A365770 -id:A365770 - 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)

A365772 Expansion of g.f. A(x) satisfying A(x) = 1 + xA(x)/(1 - 2x*A(x))^2.

Original entry on oeis.org

1, 1, 5, 25, 137, 801, 4893, 30857, 199377, 1313089, 8782389, 59491257, 407308377, 2814044897, 19594237133, 137364464681, 968743846561, 6868059398273, 48921561805413, 349942779608153, 2512722402972457, 18104571857859233, 130856263145140861, 948520413875412681

Offset: 0

Paul D. Hanna, Oct 04 2023

nonn

Related identities which hold formally for all Maclaurin series F(x):
(1) F(x) = (1/x) * Sum{n>=1} n^(n-1) * x^n * F(x)^n / (1 + n*x*F(x))^(n+1),
(2) F(x) = (2/x) * Sum{n>=1} n*(n+1)^(n-2) * x^n * F(x)^n / (1 + (n+1)*x*F(x))^(n+1),
(3) F(x) = (3/x) * Sum{n>=1} n*(n+2)^(n-2) * x^n * F(x)^n / (1 + (n+2)*x*F(x))^(n+1),
(4) F(x) = (4/x) * Sum{n>=1} n*(n+3)^(n-2) * x^n * F(x)^n / (1 + (n+3)*x*F(x))^(n+1),
(5) F(x) = (k/x) * Sum{n>=1} n*(n+k-1)^(n-2) * x^n * F(x)^n / (1 + (n+k-1)*x*F(x))^(n+1) for all fixed nonzero k.

			G.f.: A(x) = 1 + x + 5*x^2 + 25*x^3 + 137*x^4 + 801*x^5 + 4893*x^6 + 30857*x^7 + 199377*x^8 + 1313089*x^9 + 8782389*x^10 + ...
where A(x) satisfies A(x) = 1 + x*A(x)/(1 - 2*x*A(x))^2
also
A(x) = 1 + x*A(x)/(1 + (-1)*x*A(x))^2 + 2*x^2*A(x)^2/(1 + 0*x*A(x))^3 + 3^2*x^3*A(x)^3/(1 + 1*x*A(x))^4 + 4^3*x^4*A(x)^4/(1 + 2*x*A(x))^5 + 5^4*x^5*A(x)^5/(1 + 3*x*A(x))^6 + 6^5*x^6*A(x)^6/(1 + 4*x*A(x))^7 + ...
and
A(x) = 1 + 3*1*3^(-1)*x*A(x)/(1 + 1*x*A(x))^2 + 3*2*4^0*x^2*A(x)^2/(1 + 2*x*A(x))^3 + 3*3*5^1*x^3*A(x)^3/(1 + 3*x*A(x))^4 + 3*4*6^2*x^4*A(x)^4/(1 + 4*x*A(x))^5 + 3*5*7^3*x^5*A(x)^5/(1 + 5*x*A(x))^6 + ...

Paul D. Hanna, Table of n, a(n) for n = 0..400

Cf. A365770, A109081, A365773, A365774, A365775.

Cf. A366232 (dual).

nmax = 30; A[] = 0; Do[A[x] = 1 + x*A[x]/(1 - 2*x*A[x])^2 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] (* Vaclav Kotesovec, Oct 05 2023 *)

{a(n) = sum(k=0, n, binomial(n+1, n-k)/(n+1) * binomial(2*n-k-1, k) * 2^k)}
for(n=0,30, print1(a(n),", "))

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

a(n) = Sum{k=0..n} binomial(n+1, n-k)/(n+1) * binomial(2*n-k-1, k) * 2^k.
Let A(x)^m = Sum_{n>=0} a(n,m) * x^n then a(n,m) = Sum_{k=0..n} binomial(n+m, n-k)*m/(n+m) * binomial(2*n-k-1, k) * 2^k.
G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) A(x) = 1 + x*A(x)/(1 - 2*x*A(x))^2.
(2) A(x) = (1/x) * Series_Reversion( x/(1 + x/(1 - 2*x)^2) ).
(3) A( x/(1 + x/(1 - 2*x)^2) ) = 1 + x/(1 - 2*x)^2.
(4) A(x) = 1 + (m+1) * Sum{n>=1} n*(n+m)^(n-2) * x^n * A(x)^n / (1 + (n+m-2)*x*A(x))^(n+1) for all fixed nonnegative m.
(4.a) A(x) = 1 + Sum{n>=1} n^(n-1) * x^n * A(x)^n / (1 + (n-2)*x*A(x))^(n+1).
(4.b) A(x) = 1 + 2 * Sum{n>=1} n*(n+1)^(n-2) * x^n * A(x)^n / (1 + (n-1)*x*A(x))^(n+1).
(4.c) A(x) = 1 + 3 * Sum{n>=1} n*(n+2)^(n-2) * x^n * A(x)^n / (1 + n*x*A(x))^(n+1).
(4.d) A(x) = 1 + 4 * Sum{n>=1} n*(n+3)^(n-2) * x^n * A(x)^n / (1 + (n+1)*x*A(x))^(n+1).
a(n) ~ 7^(n + 3/2) * sqrt(3/((-1916 + (1833997600 - 95194848*sqrt(69))^(1/3) + 2^(5/3)*(57312425 + 2974839*sqrt(69))^(1/3))*Pi)) / (2 * n^(3/2) * (1 - 53*(2/(3*(-45 + 161*sqrt(69))))^(1/3) + ((-45 + 161*sqrt(69))/2)^(1/3)/3^(2/3))^n). - Vaclav Kotesovec, Oct 05 2023

A365773 Expansion of g.f. A(x) satisfying A(x) = 1 + xA(x)/(1 - 3x*A(x))^2.

Original entry on oeis.org

1, 1, 7, 46, 325, 2446, 19234, 156115, 1298077, 11000584, 94668508, 825087418, 7267943962, 64602794647, 578726742481, 5219620390558, 47357456920165, 431941341136552, 3958215409319608, 36425213089790932, 336475535026075180, 3118885520601252016, 29000562051786329512

Offset: 0

Paul D. Hanna, Oct 04 2023

nonn

Related identities which hold formally for all Maclaurin series F(x):
(1) F(x) = (1/x) * Sum{n>=1} n^(n-1) * x^n * F(x)^n / (1 + n*x*F(x))^(n+1),
(2) F(x) = (2/x) * Sum{n>=1} n*(n+1)^(n-2) * x^n * F(x)^n / (1 + (n+1)*x*F(x))^(n+1),
(3) F(x) = (3/x) * Sum{n>=1} n*(n+2)^(n-2) * x^n * F(x)^n / (1 + (n+2)*x*F(x))^(n+1),
(4) F(x) = (4/x) * Sum{n>=1} n*(n+3)^(n-2) * x^n * F(x)^n / (1 + (n+3)*x*F(x))^(n+1),
(5) F(x) = (k/x) * Sum{n>=1} n*(n+k-1)^(n-2) * x^n * F(x)^n / (1 + (n+k-1)*x*F(x))^(n+1) for all fixed nonzero k.

			G.f.: A(x) = 1 + x + 7*x^2 + 46*x^3 + 325*x^4 + 2446*x^5 + 19234*x^6 + 156115*x^7 + 1298077*x^8 + 11000584*x^9 + 94668508*x^10 + ...
where A(x) satisfies A(x) = 1 + x*A(x)/(1 - 3*x*A(x))^2
also
A(x) = 1 + 1^0*x^1*A(x)^1/(1 + (-2)*x*A(x))^2 + 2^1*x^2*A(x)^2/(1 + (-1)*x*A(x))^3 + 3^2*x^3*A(x)^3/(1 + 0*x*A(x))^4 + 4^3*x^4*A(x)^4/(1 + 1*x*A(x))^5 + 5^4*x^5*A(x)^5/(1 + 2*x*A(x))^6 + 6^5*x^6*A(x)^6/(1 + 3*x*A(x))^7 + ...
and
A(x) = 1 + 4*1*4^(-1)*x*A(x)/(1 + 1*x*A(x))^2 + 4*2*5^0*x^2*A(x)^2/(1 + 2*x*A(x))^3 + 4*3*6^1*x^3*A(x)^3/(1 + 3*x*A(x))^4 + 4*4*7^2*x^4*A(x)^4/(1 + 4*x*A(x))^5 + 4*5*8^3*x^5*A(x)^5/(1 + 5*x*A(x))^6 + ...

Paul D. Hanna, Table of n, a(n) for n = 0..400

Cf. A365770, A109081, A365772, A365774, A365775.

Cf. A366233 (dual).

{a(n) = sum(k=0, n, binomial(n+1, n-k)/(n+1) * binomial(2*n-k-1, k) * 3^k)}
for(n=0,30, print1(a(n),", "))

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

a(n) = Sum{k=0..n} binomial(n+1, n-k)/(n+1) * binomial(2*n-k-1, k) * 3^k.
Let A(x)^m = Sum_{n>=0} a(n,m) * x^n then a(n,m) = Sum_{k=0..n} binomial(n+m, n-k)*m/(n+m) * binomial(2*n-k-1, k) * 3^k.
G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) A(x) = 1 + x*A(x)/(1 - 3*x*A(x))^2.
(2) A(x) = (1/x) * Series_Reversion( x/(1 + x/(1 - 3*x)^2) ).
(3) A( x/(1 + x/(1 - 3*x)^2) ) = 1 + x/(1 - 3*x)^2.
(4) A(x) = 1 + (m+1) * Sum{n>=1} n*(n+m)^(n-2) * x^n * A(x)^n / (1 + (n+m-3)*x*A(x))^(n+1) for all fixed nonnegative m.
(4.a) A(x) = 1 + Sum{n>=1} n^(n-1) * x^n * A(x)^n / (1 + (n-3)*x*A(x))^(n+1).
(4.b) A(x) = 1 + 2 * Sum{n>=1} n*(n+1)^(n-2) * x^n * A(x)^n / (1 + (n-2)*x*A(x))^(n+1).
(4.c) A(x) = 1 + 3 * Sum{n>=1} n*(n+2)^(n-2) * x^n * A(x)^n / (1 + (n-1)*x*A(x))^(n+1).
(4.d) A(x) = 1 + 4 * Sum{n>=1} n*(n+3)^(n-2) * x^n * A(x)^n / (1 + n*x*A(x))^(n+1).
(4.e) A(x) = 1 + 5 * Sum{n>=1} n*(n+4)^(n-2) * x^n * A(x)^n / (1 + (n+1)*x*A(x))^(n+1).
a(n) ~ 3^(1 + 3*n) * 11^(3/2 + n) / (2*sqrt((65 - 288/(1031 + 121*sqrt(73))^(1/3) + 16*(1031 + 121*sqrt(73))^(1/3)) * Pi) * n^(3/2) * (52 - (5182*2^(2/3)) / (-174721 + 65043*sqrt(73))^(1/3) + (2*(-174721 + 65043*sqrt(73)))^(1/3))^(n + 1/2)). - Vaclav Kotesovec, Nov 16 2023

A365774 Expansion of g.f. A(x) satisfying A(x) = 1 + xA(x)/(1 - 4x*A(x))^2.

Original entry on oeis.org

1, 1, 9, 73, 625, 5681, 53945, 528697, 5307489, 54298849, 564079337, 5934390441, 63098046929, 676976915473, 7319925023897, 79684985945753, 872620958369473, 9606337027601345, 106249046704511945, 1180096759408431881, 13156993620315230001, 147193406523115480049

Offset: 0

Paul D. Hanna, Oct 04 2023

nonn

Related identities which hold formally for all Maclaurin series F(x):
(1) F(x) = (1/x) * Sum{n>=1} n^(n-1) * x^n * F(x)^n / (1 + n*x*F(x))^(n+1),
(2) F(x) = (2/x) * Sum{n>=1} n*(n+1)^(n-2) * x^n * F(x)^n / (1 + (n+1)*x*F(x))^(n+1),
(3) F(x) = (3/x) * Sum{n>=1} n*(n+2)^(n-2) * x^n * F(x)^n / (1 + (n+2)*x*F(x))^(n+1),
(4) F(x) = (4/x) * Sum{n>=1} n*(n+3)^(n-2) * x^n * F(x)^n / (1 + (n+3)*x*F(x))^(n+1),
(5) F(x) = (k/x) * Sum{n>=1} n*(n+k-1)^(n-2) * x^n * F(x)^n / (1 + (n+k-1)*x*F(x))^(n+1) for all fixed nonzero k.

			G.f.: A(x) = 1 + x + 9*x^2 + 73*x^3 + 625*x^4 + 5681*x^5 + 53945*x^6 + 528697*x^7 + 5307489*x^8 + 54298849*x^9 + 564079337*x^10 + ...
where A(x) satisfies A(x) = 1 + x*A(x)/(1 - 4*x*A(x))^2
also
A(x) = 1 + 1^0*x*A(x)/(1 + (-3)*x*A(x))^2 + 2^1*x^2*A(x)^2/(1 + (-2)*x*A(x))^3 + 3^2*x^3*A(x)^3/(1 + (-1)*x*A(x))^4 + 4^3*x^4*A(x)^4/(1 + 0*x*A(x))^5 + 5^4*x^5*A(x)^5/(1 + 1*x*A(x))^6 + 6^5*x^6*A(x)^6/(1 + 2*x*A(x))^7 + ...
and
A(x) = 1 + 5*1*5^(-1)*x*A(x)/(1 + 1*x*A(x))^2 + 5*2*6^0*x^2*A(x)^2/(1 + 2*x*A(x))^3 + 5*3*7^1*x^3*A(x)^3/(1 + 3*x*A(x))^4 + 5*4*8^2*x^4*A(x)^4/(1 + 4*x*A(x))^5 + 5*5*9^3*x^5*A(x)^5/(1 + 5*x*A(x))^6 + ...

Paul D. Hanna, Table of n, a(n) for n = 0..400

Cf. A365770, A109081, A365772, A365773, A365775.

Cf. A366234 (dual).

{a(n) = sum(k=0, n, binomial(n+1, n-k)/(n+1) * binomial(2*n-k-1, k) * 4^k)}
for(n=0,30, print1(a(n),", "))

{a(n) = my(A = (1/x) * serreverse( x/(1 + x/(1 - 4*x +O(x^(n+2)) )^2) ) ); polcoeff(A,n)}
for(n=0,30, print1(a(n),", "))

a(n) = Sum{k=0..n} binomial(n+1, n-k)/(n+1) * binomial(2*n-k-1, k) * 4^k.
Let A(x)^m = Sum_{n>=0} a(n,m) * x^n then a(n,m) = Sum_{k=0..n} binomial(n+m, n-k)*m/(n+m) * binomial(2*n-k-1, k) * 4^k.
G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) A(x) = 1 + x*A(x)/(1 - 4*x*A(x))^2.
(2) A(x) = (1/x) * Series_Reversion( x/(1 + x/(1 - 4*x)^2) ).
(3) A( x/(1 + x/(1 - 4*x)^2) ) = 1 + x/(1 - 4*x)^2.
(4) A(x) = 1 + (m+1) * Sum{n>=1} n*(n+m)^(n-2) * x^n * A(x)^n / (1 + (n+m-4)*x*A(x))^(n+1) for all fixed nonnegative m.
(4.a) A(x) = 1 + Sum{n>=1} n^(n-1) * x^n * A(x)^n / (1 + (n-4)*x*A(x))^(n+1).
(4.b) A(x) = 1 + 2 * Sum{n>=1} n*(n+1)^(n-2) * x^n * A(x)^n / (1 + (n-3)*x*A(x))^(n+1).
(4.c) A(x) = 1 + 3 * Sum{n>=1} n*(n+2)^(n-2) * x^n * A(x)^n / (1 + (n-2)*x*A(x))^(n+1).
(4.d) A(x) = 1 + 4 * Sum{n>=1} n*(n+3)^(n-2) * x^n * A(x)^n / (1 + (n-1)*x*A(x))^(n+1).
(4.e) A(x) = 1 + 5 * Sum{n>=1} n*(n+4)^(n-2) * x^n * A(x)^n / (1 + n*x*A(x))^(n+1).

A365775 Expansion of g.f. A(x) satisfying A(x) = 1 + xA(x)/(1 - 5x*A(x))^2.

Original entry on oeis.org

1, 1, 11, 106, 1061, 11226, 124026, 1414211, 16515981, 196551736, 2375042076, 29062573926, 359407971786, 4484868410231, 56399986492661, 714067825064426, 9094408567049701, 116436367409647736, 1497734068943432856, 19346547929074098836, 250851388061224003276

Offset: 0

Paul D. Hanna, Oct 04 2023

nonn

Related identities which hold formally for all Maclaurin series F(x):
(1) F(x) = (1/x) * Sum{n>=1} n^(n-1) * x^n * F(x)^n / (1 + n*x*F(x))^(n+1),
(2) F(x) = (2/x) * Sum{n>=1} n*(n+1)^(n-2) * x^n * F(x)^n / (1 + (n+1)*x*F(x))^(n+1),
(3) F(x) = (3/x) * Sum{n>=1} n*(n+2)^(n-2) * x^n * F(x)^n / (1 + (n+2)*x*F(x))^(n+1),
(4) F(x) = (4/x) * Sum{n>=1} n*(n+3)^(n-2) * x^n * F(x)^n / (1 + (n+3)*x*F(x))^(n+1),
(5) F(x) = (k/x) * Sum{n>=1} n*(n+k-1)^(n-2) * x^n * F(x)^n / (1 + (n+k-1)*x*F(x))^(n+1) for all fixed nonzero k.

			G.f.: A(x) = 1 + x + 11*x^2 + 106*x^3 + 1061*x^4 + 11226*x^5 + 124026*x^6 + 1414211*x^7 + 16515981*x^8 + 196551736*x^9 + 2375042076*x^10 + ...
where A(x) satisfies A(x) = 1 + x*A(x)/(1 - 5*x*A(x))^2
also
A(x) = 1 + 1^0*x*A(x)/(1 + (-4)*x*A(x))^2 + 2^1*x^2*A(x)^2/(1 + (-3)*x*A(x))^3 + 3^2*x^3*A(x)^3/(1 + (-2)*x*A(x))^4 + 4^3*x^4*A(x)^4/(1 + (-1)*x*A(x))^5 + 5^4*x^5*A(x)^5/(1 + 0*x*A(x))^6 + 6^5*x^6*A(x)^6/(1 + 1*x*A(x))^7 + ...
and
A(x) = 1 + 6*1*6^(-1)*x*A(x)/(1 + 1*x*A(x))^2 + 6*2*7^0*x^2*A(x)^2/(1 + 2*x*A(x))^3 + 6*3*8^1*x^3*A(x)^3/(1 + 3*x*A(x))^4 + 6*4*9^2*x^4*A(x)^4/(1 + 4*x*A(x))^5 + 6*5*10^3*x^5*A(x)^5/(1 + 5*x*A(x))^6 + ...

Paul D. Hanna, Table of n, a(n) for n = 0..400

Cf. A365770, A109081, A365772, A365773, A365774.

Cf. A366235 (dual).

{a(n) = sum(k=0, n, binomial(n+1, n-k)/(n+1) * binomial(2*n-k-1, k) * 5^k)}
for(n=0,30, print1(a(n),", "))

{a(n) = my(A = (1/x) * serreverse( x/(1 + x/(1 - 5*x +O(x^(n+2)) )^2) ) ); polcoeff(A,n)}
for(n=0,30, print1(a(n),", "))

a(n) = Sum{k=0..n} binomial(n+1, n-k)/(n+1) * binomial(2*n-k-1, k) * 5^k.
Let A(x)^m = Sum_{n>=0} a(n,m) * x^n then a(n,m) = Sum_{k=0..n} binomial(n+m, n-k)*m/(n+m) * binomial(2*n-k-1, k) * 5^k.
G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) A(x) = 1 + x*A(x)/(1 - 5*x*A(x))^2.
(2) A(x) = (1/x) * Series_Reversion( x/(1 + x/(1 - 5*x)^2) ).
(3) A( x/(1 + x/(1 - 5*x)^2) ) = 1 + x/(1 - 5*x)^2.
(4) A(x) = 1 + (m+1) * Sum{n>=1} n*(n+m)^(n-2) * x^n * A(x)^n / (1 + (n+m-5)*x*A(x))^(n+1) for all fixed nonnegative m.
(4.a) A(x) = 1 + Sum{n>=1} n^(n-1) * x^n * A(x)^n / (1 + (n-5)*x*A(x))^(n+1).
(4.b) A(x) = 1 + 2 * Sum{n>=1} n*(n+1)^(n-2) * x^n * A(x)^n / (1 + (n-4)*x*A(x))^(n+1).
(4.c) A(x) = 1 + 3 * Sum{n>=1} n*(n+2)^(n-2) * x^n * A(x)^n / (1 + (n-3)*x*A(x))^(n+1).
(4.d) A(x) = 1 + 4 * Sum{n>=1} n*(n+3)^(n-2) * x^n * A(x)^n / (1 + (n-2)*x*A(x))^(n+1).
(4.e) A(x) = 1 + 5 * Sum{n>=1} n*(n+4)^(n-2) * x^n * A(x)^n / (1 + (n-1)*x*A(x))^(n+1).
(4.f) A(x) = 1 + 6 * Sum{n>=1} n*(n+5)^(n-2) * x^n * A(x)^n / (1 + n*x*A(x))^(n+1).
a(n) ~ sqrt(3) * 5^(2*n) * (19^(3/2 + n) / (2*sqrt((113 + (28*(47225 + 1083*sqrt(1905))^(1/3))/5^(2/3) - 2632/(5*(47225 + 1083*sqrt(1905)))^(1/3))*Pi) * n^(3/2) * (68 + (2*(-1496331 + 60325*sqrt(1905)))^(1/3)/3^(2/3) - 9214*2^(2/3)/(3*(-1496331 + 60325*sqrt(1905)))^(1/3))^(n + 1/2))). - Vaclav Kotesovec, Oct 06 2023

A365771 a(n) = binomial(2n+1, n)/(2n+1) * binomial(3*n-1, n) for n >= 0.

Original entry on oeis.org

1, 2, 20, 280, 4620, 84084, 1633632, 33256080, 701149020, 15191562100, 336424047960, 7584833081280, 173575987821600, 4022766574898400, 94247674040476800, 2228957491057276320, 53150802525726081660, 1276661433215969608500, 30863850087221160009000

Offset: 0

Paul D. Hanna, Oct 10 2023

Equals the central terms of triangle A365770.
Conjectures: given A033042 is the sums of distinct powers of 5, then
(1) a(5*A033042(n)) == 4 (mod 5) for n > 0,
(2) a(5*A033042(n) + 1) == 2 (mod 5) for n > 0,
(3) a(n) == 0 (mod 5) for n > 0 except when n or n-1 equals 5*A033042(k) for some k >= 0.

Paolo Xausa, Table of n, a(n) for n = 0..500

Cf. A007004, A000108, A165817, A033042, A319578, A356770.

A365771[n_] := Binomial[2*n + 1, n]/(2*n + 1)*Binomial[3*n - 1, n];
Array[A365771, 20, 0] (* Paolo Xausa, Oct 12 2024 *)

{a(n) = binomial(2*n+1, n)/(2*n+1) * binomial(3*n-1, n)}
for(n=0,30,print1(a(n),", "))

from math import comb
def A365771(n): return comb(m:=(n<<1)+1,n)*comb(m+n-2,n)//m if n else 1 # Chai Wah Wu, Oct 11 2023

a(n) = A365770(2*n,n) for n >= 0.
a(n) = A000108(n) * A165817(n) for n >= 0.
a(n) = 2*A319578(n) = (2/3) * A007004(n) for n >= 1. - Peter Bala, Aug 25 2025

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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A365772 Expansion of g.f. A(x) satisfying A(x) = 1 + x*A(x)/(1 - 2*x*A(x))^2.

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A365773 Expansion of g.f. A(x) satisfying A(x) = 1 + x*A(x)/(1 - 3*x*A(x))^2.

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A365774 Expansion of g.f. A(x) satisfying A(x) = 1 + x*A(x)/(1 - 4*x*A(x))^2.

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A365775 Expansion of g.f. A(x) satisfying A(x) = 1 + x*A(x)/(1 - 5*x*A(x))^2.

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A365771 a(n) = binomial(2*n+1, n)/(2*n+1) * binomial(3*n-1, n) for n >= 0.

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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).

A365772 Expansion of g.f. A(x) satisfying A(x) = 1 + xA(x)/(1 - 2x*A(x))^2.

A365773 Expansion of g.f. A(x) satisfying A(x) = 1 + xA(x)/(1 - 3x*A(x))^2.

A365774 Expansion of g.f. A(x) satisfying A(x) = 1 + xA(x)/(1 - 4x*A(x))^2.

A365775 Expansion of g.f. A(x) satisfying A(x) = 1 + xA(x)/(1 - 5x*A(x))^2.

A365771 a(n) = binomial(2n+1, n)/(2n+1) * binomial(3*n-1, n) for n >= 0.