A106228 -id:A106228 - cp's OEIS frontend

A109081 Reversion of x(1-x)(1-x^2)(1-x^3)/(1-x^6) = x(1-x)^2/(1-x+x^2).

1, 1, 3, 10, 37, 146, 602, 2563, 11181, 49720, 224540, 1027038, 4748042, 22150519, 104146733, 493012682, 2347796965, 11239697816, 54061835288, 261130778516, 1266125122956, 6160158505040, 30065608532008, 147161532388934

Offset: 1

Michael Somos, Jun 17 2005

nonn

From David Callan, Mar 30 2007: (Start)
a(n) is the number of vertex-labeled ordered trees (A000108) on n vertices, in which each non-leaf vertex is labeled with a positive integer <= its outdegree. Example. a(3)=3 counts the trees on 3 vertices with labels as shown (the 2 edges in each tree are shown, you have to visualize the vertices).
.
     1     2    1
    / \   / \   |1
                |
.
Proof. Let F(x) = x + x^2 + 3x^3 + ... denote the g.f. for these trees, with x marking number of vertices. Counting these trees by degree of the root leads to F = x + Sum_{k>=1} k*x*F^k, or F = x + x*F/(1-F)^2. This is the same equation as that satisfied by the reversion of x*(1-x)*(1-x^2)*(1-x^3)/(1-x^6) = x*(1-x)^2/(1-x+x^2). (End)
(1 + 3x + 10x^2 + ...) = (1 + 2x + 6x^2 + ...)*(1 + x + 2x^2 + 6x^3 + ...), where A106228 = (1, 1, 2, 6, 21, ...). - Gary W. Adamson, Nov 15 2011
Reversion of x/(1 + sum(k>=1, k*x^k )) (cf. A028310). - Joerg Arndt, Aug 19 2012
a(n) is the number of Motzkin paths of length 2n-3 with no downsteps in even position (n>=2). Example: a(3)=3 counts FFF, FUD, UFD, where U denotes an upstep (1,1), F a flatstep (1,0), and D a downstep (1,-1). - David Callan, May 20 2015
a(n) is the number of peakless Motzkin paths of length 2n-2 where every pair of matching up and down edges occupies positions of the same parity. Equivalently, the number of RNA secondary structures on 2n-2 vertices where only vertices of the same parity can be matched. - Alexander Burstein, May 17 2021

			a(5) = 37 = the upper left term of M^4: (37, 26, 12, 4, 1); where (37 + 26 + 12 + 4 + 1) = 80 = A106228(5). - _Gary W. Adamson_, Nov 15 2011
G.f. = x + x^2 + 3*x^3 + 10*x^4 + 37*x^5 + 146*x^6 + 602*x^7 + 2563*x^8 + ...

Robert Israel, Table of n, a(n) for n = 1..1400
Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group, arXiv:2112.11595 [math.CO], 2021.
Nancy S. S. Gu, Nelson Y. Li, and Toufik Mansour, 2-Binary trees: bijections and related issues, Discr. Math., 308 (2008), 1209-1221.
Tian-Xiao He and Louis W. Shapiro, Row sums and alternating sums of Riordan arrays, Linear Algebra and its Applications, Volume 507, 15 October 2016, Pages 77-95. See R_2^{-}.
JiSun Huh, Sangwook Kim, Seunghyun Seo, and Heesung Shin, Bijections on pattern avoiding inversion sequences and related objects, arXiv:2404.04091 [math.CO], 2024. See p. 22.
Markus Kuba and Alois Panholzer, Enumeration formulas for pattern restricted Stirling permutations, Discrete Math. 312 (2012), no. 21, 3179--3194. MR2957938. - From _N. J. A. Sloane_, Sep 25 2012
Hanna Mularczyk, Lattice Paths and Pattern-Avoiding Uniquely Sorted Permutations, arXiv:1908.04025 [math.CO], 2019.
Jun Yan, Results on pattern avoidance in parking functions, arXiv:2404.07958 [math.CO], 2024. See p. 11.
Jun Yan, Lattice paths enumerations weighted by ascent lengths, arXiv:2501.01152 [math.CO], 2025. See p. 7.

Cf. A000108, A028310, A106228, A365770.

[&+[Binomial(n, k)/(n-k+1)*Binomial(n+k-1, n-k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Sep 23 2015

S:= series(RootOf(-x*z^2+z^3+x*z-2*z^2-x+z, z), x, 101):
seq(coeff(S,x,j),j=1..100); # Robert Israel, Nov 19 2015

a[ n_] := If[ n < 2, Boole[n == 1], (n - 1) HypergeometricPFQ[ {n, 1 - n, 2 - n}, {3/2, 2}, 1/4]]; (* Michael Somos, May 28 2014 *)
Join[{1}, Table[Sum[ Binomial[n,k] / (n-k+1) Binomial[n+k-1, n-k], {k, n}], {n, 25}]] (* Vincenzo Librandi, Sep 23 2015 *)

{a(n) = if( n<1, 0, polcoeff( serreverse( x * (1 - x) * (1 - x^2) * (1 - x^3) / (1 - x^6) + x * O(x^n)), n))};

{a(n)=sum(k=0,n,binomial(n,k)/(n-k+1)*binomial(n+k-1,n-k))} \\ Paul D. Hanna, Jun 19 2009

def A109081(n) :
    return (n-1)*hypergeometric([n,1-n,2-n],[3/2, 2],1/4) if n > 1 else 1
[simplify(A109081(n)) for n in (1..24)] # Peter Luschny, Aug 02 2012, Nov 13 2014

G.f. A(x) satisfies 0 = f(x, A(x)) where f(x, y) = x*(1 - y + y^2) - y*(1 - y)^2.
G.f. A(x) satisfies 0 = f(x, A(x)) where f(x, y) = y*(1 - y)*((1 - y) / x + 1) - 1.
From Paul D. Hanna, Jun 19 2009: (Start)
G.f. satisfies: A(x) = x/(1 - x/(1 - A(x))^2).
a(n) = Sum_{k=0..n} C(n,k)/(n-k+1) * C(n+k-1,n-k). (End)
From Gary W. Adamson, Nov 15 2011: (Start)
a(n) is the upper left term in M^(n-1), M = an infinite square matrix as follows:
  1, 1, 0, 0, 0, ...
  2, 1, 1, 0, 0, ...
  3, 2, 1, 1, 0, ...
  4, 3, 2, 1, 1, ...
  5, 4, 3, 2, 1, ...
  ... (End)
With different signs, g.f. = 2/(3-sqrt(1-4xC(x))) where C = g.f. for A000108 [He-Shapiro]. - N. J. A. Sloane, Apr 28 2017
From Vaclav Kotesovec, Aug 14 2018: (Start)
Recurrence: 2*n*(2*n - 1)*(19*n^2 - 85*n + 90)*a(n) = 2*(190*n^4 - 1230*n^3 + 2783*n^2 - 2595*n + 828)*a(n-1) + 2*(n-3)*(38*n^3 - 189*n^2 + 289*n - 132)*a(n-2) + 3*(n-4)*(n-3)*(19*n^2 - 47*n + 24)*a(n-3).
a(n) ~ (1 - (1-s)*s)^(n + 1/2) / (2*sqrt(Pi*(3 - 6*s + s^2)) * n^(3/2) * s^n * (1-s)^(2*n-2)), where s = 0.3611030805286473776346465621590281395264149... is the real root of the equation (s^2 - s + 3)*s = 1. (End)

A219537 G.f. satisfies A(x) = 1 + x*(A(x)^2 - A(x)^3 + A(x)^4).

Original entry on oeis.org

1, 1, 3, 13, 66, 366, 2148, 13115, 82449, 530095, 3469401, 23037642, 154820262, 1050999343, 7196493255, 49644745965, 344704716018, 2407157839593, 16895247295947, 119121868831235, 843306880720218, 5992060655349521, 42718501097385207, 305476181765843358

Offset: 0

Paul D. Hanna, Nov 21 2012

nonn

a(n) is the number of noncrossing partial matchings on points 1, 2, ... , 3*n where point 1 is unmatched if n>0 and only points congruent modulo 3 can be matched. See Example 57 on p. 47 of the Burstein-Shapiro reference. - Alexander Burstein, Jun 03 2022

			G.f.: A(x) = 1 + x + 3*x^2 + 13*x^3 + 66*x^4 + 366*x^5 + 2148*x^6 +...
Related expansions:
A(x)^2 = 1 + 2*x + 7*x^2 + 32*x^3 + 167*x^4 + 942*x^5 + 5593*x^6 +...
A(x)^3 = 1 + 3*x + 12*x^2 + 58*x^3 + 312*x^4 + 1794*x^5 + 10794*x^6 +...
A(x)^4 = 1 + 4*x + 18*x^2 + 92*x^3 + 511*x^4 + 3000*x^5 + 18316*x^6 +...
A(x)^5 = 1 + 5*x + 25*x^2 + 135*x^3 + 775*x^4 + 4651*x^5 + 28845*x^6 +...
A(x)^6 = 1 + 6*x + 33*x^2 + 188*x^3 + 1116*x^4 + 6852*x^5 + 43204*x^6 +...
where A(x) = 1 + x*(A(x)^2 - A(x)^3 + A(x)^4),
and A(x)^2 = 1 + x*(A(x)^2 + A(x)^5),
and A(x)^3 = 1 + x*(A(x)^2 + A(x)^4 + A(x)^6),
and A(x)^4 = 1 + x*(A(x)^2 + A(x)^4 + A(x)^5 + A(x)^7),
and A(x)^5 = 1 + x*(A(x)^2 + A(x)^4 + A(x)^5 + A(x)^6 + A(x)^8), etc.
The g.f. satisfies A(x) = F(x*A(x)^2) and F(x) = A(x/F(x)^2) where
F(x) = 1 + x + x^2 + 2*x^3 + 4*x^4 + 9*x^5 + 21*x^6 + 51*x^7 +...+ A001006(n-1)*x^n +...
is a g.f. of the Motzkin numbers (A001006, shifted right 1 place).
The g.f. satisfies A(x) = G(x*A(x)) and G(x) = A(x/G(x)) where
G(x) = 1 + x + 2*x^2 + 6*x^3 + 21*x^4 + 80*x^5 + 322*x^6 +...+ A106228(n)*x^n +...
satisfies G(x) = 1 + x*G(x)/(1 - x*G(x)^2).

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Alexander Burstein and Louis W. Shapiro, Pseudo-Involutions in the Riordan Group, J. Integer Sequences 25 (2022), Article 22.3.6.

Cf. A219534, A219535, A219536, A219538, A363573, A366400.

Cf. A001006, A106228, A271469, A364765.

rec := {(36*n^4+126*n^3+126*n^2+36*n)*a(n)+(-276*n^4-1548*n^3-3198*n^2-2898*n-972)*a(n+1)+(940*n^4+7090*n^3+19916*n^2+24650*n+11316)*a(n+2)+(-845*n^4-9000*n^3-34159*n^2-53004*n-26136)*a(n+3)+(-260*n^4-5200*n^3-37454*n^2-116538*n-133128)*a(n+4)+(459*n^4+9774*n^3+77955*n^2+276012*n+366060)*a(n+5)+(-54*n^4-1242*n^3-10686*n^2-40758*n-58140)*a(n+6), a(0) = 1, a(1) = 1, a(2) = 3, a(3) = 13, a(4) = 66, a(5) = 366}:
f:= gfun:-rectoproc(rec,a(n),remember):
map(f, [$0..50]); # Robert Israel, Feb 25 2018

nmax = 23; sol = {a[0] -> 1};
Do[A[x_] = Sum[a[k] x^k, {k, 0, n}] /. sol; eq = CoefficientList[A[x] - (1 + x (A[x]^2 - A[x]^3 + A[x]^4)) + O[x]^(n + 1), x] == 0 /. sol; sol = sol ~Join~ Solve[eq][[1]], {n, 1, nmax}];
sol /. Rule -> Set;
a /@ Range[0, nmax] (* Jean-François Alcover, Nov 01 2019 *)

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

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

G.f. A(x) satisfies [from Paul D. Hanna, Mar 21 2016]: (Start)
(1) A(x)^2 = 1 + x*(A(x)^2 + A(x)^5).
(2) A(x)^3 = 1 + x*(A(x)^2 + A(x)^4 + A(x)^6).
Let F(x) = (1+x - sqrt(1 - 2*x - 3*x^2)) / (2*x), then g.f. A(x) satisfies:
(3) A(x) = sqrt( (1/x)*Series_Reversion(x/F(x)^2) ),
(4) A(x) = F(x*A(x)^2) and F(x) = A(x/F(x)^2),
where F(x) = 1 + x*M(x) such that M(x) = 1 + x*M(x) + x^2*M(x)^2 is the g.f. of the Motzkin numbers (A001006).
Let G(x) = 1 + x*G(x)/(1 - x*G(x)^2), then g.f. A(x) satisfies:
(5) A(x) = (1/x)*Series_Reversion(x/G(x)),
(6) A(x) = G(x*A(x)) and G(x) = A(x/G(x)).
where G(x) is the g.f. of A106228. (End)
Recurrence: 3*n*(3*n-1)*(3*n+1)*(5*n-11)*(5*n-8)*(5*n-6)*a(n) = 6*(5*n-11)*(900*n^5 - 3870*n^4 + 6033*n^3 - 4165*n^2 + 1238*n - 120)*a(n-1) - 2*(n-2)*(5*n-1)*(950*n^4 - 5510*n^3 + 11199*n^2 - 9207*n + 2430)*a(n-2) + 6*(n-3)*(n-2)*(2*n-5)*(5*n-6)*(5*n-3)*(5*n-1)*a(n-3). - Vaclav Kotesovec, Aug 19 2013
a(n) ~ sqrt(300+75*10^(2/3)+30*10^(1/3))/90 * (5/9*10^(2/3)+10/9*10^(1/3)+8/3)^n / (sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Aug 19 2013
Recurrence: 18*n*(2*n+1)*(n+2)*(n+1)*a(n)-(6*(n+1))*(n+2)*(46*n^2+120*n+81)*a(n+1)+(2*(n+2))*(470*n^3+2605*n^2+4748*n+2829)*a(n+2)-(n+3)*(845*n^3+6465*n^2+14764*n+8712)*a(n+3)-(2*(n+4))*(130*n^3+2080*n^2+10407*n+16641)*a(n+4)+(3*(n+5))*(153*n^3+2493*n^2+13520*n+24404)*a(n+5)-(6*(n+5))*(3*n+17)*(3*n+19)*(n+6)*a(n+6) = 0. - Robert Israel, Feb 25 2018
G.f. A(x) satisfies: A(-x*A(x)^5) = 1/A(x). - Alexander Burstein, Jun 03 2022
a(n) = (1/n) * Sum_{k=0..n-1} binomial(n,k) * binomial(2*n+k,n-1-k) for n > 0. - Seiichi Manyama, Aug 05 2023
a(n) = (1/n) * Sum_{k=0..n-1} (-1)^k * binomial(n,k) * binomial(4*n-2*k,n-1-k) for n > 0. - Seiichi Manyama, Aug 06 2023
G.f.: A(x) = sqrt(B(x)) where B(x) is the g.f. of A366400. - Seiichi Manyama, Mar 31 2024
a(n) = (1/n) * Sum_{k=0..floor((n-1)/2)} 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(n+3*k/2+1/2,n)/(2*n+3*k+1). - Seiichi Manyama, Apr 04 2024
G.f.: Sum_{k>=0} binomial(5*k/2 + 1/2, k)*x^k/((5*k + 1)*(1 - x)^((5*k + 1)/2)). - Miles Wilson, Feb 02 2025

A271469 G.f. satisfies A(x) = 1 + x*(A(x)^3 - A(x)^4 + A(x)^5).

Original entry on oeis.org

1, 1, 4, 23, 155, 1142, 8910, 72350, 605056, 5175866, 45077560, 398348733, 3562916317, 32192775763, 293410452560, 2694283228653, 24902681767987, 231496130358758, 2162985033344112, 20301976721356134, 191336242071696514, 1809916398759630481, 17178063381786563194, 163536967014934201972, 1561247114394683682834, 14943175106109268856975

Offset: 0

Paul D. Hanna, Apr 08 2016

nonn

			G.f.: A(x) = 1 + x + 4*x^2 + 23*x^3 + 155*x^4 + 1142*x^5 + 8910*x^6 +...
Related expansions:
A(x)^2 = 1 + 2*x + 9*x^2 + 54*x^3 + 372*x^4 + 2778*x^5 + 21873*x^6 +...
A(x)^3 = 1 + 3*x + 15*x^2 + 94*x^3 + 663*x^4 + 5025*x^5 + 39970*x^6 +...
A(x)^4 = 1 + 4*x + 22*x^2 + 144*x^3 + 1041*x^4 + 8016*x^5 + 64470*x^6 +...
A(x)^5 = 1 + 5*x + 30*x^2 + 205*x^3 + 1520*x^4 + 11901*x^5 + 96850*x^6 +...
A(x)^6 = 1 + 6*x + 39*x^2 + 278*x^3 + 2115*x^4 + 16848*x^5 + 138816*x^6 +...
A(x)^7 = 1 + 7*x + 49*x^2 + 364*x^3 + 2842*x^4 + 23044*x^5 + 192325*x^6 +...
where A(x) = 1 + x*(A(x)^3 - A(x)^4 + A(x)^5),
and A(x)^2 = 1 + x*(A(x)^3 + A(x)^6),
and A(x)^3 = 1 + x*(A(x)^3 + A(x)^5 + A(x)^7),
and A(x)^4 = 1 + x*(A(x)^3 + A(x)^5 + A(x)^6 + A(x)^8),
and A(x)^5 = 1 + x*(A(x)^3 + A(x)^5 + A(x)^6 + A(x)^7 + A(x)^9), etc.
The g.f. satisfies A(x) = F(x*A(x)^3) and F(x) = A(x/F(x)^3) where
F(x) = 1 + x + x^2 + 2*x^3 + 4*x^4 + 9*x^5 + 21*x^6 + 51*x^7 +...+ A001006(n-1)*x^n +...
is a g.f. of the Motzkin numbers (A001006, shifted right 1 place).
The g.f. satisfies A(x) = G(x*A(x)) and G(x) = A(x/G(x)) where
G(x) = 1 + x + 3*x^2 + 13*x^3 + 66*x^4 + 366*x^5 + 2148*x^6 +...+ A219537(n)*x^n +...
satisfies G(x) = 1 + x*(G(x)^2 - G(x)^3 + G(x)^4).

Seiichi Manyama, Table of n, a(n) for n = 0..998

Cf. A363573, A001006, A370474.
Cf. A001006, A106228, A219537, A364765, A365194.
Cf. A001764, A370476.

CoefficientList[(1/x*InverseSeries[Series[8*x^4/(1 + x - Sqrt[1 - 2*x - 3*x^2])^3, {x, 0, 20}], x])^(1/3), x] (* Vaclav Kotesovec, Apr 16 2016 *)

/* Formula A(x) = 1 + x*(A(x)^3 - A(x)^4 + A(x)^5): */
{a(n)=local(A=1); for(i=1, n, A=1+x*(A^3-A^4+A^5) +x*O(x^n)); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

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

G.f. A(x) satisfies:
(1) A(x)^2 = 1 + x*(A(x)^3 + A(x)^6).
(2) A(x)^3 = 1 + x*(A(x)^3 + A(x)^5 + A(x)^7).
Let F(x) = (1+x - sqrt(1 - 2*x - 3*x^2)) / (2*x), then g.f. A(x) satisfies:
(3) A(x) = ( (1/x)*Series_Reversion(x/F(x)^3) )^(1/3),
(4) A(x) = F(x*A(x)^3) and F(x) = A(x/F(x)^3),
where F(x) = 1 + x*M(x) such that M(x) = 1 + x*M(x) + x^2*M(x)^2 is the g.f. of the Motzkin numbers (A001006).
Let G(x) =  1 + x*(G(x)^2 - G(x)^3 + G(x)^4), then g.f. A(x) satisfies:
(5) A(x) = (1/x)*Series_Reversion(x/G(x)),
(6) A(x) = G(x*A(x)) and G(x) = A(x/G(x)),
where G(x) is the g.f. of A219537.
a(n) ~ sqrt((34 + (34102 - 8262*sqrt(17))^(1/3) + (34102 + 8262*sqrt(17))^(1/3)) / 1632) * ((28 + (513243 - 4131*sqrt(17))^(1/3)/3 + (19009 + 153*sqrt(17))^(1/3)) / 8)^n / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Apr 16 2016
D-finite recurrence: 8*n*(2*n-1)*(4*n-1)*(4*n+1)*(204*n^4 - 1341*n^3 + 3191*n^2 - 3286*n + 1242)*a(n) = 12*(45696*n^8 - 391776*n^7 + 1376164*n^6 - 2580579*n^5 + 2808064*n^4 - 1797694*n^3 + 651566*n^2 - 119476*n + 8160)*a(n-1) - 6*(n-2)*(29376*n^7 - 237168*n^6 + 760044*n^5 - 1236774*n^4 + 1082233*n^3 - 496791*n^2 + 108530*n - 8400)*a(n-2) + 9*(n-3)*(n-2)*(3*n-8)*(3*n-4)*(204*n^4 - 525*n^3 + 392*n^2 - 111*n + 10)*a(n-3). - Vaclav Kotesovec, Apr 16 2016
From Seiichi Manyama, Aug 06 2023: (Start)
a(n) = (1/n) * Sum_{k=0..n-1} binomial(n,k) * binomial(3*n+k,n-1-k) for n > 0.
a(n) = (1/n) * Sum_{k=0..n-1} (-1)^k * binomial(n,k) * binomial(5*n-2*k,n-1-k) for n > 0. (End)
G.f.: A(x) = sqrt(B(x)) where B(x) is the g.f. of A370474. - Seiichi Manyama, Mar 31 2024
a(n) = (1/n) * Sum_{k=0..floor((n-1)/2)} binomial(n,k) * binomial(4*n-k,n-1-2*k) for n > 0. - Seiichi Manyama, Apr 01 2024
a(n) = Sum_{k=0..n} binomial(n,k) * binomial(3*n/2+3*k/2+1/2,n)/(3*n+3*k+1). - Seiichi Manyama, Apr 04 2024

A082759 a(n) = Sum_{k = 0..n} binomial(n,k)*trinomial(n,k), where trinomial(n,k) = trinomial coefficients.

Original entry on oeis.org

1, 2, 8, 35, 160, 752, 3599, 17446, 85376, 420884, 2087008, 10398016, 52010479, 261021854, 1313707256, 6628095035, 33512880640, 169768235840, 861450392708, 4377796514152, 22277498220160, 113502759811000, 578931209245760, 2955873376166144, 15105883318474991

Offset: 0

Emanuele Munarini, May 21 2003

Central coefficients of A115990. - Paul Barry, Feb 25 2011

			G.f. = 1 + 2*x + 8*x^2 + 35*x^3 + 160*x^4 + 752*x^5 + 3599*x^6 + 17446*x^7 + ...

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

Cf. A037011, A106228, A115990.

a := n -> hypergeom([-n, -n, n + 1], [1/2, 1], 1/4):
seq(simplify(a(n)), n = 0..24);  # Peter Luschny, Jan 04 2025

Table[Sum[Binomial[2 n - k, k] Binomial[n, k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 24 2012; typo fixed by Vincenzo Librandi, May 07 2013 *)

a(n)=sum(k=0,n,binomial(n+k,n-k)*binomial(n,k))

a(n) = Sum_{k = 0..n} C(n+k, n-k)*C(n, k). - Benoit Cloitre, Jun 20 2003
2*n*(2*n - 1)*(38*n - 53)*a(n) + ( - 760*n^3 + 1820*n^2 - 1252*n + 252)*a(n - 1) - 8*(n - 1)*(19*n^2 - 36*n + 9)*a(n - 2) - 3*(38*n - 15)*(n - 1)*(n - 2)*a(n - 3) = 0. - Vladeta Jovovic, Jul 15 2004
a(n) = Sum_{k = 0..n} C(2*n - k, k)*C(n, k). - Paul Barry, Jan 20 2005
a(n) ~ c * d^n / sqrt(Pi*n), where d = 5.21913624874158651... = (((1261 + 57*sqrt(57))^(2/3) + 112 + 10*(1261 + 57*sqrt(57))^(1/3))/(6*(1261 + 57*sqrt(57))^(1/3))) is the real root of the equation 4*d^3 - 20*d^2 - 4*d - 3 = 0 and c = 0.79036380822702870439029... = 1/114*sqrt(57)*sqrt((9747 + 57*sqrt(57))^(1/3)*(2*(9747 + 57*sqrt(57))^(2/3) + 912 + 57*(9747 + 57*sqrt(57))^(1/3)))/((9747 + 57*sqrt(57))^(1/3)) is the positive real root of the equation 1216*c^6 - 912*c^4 + 100*c^2 - 3 = 0. - Vaclav Kotesovec, Oct 24 2012 (updated Oct 16 2016, following a suggestion of Michael Somos)
G.f.: A(x) = x*B'(x)/B(x), where B(x) satisfies B(x) = x*(1 + 2*B(x) + 2*B(x)^2 + B(x)^3). - Vladimir Kruchinin, Jan 14 2015
a(n) = Sum_{k = 0..n} (-1)^k*C(n, k)*C(3*n - 2*k, n - k). - Peter Bala, Jul 13 2016
G.f. y = A(x) satisfies  0 = 1 + y*(3-2*x) + y^3*(-4+20*x+4*x^2+3*x^3). - Michael Somos, Oct 15 2016
From Peter Bala, Jan 09 2022: (Start)
a(n) = [x^n] (1 + 2*x + 2*x^2 + x^3)^n.
The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for prime p and positive integers n and k. (End)
a(n) = hypergeom([-n, -n, n + 1], [1/2, 1], 1/4). - Peter Luschny, Jan 04 2025

A300048 G.f. A(x) satisfies A(x)^3 = 1 + xA(x) + xA(x)^2 + x*A(x)^6.

Original entry on oeis.org

1, 1, 2, 7, 29, 131, 627, 3124, 16032, 84162, 449828, 2439550, 13391105, 74256824, 415357737, 2340775363, 13278009018, 75753246286, 434392031856, 2502289328542, 14473290097526, 84023214062635, 489424396591995, 2859551104564120, 16754209625090980, 98415932763515679, 579475837597933632, 3419452319373566239, 20219028961691299994

Offset: 0

Paul D. Hanna, Feb 25 2018

nonn

Compare to: G(x)^3 = 1 + x*G(x) + x*G(x)^2 + x*G(x)^3 holds when G(x) = 1/(1-x).

			G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 29*x^4 + 131*x^5 + 627*x^6 + 3124*x^7 + 16032*x^8 + 84162*x^9 + 449828*x^10 + 2439550*x^11 + 13391105*x^12 + ...
RELATED SERIES.
A(x)^2 = 1 + 2*x + 5*x^2 + 18*x^3 + 76*x^4 + 348*x^5 + 1681*x^6 + 8432*x^7 + 43495*x^8 + 229260*x^9 + 1229371*x^10 + ...
A(x)^6 = 1 + 6*x + 27*x^2 + 122*x^3 + 579*x^4 + 2862*x^5 + 14588*x^6 + 76146*x^7 + 405039*x^8 + 2187756*x^9 + 11967426*x^10 + ...
A(x)^3 = 1 + 3*x + 9*x^2 + 34*x^3 + 147*x^4 + 684*x^5 + 3341*x^6 + 16896*x^7 + 87702*x^8 + 464566*x^9 + 2501178*x^10 + ...

Paul D. Hanna, Table of n, a(n) for n = 0..300

Cf. A000108, A106228, A219537.

{a(n) = my(A=1); for(i=1,n, A = (1 + x*A + x*A^2 + x*A^6 +x*O(x^n))^(1/3) ); polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

a(n) ~ sqrt(1 + sqrt((9 + 40*sqrt(3))/13)) * (9 + 6*sqrt(3) + sqrt(153 + 100*sqrt(3)))^n / (sqrt(Pi) * n^(3/2) * 2^(n + 3/2) * 3^(n + 3/4)). - Vaclav Kotesovec, Aug 11 2021

a(n) = (1/n) * Sum_{k=0..n-1} binomial(n,k) * binomial(n+2*k,n-1-k) for n > 0. - Seiichi Manyama, Aug 05 2023

A364765 G.f. A(x) satisfies A(x) = 1 + xA(x)^4 / (1 - xA(x)^5).

Original entry on oeis.org

1, 1, 5, 36, 304, 2808, 27475, 279845, 2935987, 31511097, 344344868, 3818320487, 42855633210, 485923475563, 5557803724920, 64046876264292, 742908320701832, 8667090253409215, 101631581618367133, 1197190915359577973, 14160413911721178800

Offset: 0

Seiichi Manyama, Aug 06 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..909

Cf. A001006, A106228, A219537, A271469.
Cf. A349331, A364747.
Cf. A378952, A378954.

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

G.f. satisfies A(x) = 1 + x*A(x)^6 / (1 + x*A(x)^4).
a(n) = (1/n) * Sum_{k=0..n-1} binomial(n,k) * binomial(4*n+k,n-1-k) for n > 0.
a(n) = (1/n) * Sum_{k=0..n-1} (-1)^k * binomial(n,k) * binomial(6*n-2*k,n-1-k) for n > 0.
a(n) = (1/n) * Sum_{k=0..floor((n-1)/2)} binomial(n,k) * binomial(5*n-k,n-1-2*k) for n > 0. - Seiichi Manyama, Apr 01 2024

A364759 G.f. satisfies A(x) = 1 + xA(x)^5 / (1 + xA(x)).

Original entry on oeis.org

1, 1, 4, 25, 182, 1447, 12175, 106575, 960579, 8854622, 83089537, 791063172, 7622317663, 74191096721, 728389554533, 7204640725610, 71727367291455, 718195853746770, 7227785937663908, 73069500402699226, 741712341691454837, 7556704348506425398

Offset: 0

Seiichi Manyama, Aug 05 2023

nonn

Cf. A090192, A106228, A364758.

Cf. A364748.

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

a(n) = (1/n) * Sum_{k=0..n-1} (-1)^k * binomial(n,k) * binomial(5*n-4*k,n-1-k) for n > 0.

A336708 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(0,k) = 1 and T(n,k) = (1/n) * Sum_{j=1..n} (-1)^(n-j) * binomial(n,j) * binomial(n+(k-1)*j,j-1) for n > 0.

Original entry on oeis.org

1, 1, 1, 1, 1, -1, 1, 1, 0, 0, 1, 1, 1, -1, 2, 1, 1, 2, 1, 0, -3, 1, 1, 3, 6, 1, 2, -1, 1, 1, 4, 14, 21, 1, 0, 11, 1, 1, 5, 25, 76, 80, 1, -5, -15, 1, 1, 6, 39, 182, 450, 322, 1, 0, -13, 1, 1, 7, 56, 355, 1447, 2818, 1347, 1, 14, 77, 1, 1, 8, 76, 611, 3532, 12175, 18352, 5798, 1, 0, -86

Offset: 0

Seiichi Manyama, Aug 01 2020

			Square array begins:
   1,  1, 1,   1,    1,     1,     1, ...
   1,  1, 1,   1,    1,     1,     1, ...
  -1,  0, 1,   2,    3,     4,     5, ...
   0, -1, 1,   6,   14,    25,    39, ...
   2,  0, 1,  21,   76,   182,   355, ...
  -3,  2, 1,  80,  450,  1447,  3532, ...
  -1,  0, 1, 322, 2818, 12175, 37206, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0-3 give: A007440, A090192, A000012, A106228.
Main diagonal gives A336713.
Cf. A336575, A336706, A336707, A336709.

T[0, k_] := 1; T[n_, k_] := Sum[(-1)^(n - j) * Binomial[n, j] * Binomial[n + (k - 1)*j, j - 1], {j, 1, n}] / n; Table[T[k, n - k], {n, 0, 11}, {k, 0, n}] // Flatten (* Amiram Eldar, Aug 01 2020 *)

{T(n, k) = if(n==0, 1, sum(j=1, n, (-1)^(n-j)*binomial(n, j)*binomial(n+(k-1)*j, j-1))/n)}

{T(n, k) = local(A=1+x*O(x^n)); for(i=0, n, A=1+x*A^k/(1+x*A)); polcoef(A, n)}

G.f. A_k(x) of column k satisfies A_k(x) = 1 + x * A_k(x)^k / (1 + x * A_k(x)).

Typo in name corrected by Georg Fischer, Sep 19 2023

A364758 G.f. A(x) satisfies A(x) = 1 + xA(x)^4 / (1 + xA(x)).

Original entry on oeis.org

1, 1, 3, 14, 76, 450, 2818, 18352, 123028, 843345, 5884227, 41650479, 298352365, 2158751879, 15754446893, 115830820439, 857147952469, 6379136387303, 47715901304501, 358529599468636, 2704884469806606, 20481615947325089, 155605509972859999, 1185779099027494848

Offset: 0

Seiichi Manyama, Aug 05 2023

nonn

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

Cf. A090192, A106228, A364759, A378919.
Cf. A001764, A219537, A364865, A365224.
Cf. A300048, A364747, A378889.

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

a(n, r=1, s=-1, t=4, u=1) = r*sum(k=0, n, binomial(t*k+u*(n-k)+r, k)*binomial(s*k, n-k)/(t*k+u*(n-k)+r)); \\ Seiichi Manyama, Dec 11 2024

a(n) = (1/n) * Sum_{k=0..n-1} (-1)^k * binomial(n,k) * binomial(4*n-3*k,n-1-k) for n > 0.
From Seiichi Manyama, Dec 11 2024: (Start)
G.f. A(x) satisfies A(x)^3 = 1 + x*A(x) + x*A(x)^5 + x*A(x)^6.
G.f. A(x) satisfies A(x) = 1/(1 - x*A(x)^3/(1 + x*A(x))).
If g.f. satisfies A(x) = ( 1 + x*A(x)^(t/r) * (1 + x*A(x)^(u/r))^s )^r, then a(n) = r * Sum_{k=0..n} binomial(t*k+u*(n-k)+r,k) * binomial(s*k,n-k)/(t*k+u*(n-k)+r). (End)

A370472 G.f. satisfies A(x) = 1 + x * A(x) * (1 - A(x) + A(x)^2 - A(x)^3 + A(x)^4).

Original entry on oeis.org

1, 1, 3, 15, 88, 565, 3844, 27228, 198670, 1482981, 11271117, 86926262, 678568982, 5351340410, 42570335161, 341201704970, 2752693408051, 22335989938093, 182166978172055, 1492496248447713, 12278191839580716, 101382009468089580, 839932374157895727

Offset: 0

Seiichi Manyama, Mar 31 2024

nonn

Cf. A106228, A370471.

Cf. A370473, A370476.

a(n) = sum(k=0, n, binomial(n, k)*binomial(n/2+5*k/2+1/2, n)/(n+5*k+1));

G.f. A(x) satisfies:
(1) A(x)^2 = 1 + x * A(x) * (1 + A(x)^5).
(2) A(x) = sqrt(B(x)) where B(x) is the g.f. of A370471.
a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n/2+5*k/2+1/2,n)/(n+5*k+1).

cp's OEIS Frontend

A109081 Reversion of x*(1-x)*(1-x^2)*(1-x^3)/(1-x^6) = x*(1-x)^2/(1-x+x^2).

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A219537 G.f. satisfies A(x) = 1 + x*(A(x)^2 - A(x)^3 + A(x)^4).

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A271469 G.f. satisfies A(x) = 1 + x*(A(x)^3 - A(x)^4 + A(x)^5).

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A082759 a(n) = Sum_{k = 0..n} binomial(n,k)*trinomial(n,k), where trinomial(n,k) = trinomial coefficients.

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A300048 G.f. A(x) satisfies A(x)^3 = 1 + x*A(x) + x*A(x)^2 + x*A(x)^6.

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A364765 G.f. A(x) satisfies A(x) = 1 + x*A(x)^4 / (1 - x*A(x)^5).

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A364759 G.f. satisfies A(x) = 1 + x*A(x)^5 / (1 + x*A(x)).

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A109081 Reversion of x(1-x)(1-x^2)(1-x^3)/(1-x^6) = x(1-x)^2/(1-x+x^2).

A300048 G.f. A(x) satisfies A(x)^3 = 1 + xA(x) + xA(x)^2 + x*A(x)^6.

A364765 G.f. A(x) satisfies A(x) = 1 + xA(x)^4 / (1 - xA(x)^5).

A364759 G.f. satisfies A(x) = 1 + xA(x)^5 / (1 + xA(x)).

A364758 G.f. A(x) satisfies A(x) = 1 + xA(x)^4 / (1 + xA(x)).