum_{n>=0} x^(n*(n+1)/2) * (C(x)^n + 1/C(x)^(n+1)).
(4) A(x) = 1 + Sum_{n>=0} x^(n*(n+1)/2) * ( [y^n] (1 - 2*y*x)/(1-y + x*y^2) ).
(5) A(x) = 1 + Sum_{n>=1} x^(n*(n-1)/2) * Sum_{k=0..n} A244422(n,k) * x^k.

A381817 Expansion of (1/x) * Series_Reversion( x * (1-x) / C(x) ), where C(x) is the g.f. of A000108.

Original entry on oeis.org

1, 2, 8, 41, 239, 1507, 10016, 69123, 490676, 3560150, 26285896, 196862679, 1491921261, 11420072162, 88166571504, 685724643699, 5367842153463, 42259058503891, 334373741310812, 2657683458672907, 21209720057079565, 169886023881795700, 1365290865904393560

Offset: 0

Seiichi Manyama, Mar 07 2025

nonn

Cf. A381818, A381819, A381820.

Cf. A000108.

my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x)*2*x/(1-sqrt(1-4*x)))/x)

G.f. A(x) satisfies A(x) = C(x*A(x)) / (1 - x*A(x)).
a(n) = Sum_{k=0..n} binomial(n+2*k+1,k) * binomial(2*n-k,n-k)/(n+2*k+1).
D-finite with recurrence 270*n*(n-1)*(2*n+1)*(4886806261*n -12359738163)*(n+1)*a(n) +36*n*(n-1)*(73302093915*n^3 -4013759132354*n^2 +11228589268975*n -4731576382254)*a(n-1) -6*(n-1)*(78948725805818*n^4 -721014042837927*n^3 +2114039183987386*n^2 -2373558292742247*n +834825525358878)*a(n-2) +(3703469060597227*n^5 -40768871113864973*n^4 +173554734639707111*n^3 -360669855974794759*n^2 +370762762031723274*n -153683482287306096)*a(n-3) +6*(-2284895393144753*n^5 +28245013068548213*n^4 -138588666805096327*n^3 +341806596235129383*n^2 -433338949590369664*n +232825263110939100)*a(n-4) +10*(5*n-22)*(5*n-21) *(5*n-19)*(5*n-18)*(1032930487477*n -4077934418263)*a(n-5)=0. - R. J. Mathar, Mar 10 2025

A067336 a(0)=1, a(1)=2, a(n) = a(n-1)*9/2 - Catalan(n-1) where Catalan(n) = binomial(2n,n)/(n+1) = A000108(n).

Original entry on oeis.org

1, 2, 8, 34, 148, 652, 2892, 12882, 57540, 257500, 1153888, 5175700, 23231864, 104335376, 468766292, 2106773874, 9470787588, 42583186476, 191494694352, 861248485884, 3873850923288, 17425765034376, 78391476387672, 352670161180884, 1586672665700328, 7138737091504152

Offset: 0

Henry Bottomley, Jan 15 2002

nonn

Note that while a(n) is even (for n > 0), it is a multiple of 4 except when n = 2^m-1, i.e., when Catalan(n) is odd.
Result of applying the Riordan matrix ((1+sqrt(1-4*x))/2, (1-sqrt(1-4*x))/2) (inverse of (1/(1-x), x*(1-x))) to 3^n. - Paul Barry, Mar 12 2005
Hankel transform is A001787(n+1). - Paul Barry, Mar 15 2010

			a(2) =   2*9/2 -  1 =   8;
a(3) =   8*9/2 -  2 =  34;
a(4) =  34*9/2 -  5 = 148;
a(5) = 148*9/2 - 14 = 652.

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

Cf. A000108, A001045, A039599, A067337, A088218.

CoefficientList[Series[(1+Sqrt[1-4*x])/(3*Sqrt[1-4*x]-1), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 13 2014 *)

a(n) = A067337(2n, n).
G.f.: (1+sqrt(1-4*x))/(3*sqrt(1-4*x)-1). - Paul Barry, Mar 12 2005
a(n) = Sum_{k=0..n} A039599(n,k)*A001045(k+1). - Philippe Deléham, Jun 10 2007
G.f.: (1-x*c(x))/(1-3*x*c(x)), where c(x) is the g.f. of A000108. - Paul Barry, Mar 15 2010
Conjecture: 2*n*a(n) + (-17*n+12)*a(n-1) + 18*(2*n-3)*a(n-2) = 0. - R. J. Mathar, Nov 30 2012
The above conjecture is true. - Nguyen Tuan Anh, Mar 15 2025
G.f.: 1 + 2*x/(Q(0)-3*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
a(n) ~ 3^(2*n-1) / 2^n. - Vaclav Kotesovec, Feb 13 2014

A073191 Number of separate orbits/cycles to which the Catalan bijections A072796/A072797 partition each A000108(n) structures encoded in the range [A014137(n-1)..A014138(n-1)] of the sequence A014486/A063171.

Original entry on oeis.org

1, 1, 2, 4, 11, 31, 96, 305, 1007, 3389, 11636, 40498, 142714, 507870, 1823040, 6591885, 23989419, 87795473, 322922652, 1193058230, 4425547638, 16475756738, 61539293424, 230548633954, 866095934598, 3261868457698, 12313423931624

Offset: 0

Antti Karttunen, Jun 25 2002

nonn

Occurs for first time in A073201 as row 1.

a(n) = (A000108(n)+A073190(n))/2.

A073193 Number of separate orbits/cycles to which the Catalan bijection A057508 partitions each A000108(n) structures encoded in the range [A014137(n-1)..A014138(n-1)] of the sequence A014486/A063171.

Original entry on oeis.org

1, 1, 2, 4, 11, 30, 93, 292, 965, 3238, 11126, 38708, 136486, 485820, 1744677, 6310584, 22973793, 84103302, 309429066, 1143487428, 4242631626, 15798011604, 59018856522, 221143860936, 830895360978, 3129747395548, 11816242209260

Offset: 0

Antti Karttunen, Jun 25 2002

nonn

Occurs for first time in A073201 as row 168.

a(n) = (A000108(n)+A073192(n))/2

A076026 Expansion of g.f.: (1-4xC)/(1-5xC) where C = (1/2-1/2(1-4x)^(1/2))/x = g.f. for Catalan numbers A000108.

Original entry on oeis.org

1, 1, 6, 37, 230, 1434, 8952, 55917, 349374, 2183230, 13643972, 85270626, 532926716, 3330739972, 20816939100, 130105200765, 813155081070, 5082210417270, 31763782696740, 198523522444950, 1240771573465140, 7754820693127020, 48467623215477120, 302922622226091090

Offset: 0

N. J. A. Sloane, Oct 29 2002

a(n) is the number of Motzkin paths of length n-1 in which the (1,0)-steps at level 0 come in 6 colors and those at a higher level come in 2 colors. Example: a(4)=230 because, denoting U=(1,1), H=(1,0), and D=(1,-1), we have 6^3 = 216 paths of shape HHH, 6 paths of shape HUD, 6 paths of shape UDH, and 2 paths of shape UHD. - Emeric Deutsch, May 02 2011

L. W. Shapiro and C. J. Wang, Generating identities via 2 X 2 matrices, Congressus Numerantium, 205 (2010), 33-46.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Richard Ehrenborg, Gábor Hetyei, and Margaret Readdy, Catalan-Spitzer permutations, arXiv:2310.06288 [math.CO], 2023. See p. 20.

Cf. A000108, A001700, A049027, A076025.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (2- 4*Sqrt(1-4*x))/(3-5*Sqrt(1-4*x)) )); // G. C. Greubel, May 04 2019

CoefficientList[Series[(2-4*Sqrt[1-4*x])/(3-5*Sqrt[1-4*x]), {x, 0, 30}], x] (* Vaclav Kotesovec, Dec 09 2013 *)
Flatten[{1,Table[FullSimplify[(2*n)!*Hypergeometric2F1Regularized[1, n+1/2, n+2, 16/25] / (25*n!) + 3*5^(2*n-1)/4^(n+1)], {n,1,30}]}] (* Vaclav Kotesovec, Dec 09 2013 *)

my(x='x+O('x^30)); Vec((2-4*sqrt(1-4*x))/(3-5*sqrt(1-4*x))) \\ G. C. Greubel, May 04 2019

((2-4*sqrt(1-4*x))/(3-5*sqrt(1-4*x))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 04 2019

a(n+1) = Sum_{k=0..n} A039598(n,k)*4^k. - Philippe Deléham, Mar 21 2007
a(n) = Sum_{k=0..n} A039599(n,k)*A015521(k), for n >= 1. - Philippe Deléham, Nov 22 2007
Let A be the Toeplitz matrix of order n defined by: A[i,i-1]=-1, A[i,j]=Catalan(j-i), (i<=j), and A[i,j]=0, otherwise. Then, for n >= 1, a(n+1)=(-1)^n*charpoly(A,-5). - Milan Janjic, Jul 08 2010
From Gary W. Adamson, Jul 25 2011: (Start)
a(n) = upper left term in M^(n-1), M = an infinite square production matrix as follows:
  6, 1, 0, 0, 0, ...
  1, 1, 1, 0, 0, ...
  1, 1, 1, 1, 0, ...
  1, 1, 1, 1, 1, ...
  ... (End)
D-finite with recurrence: 4*n*a(n) = (41*n-24)*a(n-1) - 50*(2*n-3)*a(n-2). - Vaclav Kotesovec, Dec 09 2013
a(n) ~ 3*5^(2*n-1)/4^(n+1). - Vaclav Kotesovec, Dec 09 2013
O.g.f. A(x) = (1 - *Sum_{n >= 1} binomial(2*n,n)*x^n)/(1 - (3/2)*Sum_{n >= 1} binomial(2*n,n)*x^n). - Peter Bala, Sep 01 2016

cp's OEIS Frontend

A126984 Expansion of 1/(1+2*x*c(x)), c(x) the g.f. of Catalan numbers A000108.

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A141223 Expansion of 1/(sqrt(1-4*x)*(1-3*x*c(x))), where c(x) is the g.f. of A000108.

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A158832 Main diagonal in the array A158825 of coefficients of successive iterations of x*C(x), where C(x) is the Catalan function (A000108).

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A161629 E.g.f. satisfies: A(x) = exp( x*Catalan(x*A(x)) ), where Catalan(x) = (1-sqrt(1-4*x))/(2*x) is the g.f. of A000108.

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A355341 G.f.: A(x) = Sum_{n=-oo..+oo} x^(n*(n+1)/2) * C(x)^n, where C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers (A000108).

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A381817 Expansion of (1/x) * Series_Reversion( x * (1-x) / C(x) ), where C(x) is the g.f. of A000108.

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A067336 a(0)=1, a(1)=2, a(n) = a(n-1)*9/2 - Catalan(n-1) where Catalan(n) = binomial(2n,n)/(n+1) = A000108(n).

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A126984 Expansion of 1/(1+2xc(x)), c(x) the g.f. of Catalan numbers A000108.

A141223 Expansion of 1/(sqrt(1-4x)(1-3xc(x))), where c(x) is the g.f. of A000108.

A161629 E.g.f. satisfies: A(x) = exp( xCatalan(xA(x)) ), where Catalan(x) = (1-sqrt(1-4x))/(2x) is the g.f. of A000108.

A355341 G.f.: A(x) = Sum_{n=-oo..+oo} x^(n(n+1)/2) C(x)^n, where C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers (A000108).

A076026 Expansion of g.f.: (1-4xC)/(1-5xC) where C = (1/2-1/2(1-4x)^(1/2))/x = g.f. for Catalan numbers A000108.