A216604 -id:A216604 - cp's OEIS frontend

A105633 Row sums of triangle A105632.

1, 2, 4, 9, 22, 57, 154, 429, 1223, 3550, 10455, 31160, 93802, 284789, 871008, 2681019, 8298933, 25817396, 80674902, 253106837, 796968056, 2517706037, 7977573203, 25347126630, 80738862085, 257778971504, 824798533933

Offset: 0

Paul D. Hanna, Apr 17 2005

nonn

Binomial transform of A007477. INVERT transform of A082582. First differences give A086581 and A025242 (offset 1). Is this sequence equal to A057580?
a(n) = the number of Dyck paths of semilength n+1 avoiding UUDU. a(n) = the number of Dyck paths of semilength n+1 avoiding UDUU = the number of binary trees without zigzag (i.e., with no node with a father, with a right son and with no left son). This sequence is the first column of the triangle A116424. E.g., a(2) = 4 because there exist four Dyck paths of semilength 3 that avoid UUDU: UDUDUD, UDUUDD, UUDDUD, UUUDDD, as well as four Dyck paths of semilength 3 that avoid UDUU: UDUDUD, UUDUDD, UUDDUD, UUUDDD. - I. Tasoulas (jtas(AT)unipi.gr), Feb 15 2006
The sequence beginning 1,1,2,4,9,... gives the diagonal sums of A130749, and has g.f. 1/(1-x-x^2/(1-x/(1-x-x^2/(1-x/(1-x-x^2/(1-... (continued fraction); and general term Sum_{k=0..floor(n/2)} Sum_{j=0..n-k} binomial(n-k,j)*A090181(j,k). Its Hankel transform is A099443(n+1). - Paul Barry, Jun 30 2009
The number of plain lambda terms presented by de Bruijn indices, see Bendkowski et al. - Kellen Myers, Jun 15 2015
a(n) = the number of Dyck paths of semilength n+1 with no pairs of
consecutive valleys at the same height. Sergi Elizalde, Feb 25 2021

			G.f.: A(x) = 1 + 2*x + 4*x^2 + 9*x^3 + 22*x^4 + 57*x^5 + 154*x^6 + 429*x^7 + ...
with A(x)^2 = 1 + 4*x + 12*x^2 + 34*x^3 + 96*x^4 + 274*x^5 + 793*x^6 + ...
where A(x) = 1 + x*(2-x)*A(x) + x^2*(1-x)*A(x)^2.
The logarithm of the g.f. begins:
log(A(x)) = (1 + (1-x))*x + (1 + 2^2*(1-x) + (1-x)^2)*x^2/2 +
(1 + 3^2*(1-x) + 3^2*(1-x)^2 + (1-x)^3)*x^3/3 +
(1 + 4^2*(1-x) + 6^2*(1-x)^2 + 4^2*(1-x)^3 + (1-x)^4)*x^4/4 +
(1 + 5^2*(1-x) + 10^2*(1-x)^2 + 10^2*(1-x)^3 + 5^2*(1-x)^4 + (1-x)^5)*x^5/5 + ...
Explicitly,
log(A(x)) = 2*x + 4*x^2/2 + 11*x^3/3 + 32*x^4/4 + 97*x^5/5 + 301*x^6/6 + 947*x^7/7 + 3008*x^8/8 + 9623*x^9/9 + 30959*x^10/10 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Andrei Asinowski, Axel Bacher, Cyril Banderier, and Bernhard Gittenberger, Analytic combinatorics of lattice paths with forbidden patterns, the vectorial kernel method, and generating functions for pushdown automata, Laboratoire d'Informatique de Paris Nord (LIPN 2019).
Marilena Barnabei, Flavio Bonetti, Niccolò Castronuovo, and Matteo Silimbani, Permutations avoiding a simsun pattern, The Electronic Journal of Combinatorics (2020) Vol. 27, Issue 3, P3.45. (See a_n in Theorem 4.)
Jean-Luc Baril, Daniela Colmenares, José L. Ramírez, Emmanuel D. Silva, Lina M. Simbaqueba, and Diana A. Toquica, Consecutive pattern-avoidance in Catalan words according to the last symbol, Univ. Bourgogne (France 2023).
Jean-Luc Baril and José Luis Ramírez, Descent distribution on Catalan words avoiding ordered pairs of Relations, arXiv:2302.12741 [math.CO], 2023. See pp. 3, 8.
Jean Luc Baril, Rigoberto Flórez, and José L. Ramirez, Generalized Narayana arrays, restricted Dyck paths, and related bijections, Univ. Bourgogne (France, 2025). See p. 10.
Paul Barry, Riordan Pseudo-Involutions, Continued Fractions and Somos 4 Sequences, arXiv:1807.05794 [math.CO], 2018. See p. 8.
Maciej Bendkowski, Quantitative aspects and generation of random lambda and combinatory logic terms, Ph.D. Thesis, Jagiellonian University, Kraków, 2017.
Maciej Bendkowski, Katarzyna Grygiel, Pierre Lescanne, and Marek Zaionc, Combinatorics of λ-terms: a natural approach, arXiv:1609.08106 [cs.LO], 2016.
Maciej Bendkowski, Katarzyna Grygiel, Pierre Lescanne, and Marek Zaionc, A Natural Counting of Lambda Terms, arXiv preprint arXiv:1506.02367 [cs.LO], 2015.
Maciej Bendkowski, K. Grygiel, and P. Tarau, Random generation of closed simply-typed lambda-terms: a synergy between logic programming and Boltzmann samplers, arXiv preprint arXiv:1612.07682 [cs.LO], 2016-2017.
David Callan, On Ascent, Repetition and Descent Sequences, arXiv:1911.02209 [math.CO], 2019. See p. 12.
Sergi Elizalde, Symmetric peaks and symmetric valleys in Dyck paths, arXiv:2008.05669 [math.CO], 2020. See p. 5.
Juan B. Gil and Michael D. Weiner, On pattern-avoiding Fishburn permutations, arXiv:1812.01682 [math.CO], 2018. See pp. 4, 6.
Katarzyna Grygiel and Pierre Lescanne, A natural counting of lambda terms, Preprint 2015.
Nancy S. S. Gu, Nelson Y. Li, and Toufik Mansour, 2-Binary trees: bijections and related issues, Discr. Math., 308 (2008), 1209-1221.
Toufik Mansour, Statistics on Dyck Paths, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.5.
A. Sapounakis et al., Ordered trees and the inorder transversal, Disc. Math., 306 (2006), 1732-1741.
Robin D. P. Zhou, Pattern avoidance in revised ascent sequences, arXiv:2505.05171 [math.CO], 2025. See pp. 4, 20.

Cf. A105632, A057580, A116424, A216604. See also A258973.

a := n -> add((-1)^i*hypergeom([(i+1)/2, i/2+1, i-n-1], [1, 2], -4), i=0..n+1):
seq(simplify(a(n)), n=0..26); # Peter Luschny, May 03 2018

CoefficientList[Series[(1 - x - Sqrt[(1 - x)^2 - 4 x^2/(1 - x)])/(2 x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Mar 15 2014 *)

{a(n)=local(X=x+x*O(x^n)); polcoeff(2/(1-X)/(1-X+sqrt((1-X)^2-4*X^2/(1-X))),n,x)}

{a(n)=polcoeff(exp(sum(m=1,n+1,x^m/m*sum(k=0,m,binomial(m,k)^2*(1-x)^(m-k) + x*O(x^n)))),n)} \\ Paul D. Hanna, Sep 12 2012

G.f.: A(x) = (1-x - sqrt((1-x)^2 - 4*x^2/(1-x)))/(2*x^2).
a(n) = 2*a(n-1) + Sum_{i=1..n-2} a(i)*(a(n-1-i) - a(n-2-i)). a(n) = Sum_{i=0..floor(n/2)} (-1)^i * binomial(n+1-i,i) * binomial(2*(n+1)-3*i, n-2*i) /(n+1-i). - I. Tasoulas (jtas(AT)unipi.gr), Feb 15 2006
G.f.: (1/(1-x)^2)c(x^2/(1-x)^3), where c(x) is the g.f. of A000108. - Paul Barry, May 22 2009
1/(1-x-x/(1-x^2/(1-x-x/(1-x^2/(1-x-x/(1-x^2/(1-... (continued fraction). - Paul Barry, Jun 30 2009
a(n) = Sum_{k=0..floor(n/2)} Sum_{j=0..n-k} binomial(n-k,j)(0^(j+k)+(1/(j+0^j))*binomial(j,k)*binomial(j,k+1)). - Paul Barry, Jun 30 2009
G.f. satisfies: A(x) = (1 + x*A(x)) * (1 + x*(1-x)*A(x)). - Paul D. Hanna, Sep 12 2012
G.f.: exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n,k)^2 * (1-x)^k ). - Paul D. Hanna, Sep 12 2012
D-finite with recurrence: (n+2)*a(n) + (-4*n-3)*a(n-1) + (2*n+1)*a(n-2) + a(n-3) + (n-3)*a(n-4) = 0. - R. J. Mathar, Nov 26 2012
The recurrence is true, since by holonomic transformation, it can be computed formally using GFUN, associated with the equation: x^3 + x^2 - 2x + (x^3 + 3 x^2 -3x +1) A(x) + (x^5 + 2x^3 -4 x^2 + x) A'(x) = 0. - Pierre Lescanne, Jun 30 2015
G.f.: (1 - 1/(G(0)-x))/x^2 where G(k) =  1 + x/(1 + x/(x^2 - 1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Dec 16 2012
a(n) ~ 2^(n/3-1/6) * 3^(n+2) * (13+3*sqrt(33))^((n+1)/3) * sqrt(4*(2879 + 561*sqrt(33))^(1/3) + 8*(7822 + 1362*sqrt(33))^(1/3) - 91 - 21*sqrt(33)) / (((26+6*sqrt(33))^(2/3) - (26+6*sqrt(33))^(1/3) - 8)^(n+3/2) * (4*(26+6*sqrt(33))^(1/3) - (26+6*sqrt(33))^(2/3) + 8) * n^(3/2) * sqrt(Pi)). - Vaclav Kotesovec, Mar 13 2014
a(n) = Sum_{i=0..n+1} (-1)^i*hypergeom([(i+1)/2, i/2+1, i-n-1], [1, 2], -4). - Peter Luschny, May 03 2018

More terms from I. Tasoulas (jtas(AT)unipi.gr), Feb 15 2006

A217615 G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k)^2 * x^k*(1-x)^(n-k).

Original entry on oeis.org

1, 1, 1, 3, 5, 7, 15, 29, 49, 95, 187, 345, 659, 1289, 2465, 4739, 9237, 17911, 34715, 67705, 132063, 257477, 503309, 984983, 1927895, 3778017, 7411237, 14544967, 28565661, 56144615, 110406527, 217225533, 427636561, 842256047, 1659600955, 3271579689, 6451913519

Offset: 0

Paul D. Hanna, Oct 09 2012

nonn

Radius of convergence of g.f. is r = 1/2.
More generally, given
A(x) = Sum_{n>=1} x^n * Sum_{k=0..n} binomial(n,k)^2 * x^k*(t-x)^(n-k),
then A(x) = 1/sqrt( (1 - t*x + 2*x^2)^2 - 4*x^2 )
and the radius of convergence r satisfies: (1-r)^2 = r*(t-r) for t > 0.
a(n) is the number of (2k-1)-element subsets of {1, 2, ..., n+1} whose k-th smallest (i.e., k-th largest) element equals 2k-1. - Darij Grinberg, Oct 09 2019

			G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 15*x^6 + 29*x^7 + 49*x^8 + ...
where the g.f. equals the series:
A(x) = 1 +
  x*((1-x) + x) +
  x^2*((1-x)^2 + 2^2*x*(1-x) + x^2) +
  x^3*((1-x)^3 + 3^2*x*(1-x)^2 + 3^2*x^2*(1-x) + x^3) +
  x^4*((1-x)^4 + 4^2*x*(1-x)^3 + 6^2*x^2*(1-x)^2 + 4^2*x^3*(1-x) + x^4) +
  x^5*((1-x)^5 + 5^2*x*(1-x)^4 + 10^2*x^2*(1-x)^3 + 10^2*x^3*(1-x)^2 + 5^2*x^4*(1-x) + x^5) + ...

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

Cf. A217616, A217617, A217461, A217464, A216604, A217661.

a := n -> `if`(n < 4, [1, 1, 1, 3][n+1], hypergeom([1/2, (1-n)/3, (2-n)/3, -n/3], [1, (1-n)/2, -n/2], -27)):
seq(simplify(a(n)), n=0..36); # Peter Luschny, Oct 09 2019

CoefficientList[Series[1/Sqrt[(1-x+2*x^2)^2-4*x^2], {x, 0, 20}], x] (* Vaclav Kotesovec, Sep 16 2013 *)

{a(n)=polcoeff(sum(m=0, n+1, x^m*sum(k=0, m, binomial(m, k)^2*x^k*(1-x)^(m-k) + x*O(x^n))), n)}
for(n=0,40,print1(a(n),", "))

a(n)={sum(k=0, n\2, binomial(2*k, k) * binomial(n-2*k, k))} \\ Andrew Howroyd, Oct 09 2019

G.f.: A(x) = 1 / sqrt( (1 - x + 2*x^2)^2 - 4*x^2 ).
G.f.: A(x) = 1 / sqrt( (1-x)*(1-2*x)*(1+x+2*x^2) ).
G.f. satisfies: A(x) = (1 + 2*x^2*Sum_{n>=0} A000108(n)*(-x*A(x))^(2*n)) / (1-x+2*x^2) where A000108(n) = binomial(2*n,n)/(n+1) forms the Catalan numbers.
a(n) ~ 2^n/sqrt(Pi*n). - Vaclav Kotesovec, Sep 16 2013
a(n) = Sum_{k=0..floor(n/2)} binomial(2*k, k) * binomial(n-2*k, k). - Darij Grinberg, Oct 09 2019
a(n) = hypergeom([1/2,(1-n)/3,(2-n)/3, -n/3], [1, (1-n)/2, -n/2], -27) for n >= 4. - Peter Luschny, Oct 09 2019

A346073 a(n) = 1 + Sum_{k=0..n-4} a(k) * a(n-k-4).

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 4, 5, 8, 13, 20, 29, 45, 73, 118, 185, 293, 475, 778, 1263, 2047, 3345, 5512, 9085, 14957, 24683, 40918, 67987, 113016, 188053, 313608, 524041, 876657, 1467797, 2460644, 4130893, 6942726, 11678687, 19663068, 33139295, 55904339, 94384167, 159470488

Offset: 0

Ilya Gutkovskiy, Jul 04 2021

nonn

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

Cf. A000108, A007317, A090344, A216604, A307971, A346074.

a[n_] := a[n] = 1 + Sum[a[k] a[n - k - 4], {k, 0, n - 4}]; Table[a[n], {n, 0, 42}]
nmax = 42; A[] = 0; Do[A[x] = 1/(1 - x) + x^4 A[x]^2 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

a(n) = sum(k=0, n\4, binomial(n-3*k, k)*binomial(2*k, k)/(k+1)); \\ Seiichi Manyama, Jan 22 2023

@CachedFunction
def a(n): # a = A346073
    if (n<4): return 1
    else: return 1 + sum(a(k)*a(n-k-4) for k in range(n-3))
[a(n) for n in range(51)] # G. C. Greubel, Nov 26 2022

G.f. A(x) satisfies: A(x) = 1 / (1 - x) + x^4 * A(x)^2.

a(n) = Sum_{k=0..floor(n/4)} binomial(n-3*k,k) * Catalan(k). - Seiichi Manyama, Jan 22 2023

A217464 L.g.f.: Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n,k)^2 * x^k*(1-x)^(n-k).

Original entry on oeis.org

1, 1, 4, 5, 6, 16, 29, 45, 94, 186, 331, 644, 1275, 2409, 4644, 9117, 17630, 34162, 66843, 130390, 254153, 497487, 974166, 1906860, 3739131, 7338839, 14406214, 28303525, 55651262, 109465176, 215431773, 424229309, 835747510, 1647145386, 3247768579, 6406290590

Offset: 1

Paul D. Hanna, Oct 03 2012

nonn

			L.g.f.: L(x) = x + x^2/2 + 4*x^3/3 + 5*x^4/4 + 6*x^5/5 + 16*x^6/6 + 29*x^7/7 +...
such that the l.g.f. equals the series:
L(x) = ((1-x) + x)*x +
((1-x)^2 + 2^2*x*(1-x) + x^2)*x^2/2 +
((1-x)^3 + 3^2*x*(1-x)^2 + 3^2*x^2*(1-x) + x^3)*x^3/3 +
((1-x)^4 + 4^2*x*(1-x)^3 + 6^2*x^2*(1-x)^2 + 4^2*x^3*(1-x) + x^4)*x^4/4 +
((1-x)^5 + 5^2*x*(1-x)^4 + 10^2*x^2*(1-x)^3 + 10^2*x^3*(1-x)^2 + 5^2*x^4*(1-x) + x^5)*x^5/5 +...
where exponentiation yields the g.f. of A216604:
exp(L(x)) = 1 + x + x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 7*x^6 + 12*x^7 + 19*x^8 + 33*x^9 + 59*x^10 + 102*x^11 + 181*x^12 +...+ A216604(n)*x^n +...

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A216604.

Rest[CoefficientList[Series[-Log[(1-x+Sqrt[(1-x)^2-4*x^3*(1-x)])/2],{x,0,20}],x]*Range[0,20]] (* Vaclav Kotesovec, Mar 06 2014 *)

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

{a(n)=n*polcoeff(log(2/(1-x+sqrt((1-x)^2-4*x^3*(1-x) +x*O(x^n)))), n)}
for(n=1,40,print1(a(n),", "))

L.g.f.: -log( (1-x + sqrt((1-x)^2 - 4*x^3*(1-x)))/2 ).

a(n) ~ 2^n/sqrt(Pi*n). - Vaclav Kotesovec, Mar 06 2014

A216434 G.f. satisfies: A(x) = (1 + x(2-x)A(x)) * (1 + x^2*A(x)).

Original entry on oeis.org

1, 2, 4, 10, 27, 74, 208, 600, 1762, 5244, 15788, 48006, 147199, 454618, 1412960, 4416016, 13869998, 43756124, 138587784, 440523892, 1404849486, 4493472836, 14411800352, 46338611632, 149338703380, 482315951104, 1560824670460, 5060345244766, 16434480777703

Offset: 0

Paul D. Hanna, Sep 10 2012

nonn

The radius of convergence of the g.f. A(x) is r = 1/(2+sqrt(2)) with A(r) = 2*(1+sqrt(2)).
More generally, if A(x) = (1 + x*(t-x)*A(x)) * (1 + x^2*A(x)), |t|>0, then
A(x) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n,k)^2 * x^k*(t-x)^(n-k) )
where the radius of convergence r of the g.f. A(x) satisfies
r = (1-r)^2/(t-r) = (1-t*r)/(2*(1-r)) with A(r) = 1/(r*(1-r)) = 2/(1-t*r).

			G.f.: A(x) = 1 + 2*x + 4*x^2 + 10*x^3 + 27*x^4 + 74*x^5 + 208*x^6 + 600*x^7 +...
The logarithm of the g.f. begins:
log(A(x)) = ((2-x) + x)*x + ((2-x)^2 + 2^2*x*(2-x) + x^2)*x^2/2 +
((2-x)^3 + 3^2*x*(2-x)^2 + 3^2*x^2*(2-x) + x^3)*x^3/3 +
((2-x)^4 + 4^2*x*(2-x)^3 + 6^2*x^2*(2-x)^2 + 4^2*x^3*(2-x) + x^4)*x^4/4 +
((2-x)^5 + 5^2*x*(2-x)^4 + 10^2*x^2*(2-x)^3 + 10^2*x^3*(2-x)^2 + 5^2*x^4*(2-x) + x^5)*x^5/5 +...
Explicitly,
log(A(x)) = 2*x + 4*x^2/2 + 14*x^3/3 + 44*x^4/4 + 132*x^5/5 + 412*x^6/6 + 1318*x^7/7 + 4236*x^8/8 + 13676*x^9/9 + 44424*x^10/10 +...

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

Cf. A216604, A216454, A192415.

CoefficientList[Series[((1 - 2*x) - Sqrt[(1 - 2*x)^2 - 4*x^3*(2 - x)])/(2*x^3*(2 - x)), {x,0,50}], x] (* G. C. Greubel, Feb 03 2017 *)

{a(n)=polcoeff(exp(sum(m=1,n+1,x^m/m*sum(k=0,m,binomial(m,k)^2*x^k*(2-x)^(m-k) + x*O(x^n)))),n)}

{a(n)=polcoeff(2/(1-2*x+sqrt((1-2*x)^2-4*x^3*(2-x) +x*O(x^n))),n)}
for(n=0,40,print1(a(n),", "))

G.f.: exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} C(n,k)^2 * x^k*(2-x)^(n-k) ).
G.f.: ((1-2*x) - sqrt( (1-2*x)^2 - 4*x^3*(2-x) )) / (2*x^3*(2-x)).
Recurrence: 2*(n+3)*a(n) = 3*(3*n+5)*a(n-1) - 6*(2*n+1)*a(n-2) + 4*(5*n-6)*a(n-3) - 4*(4*n-9)*a(n-4) + 4*(n-3)*a(n-5). - Vaclav Kotesovec, Aug 18 2013
a(n) ~ 2^(n/2 + 5/4) * (1+sqrt(2))^(n+1) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Aug 18 2013, simplified Jan 21 2023

A216454 G.f. satisfies: A(x) = (1 + x(3-x)A(x)) * (1 + x^2*A(x)).

Original entry on oeis.org

1, 3, 9, 30, 107, 396, 1503, 5820, 22907, 91371, 368523, 1500354, 6157669, 25448907, 105821865, 442409844, 1858482563, 7840766601, 33207750819, 141137445258, 601768494201, 2573246794374, 11032997250357, 47421297986868, 204286464525165, 881900059488741

Offset: 0

Paul D. Hanna, Sep 10 2012

nonn

The radius of convergence of the g.f. A(x) is r = 2/(5+sqrt(17)) with A(r) = 2/(1-3*r) = (11+3*sqrt(17))/4.
More generally, if A(x) = (1 + x*(t-x)*A(x)) * (1 + x^2*A(x)), |t|>0, then
A(x) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n,k)^2 * x^k*(t-x)^(n-k) )
where the radius of convergence r of the g.f. A(x) satisfies
r = (1-r)^2/(t-r) = (1-t*r)/(2*(1-r)) with A(r) = 1/(r*(1-r)) = 2/(1-t*r).

			G.f.: A(x) = 1 + 3*x + 9*x^2 + 30*x^3 + 107*x^4 + 396*x^5 + 1503*x^6 +...
The logarithm of the g.f. begins:
log(A(x)) = ((3-x) + x)*x + ((3-x)^2 + 2^2*x*(3-x) + x^2)*x^2/2 +
((3-x)^3 + 3^2*x*(3-x)^2 + 3^2*x^2*(3-x) + x^3)*x^3/3 +
((3-x)^4 + 4^2*x*(3-x)^3 + 6^2*x^2*(3-x)^2 + 4^2*x^3*(3-x) + x^4)*x^4/4 +
((3-x)^5 + 5^2*x*(3-x)^4 + 10^2*x^2*(3-x)^3 + 10^2*x^3*(3-x)^2 + 5^2*x^4*(3-x) + x^5)*x^5/5 +...
Explicitly,
log(A(x)) = 3*x + 9*x^2/2 + 36*x^3/3 + 149*x^4/4 + 618*x^5/5 + 2592*x^6/6 + 11007*x^7/7 + 47181*x^8/8 + 203634*x^9/9 + 883674*x^10/10 +...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A216604, A216434.

CoefficientList[Series[2/(1-3*x+Sqrt[(1-3*x)^2-4*x^3*(3-x)]), {x, 0, 20}], x] (* Vaclav Kotesovec, Sep 17 2013 *)

{a(n)=polcoeff(exp(sum(m=1,n+1,x^m/m*sum(k=0,m,binomial(m,k)^2*x^k*(3-x)^(m-k) + x*O(x^n)))),n)}

{a(n)=polcoeff(2/(1-3*x+sqrt((1-3*x)^2-4*x^3*(3-x) +x*O(x^n))),n)}
for(n=0,40,print1(a(n),", "))

G.f.: exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n,k)^2 * x^k*(3-x)^(n-k) ).
G.f.: 2 / ( 1-3*x + sqrt( (1-3*x)^2 - 4*x^3*(3-x) ) ).
Recurrence: 3*(n+3)*a(n) = (19*n+30)*a(n-1) - 3*(11*n+3)*a(n-2) + 9*(5*n-6)*a(n-3) - 6*(4*n-9)*a(n-4) + 4*(n-3)*a(n-5). - Vaclav Kotesovec, Sep 17 2013
a(n) ~ sqrt(7378+1794*sqrt(17)) * ((5+sqrt(17))/2)^n / (16*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Sep 17 2013

A216617 G.f. satisfies: A(x) = (1 + x(1-x)A(x)^2) * (1 + x^2*A(x)).

Original entry on oeis.org

1, 1, 2, 5, 13, 37, 112, 351, 1130, 3716, 12424, 42101, 144277, 499136, 1740871, 6114629, 21609654, 76786625, 274171192, 983187372, 3539498904, 12787269117, 46345303727, 168463177245, 614002351108, 2243406499930, 8215549186628, 30149687633264, 110861650218443

Offset: 0

Paul D. Hanna, Sep 10 2012

nonn

Apparently the number of Dyck paths of semilength n that avoid UUDUUD. The only Dyck path of semilength 4 that contains UUDUUD is UUDUUDdd. So a(4) = A000108(4)-1 = 13. - David Scambler, Apr 24 2013

			G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 13*x^4 + 37*x^5 + 112*x^6 + 351*x^7 +...
The logarithm of the g.f. begins:
log(A(x)) = ((1-x)*A(x) + x)*x + ((1-x)^2*A(x)^2 + 2^2*x*(1-x)*A(x) + x^2)*x^2/2 +
((1-x)^3*A(x)^3 + 3^2*x*(1-x)^2*A(x)^2 + 3^2*x^2*(1-x)*A(x) + x^3)*x^3/3 +
((1-x)^4*A(x)^4 + 4^2*x*(1-x)^3*A(x)^3 + 6^2*x^2*(1-x)^2*A(x)^2 + 4^2*x^3*(1-x)*A(x) + x^4)*x^4/4 +
((1-x)^5*A(x)^5 + 5^2*x*(1-x)^4*A(x)^4 + 10^2*x^2*(1-x)^3*A(x)^3 + 10^2*x^3*(1-x)^2*A(x)^2 + 5^2*x^4*(1-x)*A(x) + x^5)*x^5/5 +...
Explicitly,
log(A(x)) = x + 3*x^2/2 + 10*x^3/3 + 31*x^4/4 + 106*x^5/5 + 378*x^6/6 + 1359*x^7/7 + 4935*x^8/8 + 18073*x^9/9 + 66578*x^10/10 +...

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A216616, A216604, A216447.

a:= n->coeff(series(RootOf(A=(1+x*(1-x)*A^2)*(1+x^2*A), A), x, n+1), x, n):
seq(a(n), n=0..30);  # Alois P. Heinz, Apr 25 2013

m = 30; A[_] = 0;
Do[A[x_] = (1 + x (1 - x) A[x]^2) (1 + x^2 A[x]) + O[x]^m, {m}];
CoefficientList[A[x], x] (* Jean-François Alcover, Oct 02 2019 *)

{a(n)=local(A=1+x);for(i=1,n,A=(1 + x*(1-x)*A^2)*(1+x^2*A) +x*O(x^n));polcoeff(A,n)}

{a(n)=local(A=1+x);for(i=1,n,A=exp(sum(m=1,n+1,x^m/m*sum(k=0,m,binomial(m,k)^2*x^k*(1-x)^(m-k)*A^(m-k) +x*O(x^n)))));polcoeff(A,n)}
for(n=0,40,print1(a(n),", "))

G.f. satisfies: A(x) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n,k)^2 * x^k*(1-x)^(n-k) * A(x)^(n-k) ).
Recurrence: (n+1)*(n+2)*(679*n^4 - 7380*n^3 + 23045*n^2 - 9120*n - 29484)*a(n) = (n+1)*(1358*n^5 - 14081*n^4 + 33259*n^3 + 60800*n^2 - 249924*n + 117936)*a(n-1) + (7469*n^6 - 81180*n^5 + 247067*n^4 - 13482*n^3 - 719746*n^2 + 667728*n - 176904)*a(n-2) - 2*(5432*n^6 - 67188*n^5 + 260696*n^4 - 210849*n^3 - 724651*n^2 + 1374444*n - 589500)*a(n-3) + 6*(1358*n^6 - 18834*n^5 + 89081*n^4 - 133447*n^3 - 140626*n^2 + 464272*n - 173424)*a(n-4) - 2*(9506*n^6 - 146097*n^5 + 784589*n^4 - 1442697*n^3 - 1099897*n^2 + 5950320*n - 4023900)*a(n-5) + 2*(6790*n^6 - 114540*n^5 + 682867*n^4 - 1465407*n^3 - 769658*n^2 + 6637308*n - 5733000)*a(n-6) + 2*(n-6)*(n-5)*(1358*n^4 - 10007*n^3 + 8773*n^2 + 34598*n - 27540)*a(n-7) - 4*(n-7)*(n-6)*(679*n^4 - 4664*n^3 + 4979*n^2 + 17546*n - 22260)*a(n-8). - Vaclav Kotesovec, Sep 16 2013
a(n) ~ c*d^n/n^(3/2), where d = 3.8781907052914131... is the root of the equation 4 + 4*d - 16*d^2 - 8*d^3 - 12*d^4 + d^6 = 0 and c = 0.561628033... - Vaclav Kotesovec, Sep 16 2013

A346074 a(n) = 1 + Sum_{k=0..n-5} a(k) * a(n-k-5).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 9, 14, 21, 30, 41, 59, 89, 136, 205, 301, 443, 664, 1011, 1545, 2341, 3530, 5341, 8143, 12487, 19148, 29299, 44817, 68721, 105742, 163025, 251392, 387595, 597988, 924047, 1430167, 2215595, 3433788, 5323915, 8260652, 12829849

Offset: 0

Ilya Gutkovskiy, Jul 04 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A007317, A090344, A216604, A307972, A346073.

a[n_] := a[n] = 1 + Sum[a[k] a[n - k - 5], {k, 0, n - 5}]; Table[a[n], {n, 0, 44}]
nmax = 44; A[] = 0; Do[A[x] = 1/(1 - x) + x^5 A[x]^2 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

a(n) = sum(k=0, n\5, binomial(n-4*k, k)*binomial(2*k, k)/(k+1)); \\ Seiichi Manyama, Jan 22 2023

G.f. A(x) satisfies: A(x) = 1 / (1 - x) + x^5 * A(x)^2.
Conjecture D-finite with recurrence (n+5)*a(n) +2*(-n-4)*a(n-1) +(n+3)*a(n-2) +2*(-2*n+5)*a(n-5) +4*(n-3)*a(n-6)=0. - R. J. Mathar, Feb 17 2022
a(n) = Sum_{k=0..floor(n/5)} binomial(n-4*k,k) * Catalan(k). - Seiichi Manyama, Jan 22 2023

A360025 a(n) = Sum_{k=0..floor(n/3)} (-1)^k * binomial(n-2k,k) Catalan(k).

Original entry on oeis.org

1, 1, 1, 0, -1, -2, -1, 2, 7, 9, 3, -16, -39, -43, 9, 126, 247, 199, -213, -984, -1555, -756, 2525, 7518, 9593, 559, -24899, -56216, -55241, 33150, 225879, 407194, 273199, -529745, -1938549, -2822128, -833219, 6083986, 15904733, 18288966, -4172187, -61154333

Offset: 0

Seiichi Manyama, Jan 22 2023

sign

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A360024, A360026, A360027.

Cf. A000108, A216604, A349047.

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

my(N=50, x='x+O('x^N)); Vec(2/(1-x+sqrt((1-x)^2+4*x^3*(1-x))))

a(n) = 1 - Sum_{k=0..n-3} a(k) * a(n-k-3).
G.f. A(x) satisfies: A(x) = 1/(1-x) - x^3 * A(x)^2.
G.f.: 2 / ( 1-x + sqrt((1-x)^2 + 4*x^3*(1-x)) ).
D-finite with recurrence +(n+3)*a(n) +2*(-n-2)*a(n-1) +(n+1)*a(n-2) +2*(2*n-3)*a(n-3) +4*(-n+2)*a(n-4)=0. - R. J. Mathar, Jan 25 2023

A216447 G.f. satisfies: A(x) = (1 + x(1-x)A(x)^2) * (1 + x^2*A(x)^2).

Original entry on oeis.org

1, 1, 2, 6, 19, 64, 228, 841, 3181, 12277, 48156, 191400, 769168, 3120044, 12758080, 52533265, 217637308, 906511243, 3793989118, 15947205096, 67290484581, 284934164506, 1210374907352, 5156562941596, 22027291990432, 94325712634264, 404842107811880

Offset: 0

Paul D. Hanna, Sep 10 2012

nonn

			G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 19*x^4 + 64*x^5 + 228*x^6 + 841*x^7 +...
The logarithm of the g.f. begins:
log(A(x)) = ((1-x) + x)*x*A(x) + ((1-x)^2 + 2^2*x*(1-x) + x^2)*x^2*A(x)^2/2 +
((1-x)^3 + 3^2*x*(1-x)^2 + 3^2*x^2*(1-x) + x^3)*x^3*A(x)^3/3 +
((1-x)^4 + 4^2*x*(1-x)^3 + 6^2*x^2*(1-x)^2 + 4^2*x^3*(1-x) + x^4)*x^4*A(x)^4/4 +
((1-x)^5 + 5^2*x*(1-x)^4 + 10^2*x^2*(1-x)^3 + 10^2*x^3*(1-x)^2 + 5^2*x^4*(1-x) + x^5)*x^5*A(x)^5/5 +...
Explicitly,
log(A(x)) = x + 3*x^2/2 + 13*x^3/3 + 51*x^4/4 + 206*x^5/5 + 861*x^6/6 + 3641*x^7/7 + 15523*x^8/8 + 66676*x^9/9 +...+ L(n)*x^n/n +...
where L(n) = [x^n] ((1-x^4)/(1-x-x^3))^n.

Cf. A216604, A216616, A216617.

nmax=20;aa=ConstantArray[0,nmax]; aa[[1]]=1;Do[AGF=1+Sum[aa[[n]]*x^n,{n,1,j-1}]+koef*x^j; sol=Solve[Coefficient[(1+x*(1-x)*AGF^2)*(1+x^2*AGF^2)-AGF,x,j]==0,koef][[1]]; aa[[j]]=koef/.sol[[1]],{j,2,nmax}];Flatten[{1,aa}] (* Vaclav Kotesovec, Aug 18 2013 *)

{a(n)=local(A=1+x);for(i=1,n,A=(1 + x*(1-x)*A^2)*(1+x^2*A^2) +x*O(x^n));polcoeff(A,n)}

{a(n)=local(A=1+x);for(i=1,n,A=exp(sum(m=1,n+1,x^m*A^m/m*sum(k=0,m,binomial(m,k)^2*x^k*(1-x)^(m-k) +x*O(x^n)))));polcoeff(A,n)}

{a(n)=polcoeff(((1-x^4)/(1-x-x^3 +x*O(x^n)))^(n+1)/(n+1),n)}

{a(n)=polcoeff((1/x)*serreverse(x*(1-x-x^3)/(1-x^4 +x^2*O(x^n))),n)}
for(n=0,40,print1(a(n),", "))

G.f. A(x) satisfies:
(1) A(x) = exp( Sum_{n>=1} x^n*A(x)^n/n * Sum_{k=0..n} binomial(n,k)^2 * x^k*(1-x)^(n-k) ).
(2) A(x) = exp( Sum_{n>=1} L(n)*x^n/n ) where L(n) = [x^n] ((1-x^4)/(1-x-x^3))^n.
(3) A(x) = (1/x) * Series_Reversion( x*(1-x-x^3)/(1-x^4) ).
(4) A(x) = (1 - x^4*A(x)^4) / (1 - x*A(x) - x^3*A(x)^3).
(5) A( x*(1-x-x^3)/(1-x^4) )  =  (1-x^4)/(1-x-x^3).
a(n) = [x^n] ((1-x^4)/(1-x-x^3))^(n+1) / (n+1).
Recurrence: 31*(n-1)*n*(n+1)*(877*n^4 - 9157*n^3 + 32799*n^2 - 48009*n + 24592)*a(n) = 2*(n-1)*n*(95593*n^5 - 1032316*n^4 + 3953542*n^3 - 6623680*n^2 + 4768887*n - 1162262)*a(n-1) - (n-1)*(402543*n^6 - 4844150*n^5 + 22130482*n^4 - 48931238*n^3 + 54934501*n^2 - 29672070*n + 6219456)*a(n-2) + 4*(143828*n^7 - 2089338*n^6 + 12282478*n^5 - 37816765*n^4 + 65573867*n^3 - 63612965*n^2 + 31782331*n - 6379188)*a(n-3) - 16*(n-3)*(49112*n^6 - 639080*n^5 + 3201244*n^4 - 7806610*n^3 + 9609821*n^2 - 5589051*n + 1241722)*a(n-4) + 64*(n-4)*(n-3)*(10524*n^5 - 108130*n^4 + 387266*n^3 - 581563*n^2 + 354497*n - 75516)*a(n-5) - 256*(n-5)*(n-4)*(n-3)*(877*n^4 - 5649*n^3 + 10590*n^2 - 6374*n + 1102)*a(n-6). - Vaclav Kotesovec, Aug 18 2013
a(n) ~ c*d^n/n^(3/2), where d=4.54477579... is the root of the equation -256 + 512*d - 384*d^2 + 272*d^3 - 187*d^4 + 31*d^5 = 0 and c = 0.448853665050529472948816... - Vaclav Kotesovec, Aug 18 2013

cp's OEIS Frontend

A105633 Row sums of triangle A105632.

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A217615 G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k)^2 * x^k*(1-x)^(n-k).

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A346073 a(n) = 1 + Sum_{k=0..n-4} a(k) * a(n-k-4).

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A217464 L.g.f.: Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n,k)^2 * x^k*(1-x)^(n-k).

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A216434 G.f. satisfies: A(x) = (1 + x*(2-x)*A(x)) * (1 + x^2*A(x)).

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A216454 G.f. satisfies: A(x) = (1 + x*(3-x)*A(x)) * (1 + x^2*A(x)).

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A216617 G.f. satisfies: A(x) = (1 + x*(1-x)*A(x)^2) * (1 + x^2*A(x)).

A216434 G.f. satisfies: A(x) = (1 + x(2-x)A(x)) * (1 + x^2*A(x)).

A216454 G.f. satisfies: A(x) = (1 + x(3-x)A(x)) * (1 + x^2*A(x)).

A216617 G.f. satisfies: A(x) = (1 + x(1-x)A(x)^2) * (1 + x^2*A(x)).

A360025 a(n) = Sum_{k=0..floor(n/3)} (-1)^k * binomial(n-2k,k) Catalan(k).

A216447 G.f. satisfies: A(x) = (1 + x(1-x)A(x)^2) * (1 + x^2*A(x)^2).