A109081 -id:A109081 - cp's OEIS frontend

A106228 G.f. satisfies A(x) = 1 + xA(x)/(1 - xA(x)^2).

1, 1, 2, 6, 21, 80, 322, 1347, 5798, 25512, 114236, 518848, 2384538, 11068567, 51817118, 244370806, 1159883685, 5536508864, 26560581688, 127993221140, 619280193640, 3007251366000, 14651743202152, 71601107803904, 350873710447210, 1723795243004223

Offset: 0

Paul D. Hanna, May 19 2005

nonn

Number of paths from (0,0) to (3n-3,0) that stay in the first quadrant (but may touch the horizontal axis), consisting of steps u=(2,1), U=(1,2), or d=(1,-1) and have no tripledescents (ddd). Example: a(3)=6 because we have udud, Uddud, udUdd, UddUdd, uudd and Ududd (the remaining four paths contain the string ddd: uUddd, UdUddd, Uuddd and UUdddd; see A027307). - Emeric Deutsch, Jun 08 2005
a(n) = number of node-labeled ordered trees (A000108) on n vertices, each node labeled with a positive integer <= its outdegree. A node is a non-root non-leaf vertex. Example. a(3)=6 counts the 5 ordered trees on 4 vertices with all labels 1 and the tree
.|.
/ \
with its (one and only) node labeled 2. - David Callan, Jul 14 2006
a(n) = number of Schroeder (n-1)-paths with no triple descents. Example: a(4)=21 counts all 22 Schroeder 3-paths (A006318) except UUUDDD. - David Callan, Jul 14 2006
(1 + 2x + 6x^2 + ...)*(1 + x + 2x^2 + 6x^3) = (1 + 3x + 10x^2 + 37x^3 + ...), where A109081 = (1, 1, 3, 10, 37, ...). - Gary W. Adamson, Nov 15 2011
a(n) = number of Motzkin paths of length 2n-1 with no downsteps in odd position. Example: a(3)=6 counts FFFFF, FFUDF, FUFDF, UDFFF, UDUDF, UFFDF with U an upstep (1,1), F a flatstep (1,0), and D a downstep (1,-1). - David Callan, May 20 2015
Number of permutations of length n that avoid 4123, 4132, and 4213. - Jay Pantone, Oct 01 2015
Conjecturally, the number of sequences (e(1), ..., e(n)), 0 <= e(i) < i, such that there is no triple i < j < k with e(i) > e(j) and e(i) <= e(k). [Martinez and Savage, 2.21] - Eric M. Schmidt, Jul 17 2017
a(n) is the number of permutations of length n avoiding the partially ordered pattern (POP) {1>3, 1>4, 4>2} of length 4. That is, the number of length n permutations having no subsequences of length 4 in which the first element is the largest and the fourth element is larger than the second element. - Sergey Kitaev, Dec 10 2020
a(n) is the number of peakless Motzkin paths of length 2n that do not start with an up edge and where every pair of matching up and down edges occupies positions of the same parity. Equivalently, the number of RNA secondary structures on 2n vertices where the leftmost vertex is not matched and only vertices of the same parity can be matched. - Alexander Burstein, May 17 2021

			A = 1 + x*A + x^2*A^3 + x^3*A^5 + x^4*A^7 + x^5*A^9 + ...
a(4) = 21 since the top row terms of Q^3 = (10, 7, 3, 1). - _Gary W. Adamson_, Nov 15 2011
G.f. = 1 + x + 2*x^2 + 6*x^3 + 21*x^4 + 80*x^5 + 322*x^6 + 1347*x^7 + ...

Jay Pantone, Table of n, a(n) for n = 0..500
Michael H. Albert, Cheyne Homberger, Jay Pantone, Nathaniel Shar, and Vincent Vatter, Generating Permutations with Restricted Containers, arXiv:1510.00269 [math.CO], 2015.
Andrei Asinowski and Cyril Banderier, From geometry to generating functions: rectangulations and permutations, arXiv:2401.05558 [cs.DM], 2024. See page 2.
E. Barcucci, R. Pinzani and R. Sprugnoli, The Motzkin family, P.U.M.A. Ser. A, Vol. 2, 1991, No. 3-4, pp. 249-279.
Paul Barry, Riordan arrays, generalized Narayana triangles, and series reversion, Linear Algebra and its Applications, 491 (2016) 343-385.
N. R. Beaton, M. Bouvel, V. Guerrini, and S. Rinaldi, Slicings of Parallelogram Polyominoes: Catalan, Schröder, Baxter, and Other Sequences, Electr. J. Combin. 26 (2019), P3.13. Also at arXiv:1511.04854 [math.CO], 2015-2019.
Beáta Bényi, Toufik Mansour, and José L. Ramírez, Pattern Avoidance in Weak Ascent Sequences, arXiv:2309.06518 [math.CO], 2023.
Wlodzimierz Bryc, Raouf Fakhfakh, and Wojciech Mlotkowski, Cauchy-Stieltjes families with polynomial variance functions and generalized orthogonality, arXiv:1708.05343 [math.PR], 2017-2019. Also in Probability and Mathematical Statistics (2019), Vol. 39, No. 2, 237-258.
Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group, arXiv:2112.11595 [math.CO], 2021.
David Callan and Toufik Mansour, Enumeration of small Wilf classes avoiding 1342 and two other 4-letter patterns, Pure Mathematics and Applications (2018) Vol. 27, No. 1, 62-97.
Wenqin Cao, Emma Yu Jin, and Zhicong Lin, Enumeration of inversion sequences avoiding triples of relations, Discrete Applied Mathematics (2019); see also author's copy
Emeric Deutsch, Problem 10658, American Math. Monthly, 107, 2000, 368-370.
Alice L. L. Gao and Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, arXiv:1903.08946 [math.CO], 2019.
Alice L. L. Gao and Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, The Electronic Journal of Combinatorics 26(3) (2019), P3.26.
Nancy S. S. Gu, Nelson Y. Li, and Toufik Mansour, 2-Binary trees: bijections and related issues, Discr. Math., 308 (2008), 1209-1221.
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 pp. 2, 22.
Megan A. Martinez and Carla D. Savage, Patterns in Inversion Sequences II: Inversion Sequences Avoiding Triples of Relations, arXiv:1609.08106 [math.CO], 2016.
Arturo Merino and Torsten Mütze, Combinatorial generation via permutation languages. III. Rectangulations, arXiv:2103.09333 [math.CO], 2021.
Chunyan Yan and Zhicong Lin, Inversion sequences avoiding pairs of patterns, arXiv:1912.03674 [math.CO], 2019.

Cf. A027907, A027307, A001006, A086246, A109081, A164965, A262441.

a := proc(n) option remember; if n<2 then 1 elif n=2 then 2 else ((380*n^3-840*n^2+496*n-72)*a(n-1)+(76*n^3-282*n^2+302*n-84)*a(n-2)+(57*n^3-297*n^2+402*n-72)*a(n-3))/(76*n^3-54*n^2-46*n) fi end: seq(a(i),i=0..23); # Peter Luschny, Aug 03 2012

Flatten[{1,Table[1/n*Sum[Binomial[n,k]*Binomial[n+k,n-k-1],{k,0,n-1}],{n,1,20}]}] (* Vaclav Kotesovec, Sep 16 2013 *)
a[ n_] := If[ n < 0, 0, HypergeometricPFQ[{-n, 1 - n, n + 1}, {1, 3/2}, 1/4]]; (* Michael Somos, May 27 2014 *)

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

{a(n) = local(A); if( n<0, 0, n++; A = (1 + x - sqrt(1 - 2*x - 3*x^2 + x * O(x^n))) / 2; polcoeff( serreverse( x^2 / A), n))}; /* Michael Somos, Jun 18 2005 */

{a(n) = if( n<1, n==0, polcoeff( serreverse( x /  (1 + 2*x + 2*x^2 + x^3) + x * O(x^n)), n))}; /* Michael Somos, Dec 31 2014 */

from mpmath import mp
mp.dps = 32; mp.pretty = True
def A106228(n) : return int(mp.hyper([-n, 1-n, n+1], [1, 3/2], 1/4))
[A106228(n) for n in (0..23)] # Peter Luschny, Aug 02 2012

G.f.: A(x) = (1/x)*series_reversion[x/(1 + x*G001006(x))] and thus G.f. satisfies: A(x) = 1 + x*A(x)*G001006(x*A(x)) where G001006(x) is the g.f. of Motzkin numbers A001006.
G.f.: 1 + x*exp( Sum_{n>=1} A082759(n)*x^n/n ), where A082759(n) = Sum_{k=0..n} binomial(n,k)*trinomial(n,k). - Paul D. Hanna, Nov 02 2012
a(n) = (1/n)sum(binomial(n, j+1)*b(n, j), j=0..n-1), where b(n, j) are the trinomial coefficients [b(n, j)=A027907(n, j)=coefficient of x^j in (1+x+x^2)^n]. - Emeric Deutsch, Jun 08 2005
Given g.f. A(x), then B(x) = x*A(x) satisfies 0 = f(x, B(x)) where f(x, y) = y^3 - (1+y)*x*(y-x). - Michael Somos, Jun 18 2005
a(n+1) = Sum[binomial(2n-2k,n-k)*binomial(n+k,n)/(n+1),{k,0,n}]. - David Callan, Aug 16 2006
For n>0: a(n) = 1/n*sum(binomial(n,j)*sum(binomial(j,i)*binomial(n-j,2*j-n-i-1)*2^(2*n-3*j+2*i+1),i=0..n-1), j=0..n); - Vladimir Kruchinin, Dec 26 2010
a(n) = 1/(n+1)*sum(binomial(n+1,k)*binomial(n+k+1,n-k),k,0,n); - Vladimir Kruchinin, Feb 28 2010
a(n) = upper left term in M^n, M = the production matrix:
  1, 1
  1, 1, 1
  2, 2, 1, 1
  3, 3, 2, 1, 1
  4, 4, 3, 2, 1, 1
  5, 5, 4, 3, 2, 1, 1
  ...
- Gary W. Adamson, Jul 08 2011
D-finite with recurrence: 4*n*(2*n+1)*a(n) + 2*(6-5*n-10*n^2)*a(n-1) + 12*(-9*n^2+35*n-33)*a(n-2) - 2*(n-3)*(13*n-28)*a(n-3) - 15*(n-3)*(n-4)*a(n-4) = 0. - R. J. Mathar, Nov 14 2011
From Gary W. Adamson, Nov 15 2011: (Start)
a(n) is the sum of top row terms of Q^(n-1), where Q = the following infinite square production matrix:
  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)
a(n) = 3_F_2([-n, 1-n, n+1], [1, 3/2], 1/4). - Peter Luschny, Aug 02 2012
A four-term recurrence equation is given in the Maple program. Peter Luschny, Aug 03 2012
a(n) ~ 1/228*sqrt(114)*sqrt((32129+3933*sqrt(57))^(1/3) * ((32129+3933*sqrt(57))^(2/3) + 532 + 38*(32129+3933*sqrt(57))^(1/3))) / ((32129+3933*sqrt(57))^(1/3)) * (((1261+57*sqrt(57))^(2/3) + 112 + 10*(1261+57*sqrt(57))^(1/3)) / (6*(1261+57*sqrt(57))^(1/3)))^n / (sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Sep 16 2013
G.f. satisfies x*F(x)^3 - x*F(x)^2 + (x-1)*F(x) + 1 = 0. - Jay Pantone, Oct 01 2015
G.f. satisfies A(-x*A(x)^3) = 1/A(x). - Alexander Burstein, Dec 05 2019
G.f.: A(x) = sqrt(B(x)) where B(x) is the g.f. of A262441. - Seiichi Manyama, Mar 31 2024
a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n/2+3*k/2+1/2,n)/(n+3*k+1). - Seiichi Manyama, Apr 04 2024

A262441 a(n) = Sum_{k=0..n+1}(binomial(n-1,k)/(k+1)*binomial(n+k+1,n-k)).

Original entry on oeis.org

1, 2, 5, 16, 58, 226, 924, 3910, 16979, 75232, 338776, 1545886, 7132580, 33219086, 155963851, 737383488, 3507680650, 16776206680, 80622416976, 389123999656, 1885405316596, 9167409871040, 44717351734160, 218762640192838, 1073082055680180

Offset: 0

Vladimir Kruchinin, Sep 23 2015

nonn

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

Cf. A109081, A161798.

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

Join[{1}, Table[Sum[ Binomial[n-1, k] / (k+1) Binomial[ n+k+1, n-k], {k, 0, n+1}], {n, 25}]] (* Vincenzo Librandi, Sep 23 2015 *)

a(n):=sum(binomial(n,k)*binomial(n+k-2,n-k-1),k,0,n-1)/n;
A(x):=sum(a(n)*x^n,n,1,30);
taylor((1/x-1/A(x)),x,0,10);

a(n)=sum(k=0,n+1,(binomial(n-1,k)/(k+1)*binomial(n+k+1,n-k))) \\ Anders Hellström, Sep 23 2015

G.f.: 1/x-1/A(x), where A(x) is g.f. of A109081.
Recurrence: 2*(n+1)*(2*n - 1)*(19*n - 30)*a(n) = 20*(19*n^3 - 49*n^2 + 34*n - 6)*a(n-1) + 2*(n-2)*(38*n^2 - 79*n + 15)*a(n-2) + 3*(n-3)*(n-2)*(19*n - 11)*a(n-3). - Vaclav Kotesovec, Sep 23 2015
a(n) = (n + 1)*hypergeom([1 - n, -n, n + 2], [3/2, 2], 1/4). - Peter Luschny, Mar 07 2022

A161797 G.f. satisfies: A(x) = 1/(1 - x/(1 - x*A(x))^3).

Original entry on oeis.org

1, 1, 4, 16, 71, 336, 1660, 8464, 44207, 235306, 1271807, 6961307, 38508659, 214950425, 1209170536, 6848080767, 39014400171, 223439516338, 1285660965508, 7428738358924, 43087099589998, 250766507928988, 1464026402082801

Offset: 0

Paul D. Hanna, Jun 19 2009

nonn

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

Cf. A109081, A321798, A321799.

Table[Sum[Binomial[n,k]/(n-k+1) * Binomial[n+2*k-1,n-k], {k,0,n}], {n,0,30}] (* Vaclav Kotesovec, Nov 18 2017 *)

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

a(n) = Sum_{k=0..n} C(n,k)/(n-k+1) * C(n+2*k-1,n-k).
Let A(x)^m = Sum_{n>=0} a(n,m)*x^n then
a(n,m) = Sum_{k=0..n} C(n+m-1,k)*m/(n-k+m) * C(n+2*k-1,n-k).
G.f.: A(x) = (1/x)*serreverse[x/(1 + x/(1 - x)^3)].
Recurrence: 3*(n+1)*(3*n - 2)*(3*n + 2)*(2145*n^4 - 14355*n^3 + 33844*n^2 - 32668*n + 10380)*a(n) = 3*(115830*n^7 - 833085*n^6 + 2195691*n^5 - 2521863*n^4 + 998671*n^3 + 259048*n^2 - 263292*n + 41520)*a(n-1) + 3*(n-2)*(19305*n^6 - 129195*n^5 + 315651*n^4 - 367201*n^3 + 219176*n^2 - 66584*n + 7608)*a(n-2) + 3*(n-3)*(n-2)*(64350*n^5 - 334125*n^4 + 546005*n^3 - 255608*n^2 - 71320*n + 54328)*a(n-3) - 23*(n-4)*(n-3)*(n-2)*(2145*n^4 - 5775*n^3 + 3649*n^2 + 535*n - 654)*a(n-4). - Vaclav Kotesovec, Nov 18 2017
a(n) ~ sqrt(sqrt((159 + 100*sqrt(3))/13) - 2 - 5/sqrt(3)) * (3 + 2*sqrt(3) + sqrt(153 + 100*sqrt(3))/3)^(n+1) / (sqrt(Pi) * n^(3/2) * 2^(n + 5/2)). - Vaclav Kotesovec, Nov 18 2017

A321798 G.f. satisfies: A(x) = 1/(1 - x/(1 - x*A(x))^4).

Original entry on oeis.org

1, 1, 5, 23, 117, 636, 3607, 21106, 126489, 772468, 4789844, 30075937, 190851839, 1222000222, 7885041530, 51222338580, 334720178969, 2198755865424, 14511029102232, 96169424666028, 639757737711300, 4270520564506069, 28595671605541357, 192025292117465445, 1292866976587651519

Offset: 0

Ludovic Schwob, Nov 19 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000 (terms 0..199 from Ludovic Schwob)

Cf. A109081, A161797, A321799.

List([0..25],n->Sum([0..n],k->Binomial(n,k)/(n-k+1)*Binomial(n+3*k-1,n-k))); # Muniru A Asiru, Nov 24 2018

a[n_] := Sum[Binomial[n, k] * Binomial[n + 3k - 1, n - k]/(n - k + 1), {k, 0,
n}]; Array[a, 20, 0] (* Amiram Eldar, Nov 19 2018 *)

a(n) = sum(k=0, n, binomial(n,k)*binomial(n+3*k-1, n-k)/(n-k+1)); \\ Michel Marcus, Nov 19 2018

a(n) = Sum_{k=0..n} (C(n,k)/(n-k+1)) * C(n+3*k-1,n-k).

a(n) ~ sqrt((1 - r*s)*(1 + 3*r*s)/(8*Pi*(8*s - 3))) / (n^(3/2) * r^(n+1)), where r = 0.139684805934917057093949761392656080860096066578... and s = 1.76437708701490464570032194388560298744432681226... are real roots of the system of equations s*(1 - r/(1 - r*s)^4) = 1, 4*r^2*s^2 = (1 - r*s)^5. - Vaclav Kotesovec, Nov 21 2018

A365770 Expansion of g.f. A(x,y) satisfying A(x,y) = 1 + xA(x,y)/(1 - xy * A(x,y))^2, as a triangle of coefficients T(n,k) of x^n*y^k in A(x,y), read by rows n >= 0.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 6, 3, 0, 1, 12, 20, 4, 0, 1, 20, 70, 50, 5, 0, 1, 30, 180, 280, 105, 6, 0, 1, 42, 385, 1050, 882, 196, 7, 0, 1, 56, 728, 3080, 4620, 2352, 336, 8, 0, 1, 72, 1260, 7644, 18018, 16632, 5544, 540, 9, 0, 1, 90, 2040, 16800, 57330, 84084, 51480, 11880, 825, 10, 0, 1, 110, 3135, 33660, 157080, 336336, 330330, 141570, 23595, 1210, 11, 0

Offset: 0

Paul D. Hanna, Oct 10 2023

A365771(n) = T(2*n,n), the central terms.
A109081(n) = Sum_{k=0..n} T(n,k), the row sums.
A365772(n) = Sum_{k=0..n} T(n,k) * 2^k.
A365773(n) = Sum_{k=0..n} T(n,k) * 3^k.
A365774(n) = Sum_{k=0..n} T(n,k) * 4^k.
A365775(n) = Sum_{k=0..n} T(n,k) * 5^k.
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,y) = 1 + x + (1 + 2*y)*x^2 + (1 + 6*y + 3*y^2)*x^3 + (1 + 12*y + 20*y^2 + 4*y^3)*x^4 + (1 + 20*y + 70*y^2 + 50*y^3 + 5*y^4)*x^5 + (1 + 30*y + 180*y^2 + 280*y^3 + 105*y^4 + 6*y^5)*x^6 + (1 + 42*y + 385*y^2 + 1050*y^3 + 882*y^4 + 196*y^5 + 7*y^6)*x^7 + (1 + 56*y + 728*y^2 + 3080*y^3 + 4620*y^4 + 2352*y^5 + 336*y^6 + 8*y^7)*x^8 + ...
where
A(x,y) = 1 + x*A(x,y)/(1 - x*y*A(x,y))^2.
Also,
A(x,y) = 1 + 1^0*x*A(x,y)/(1 + (1-y)*x*A(x,y))^2 + 2^1*x^2*A(x,y)^2/(1 + (2-y)*x*A(x,y))^3 + 3^2*x^3*A(x,y)^3/(1 + (3-y)*x*A(x,y))^4 + 4^3*x^4*A(x,y)^4/(1 + (4-y)*x*A(x,y))^5 + 5^4*x^5*A(x,y)^5/(1 + (5-y)*x*A(x,y))^6 + ...
and
A(x,y) = 1 + (1+y)*1*(1+y)^(-1)*x*A(x,y)/(1 + 1*x*A(x,y))^2 + (1+y)*2*(2+y)^0*x^2*A(x,y)^2/(1 + 2*x*A(x,y))^3 + (1+y)*3*(3+y)^1*x^3*A(x,y)^3/(1 + 3*x*A(x,y))^4 + (1+y)*4*(4+y)^2*x^4*A(x,y)^4/(1 + 4*x*A(x,y))^5 + ...
This triangle of coefficients of x^n*y^k in A(x,y) begins:
1;
1, 0;
1, 2, 0;
1, 6, 3, 0;
1, 12, 20, 4, 0;
1, 20, 70, 50, 5, 0;
1, 30, 180, 280, 105, 6, 0;
1, 42, 385, 1050, 882, 196, 7, 0;
1, 56, 728, 3080, 4620, 2352, 336, 8, 0;
1, 72, 1260, 7644, 18018, 16632, 5544, 540, 9, 0;
1, 90, 2040, 16800, 57330, 84084, 51480, 11880, 825, 10, 0; ...

Paul D. Hanna, Table of n, a(n) for n = 0..1325

Cf. A109081 (y=1), A365772 (y=2), A365773 (y=3), A365774 (y=4), A365775 (y=5).

Cf. A365771 (central terms).

{T(n,k) = binomial(n+1, n-k)/(n+1) * binomial(2*n-k-1, k)}
for(n=0,10, for(k=0,n, print1(T(n,k),", "));print(""))

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

A163810 Expansion of (1 - x) * (1 - x^2) * (1 - x^3) / (1 - x^6) in powers of x.

Original entry on oeis.org

1, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1

Offset: 0

Michael Somos, Nov 07 2007

			G.f. = 1 - x - x^2 + x^4 + x^5 - x^7 - x^8 + x^10 + x^11 - x^13 - x^14 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,-1).

A163806(n) = -a(n) unless n=0. A106510(n) = (-1)^n * a(n).

Convolution inverse of A028310. Series reversion of A109081.

Join[{1},LinearRecurrence[{1, -1},{-1, -1},104]] (* Ray Chandler, Sep 15 2015 *)

{a(n) = (n==0) + [0, -1, -1, 0, 1, 1][n%6 + 1]};

{a(n) = (n==0) + (-1)^n * kronecker(-3, n)};

Euler transform of length 6 sequence [ -1, -1, -1, 0, 0, 1].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = 2 * u * (1 - u) * (2 - v) - (v - u^2).
a(3*n) = 0 unless n=0. a(6*n + 1) = a(6*n + 2) = -1, a(6*n + 4) = a(6*n + 5) = a(0) = 1.
a(-n) = -a(n) unless n=0. a(n+3) = -a(n) unless n=0 or n=-3.
G.f.: (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.

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

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A106228 G.f. satisfies A(x) = 1 + x*A(x)/(1 - x*A(x)^2).

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A161797 G.f. satisfies: A(x) = 1/(1 - x/(1 - x*A(x))^3).

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A321798 G.f. satisfies: A(x) = 1/(1 - x/(1 - x*A(x))^4).

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A365770 Expansion of g.f. A(x,y) satisfying A(x,y) = 1 + x*A(x,y)/(1 - x*y * A(x,y))^2, as a triangle of coefficients T(n,k) of x^n*y^k in A(x,y), read by rows n >= 0.

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A163810 Expansion of (1 - x) * (1 - x^2) * (1 - x^3) / (1 - x^6) in powers of x.

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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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A106228 G.f. satisfies A(x) = 1 + xA(x)/(1 - xA(x)^2).

A365770 Expansion of g.f. A(x,y) satisfying A(x,y) = 1 + xA(x,y)/(1 - xy * A(x,y))^2, as a triangle of coefficients T(n,k) of x^n*y^k in A(x,y), read by rows n >= 0.

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