A002294 -id:A002294 - cp's OEIS frontend

A213229 G.f. satisfies: A(x) = 1/(1 - x/A(-x*A(x)^7)^2).

1, 1, 3, 16, 93, 649, 4924, 40221, 344817, 3058115, 27798895, 257009431, 2404734586, 22679499148, 214947515333, 2042353663088, 19417906390395, 184458621283607, 1748712359825873, 16530801697256737, 155736745914813741, 1461877902947680987, 13674142992787617967

Offset: 0

Paul D. Hanna, Jun 06 2012

nonn

Compare g.f. to:
(1) G(x) = 1/(1 - x/G(-x*G(x)^3)^1) when G(x) = 1/(1 - x*G(x)^1) (A000108).
(2) G(x) = 1/(1 - x/G(-x*G(x)^5)^2) when G(x) = 1/(1 - x*G(x)^2) (A001764).
(3) G(x) = 1/(1 - x/G(-x*G(x)^7)^3) when G(x) = 1/(1 - x*G(x)^3) (A002293).
(4) G(x) = 1/(1 - x/G(-x*G(x)^9)^4) when G(x) = 1/(1 - x*G(x)^4) (A002294).

			G.f.: A(x) = 1 + x + 3*x^2 + 16*x^3 + 93*x^4 + 649*x^5 + 4924*x^6 +...
Related expansions:
A(x)^7 = 1 + 7*x + 42*x^2 + 273*x^3 + 1862*x^4 + 13531*x^5 + 104062*x^6 +...
1/A(-x*A(x)^7)^2 = 1 + 2*x + 11*x^2 + 60*x^3 + 431*x^4 + 3302*x^5 + 27421*x^6 +..

Cf. A213225, A213226, A213227, A213228, A213230, A213231, A213232, A213233.
Cf. A213091, A213092, A213093, A213094, A213095, A213096, A213098.
Cf. A213099, A213100, A213101, A213102, A213103, A213104, A213105.
Cf. A213108, A213109, A213110, A213111, A213112, A213113.

{a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=1/(1-x/subst(A^2, x, -x*subst(A^7, x, x+x*O(x^n)))) ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

A213230 G.f. satisfies: A(x) = 1/(1 - x/A(-x*A(x)^8)^2).

Original entry on oeis.org

1, 1, 3, 18, 115, 902, 7722, 70784, 678251, 6670586, 66851992, 677328214, 6903177354, 70490174298, 718856047396, 7304677030708, 73837797474235, 741722190452840, 7402780597473820, 73459355234486763, 726095774886910232, 7170907377415662763, 71063833561266044578

Offset: 0

Paul D. Hanna, Jun 06 2012

nonn

Compare g.f. to:
(1) G(x) = 1/(1 - x/G(-x*G(x)^3)^1) when G(x) = 1/(1 - x*G(x)^1) (A000108).
(2) G(x) = 1/(1 - x/G(-x*G(x)^5)^2) when G(x) = 1/(1 - x*G(x)^2) (A001764).
(3) G(x) = 1/(1 - x/G(-x*G(x)^7)^3) when G(x) = 1/(1 - x*G(x)^3) (A002293).
(4) G(x) = 1/(1 - x/G(-x*G(x)^9)^4) when G(x) = 1/(1 - x*G(x)^4) (A002294).

			G.f.: A(x) = 1 + x + 3*x^2 + 18*x^3 + 115*x^4 + 902*x^5 + 7722*x^6 +...
Related expansions:
A(x)^8 = 1 + 8*x + 52*x^2 + 368*x^3 + 2754*x^4 + 22112*x^5 + 189344*x^6 +...
1/A(-x*A(x)^8)^2 = 1 + 2*x + 13*x^2 + 78*x^3 + 634*x^4 + 5488*x^5 + 50969*x^6 +...

Cf. A213225, A213226, A213227, A213228, A213229, A213231, A213232, A213233.
Cf. A213091, A213092, A213093, A213094, A213095, A213096, A213098.
Cf. A213099, A213100, A213101, A213102, A213103, A213104, A213105.
Cf. A213108, A213109, A213110, A213111, A213112, A213113.

{a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=1/(1-x/subst(A^2, x, -x*subst(A^8, x, x+x*O(x^n)))) ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

A213231 G.f. satisfies: A(x) = 1/(1 - x/A(-x*A(x)^8)^3).

Original entry on oeis.org

1, 1, 4, 25, 176, 1431, 12526, 117850, 1167446, 12080563, 129326575, 1422908670, 15999766613, 183070661566, 2124252427416, 24929036429880, 295250330398281, 3523043486823439, 42294807342916249, 510274778010082846, 6181011777164665559, 75112032752942278141

Offset: 0

Paul D. Hanna, Jun 06 2012

nonn

Compare g.f. to:
(1) G(x) = 1/(1 - x/G(-x*G(x)^3)^1) when G(x) = 1/(1 - x*G(x)^1) (A000108).
(2) G(x) = 1/(1 - x/G(-x*G(x)^5)^2) when G(x) = 1/(1 - x*G(x)^2) (A001764).
(3) G(x) = 1/(1 - x/G(-x*G(x)^7)^3) when G(x) = 1/(1 - x*G(x)^3) (A002293).
(4) G(x) = 1/(1 - x/G(-x*G(x)^9)^4) when G(x) = 1/(1 - x*G(x)^4) (A002294).

			G.f.: A(x) = 1 + x + 4*x^2 + 25*x^3 + 176*x^4 + 1431*x^5 + 12526*x^6 +...
Related expansions:
A(x)^8 = 1 + 8*x + 60*x^2 + 480*x^3 + 3998*x^4 + 34968*x^5 + 318888*x^6 +...
1/A(-x*A(x)^8)^3 = 1 + 3*x + 18*x^2 + 121*x^3 + 987*x^4 + 8646*x^5 + 82244*x^6 +...

Cf. A213225, A213226, A213227, A213228, A213229, A213230, A213232, A213233.
Cf. A213091, A213092, A213093, A213094, A213095, A213096, A213098.
Cf. A213099, A213100, A213101, A213102, A213103, A213104, A213105.
Cf. A213108, A213109, A213110, A213111, A213112, A213113.

{a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=1/(1-x/subst(A^3, x, -x*subst(A^8, x, x+x*O(x^n)))) ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

A213232 G.f. satisfies: A(x) = 1/(1 - x/A(-x*A(x)^9)^3).

Original entry on oeis.org

1, 1, 4, 28, 215, 1983, 19789, 213698, 2426851, 28661509, 348287354, 4322627557, 54508747790, 695534616050, 8953637420349, 116002300640637, 1509724588732027, 19707310304585212, 257698683361191598, 3372154116182404890, 44121356408759264549

Offset: 0

Paul D. Hanna, Jun 06 2012

nonn

Compare g.f. to:
(1) G(x) = 1/(1 - x/G(-x*G(x)^3)^1) when G(x) = 1/(1 - x*G(x)^1) (A000108).
(2) G(x) = 1/(1 - x/G(-x*G(x)^5)^2) when G(x) = 1/(1 - x*G(x)^2) (A001764).
(3) G(x) = 1/(1 - x/G(-x*G(x)^7)^3) when G(x) = 1/(1 - x*G(x)^3) (A002293).
(4) G(x) = 1/(1 - x/G(-x*G(x)^9)^4) when G(x) = 1/(1 - x*G(x)^4) (A002294).

			G.f.: A(x) = 1 + x + 4*x^2 + 28*x^3 + 215*x^4 + 1983*x^5 + 19789*x^6 +...
Related expansions:
A(x)^9 = 1 + 9*x + 72*x^2 + 624*x^3 + 5661*x^4 + 54621*x^5 + 555837*x^6 +...
1/A(-x*A(x)^9)^3 = 1 + 3*x + 21*x^2 + 154*x^3 + 1446*x^4 + 14511*x^5 + 158838*x^6 +...

Cf. A213225, A213226, A213227, A213228, A213229, A213230, A213231, A213233.
Cf. A213091, A213092, A213093, A213094, A213095, A213096, A213098.
Cf. A213099, A213100, A213101, A213102, A213103, A213104, A213105.
Cf. A213108, A213109, A213110, A213111, A213112, A213113.

{a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=1/(1-x/subst(A^3, x, -x*subst(A^9, x, x+x*O(x^n)))) ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

A213233 G.f. satisfies: A(x) = 1/(1 - x/A(-x*A(x)^10)^4).

Original entry on oeis.org

1, 1, 5, 39, 345, 3512, 38431, 451620, 5587237, 72275004, 968509140, 13361356169, 188704259571, 2716467168169, 39716842554828, 588125693790055, 8800638181341593, 132838773216409675, 2019626662710709088, 30891440565153652705, 474899505740289874276

Offset: 0

Paul D. Hanna, Jun 06 2012

nonn

Compare g.f. to:
(1) G(x) = 1/(1 - x/G(-x*G(x)^3)^1) when G(x) = 1/(1 - x*G(x)^1) (A000108).
(2) G(x) = 1/(1 - x/G(-x*G(x)^5)^2) when G(x) = 1/(1 - x*G(x)^2) (A001764).
(3) G(x) = 1/(1 - x/G(-x*G(x)^7)^3) when G(x) = 1/(1 - x*G(x)^3) (A002293).
(4) G(x) = 1/(1 - x/G(-x*G(x)^9)^4) when G(x) = 1/(1 - x*G(x)^4) (A002294).

			G.f.: A(x) = 1 + x + 5*x^2 + 39*x^3 + 345*x^4 + 3512*x^5 + 38431*x^6 +...
Related expansions:
A(x)^10 = 1 + 10*x + 95*x^2 + 960*x^3 + 10095*x^4 + 111212*x^5 +...
1/A(-x*A(x)^10)^4 = 1 + 4*x + 30*x^2 + 256*x^3 + 2605*x^4 + 28484*x^5 +...

Cf. A213225, A213226, A213227, A213228, A213229, A213230, A213231, A213232.
Cf. A213091, A213092, A213093, A213094, A213095, A213096, A213098.
Cf. A213099, A213100, A213101, A213102, A213103, A213104, A213105.
Cf. A213108, A213109, A213110, A213111, A213112, A213113.

{a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=1/(1-x/subst(A^4, x, -x*subst(A^10, x, x+x*O(x^n)))) ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

A251585 a(n) = 5^(n-3) * (n+1)^(n-5) * (16n^3 + 87n^2 + 172*n + 125).

Original entry on oeis.org

1, 1, 7, 116, 3229, 129000, 6776875, 443200000, 34766465625, 3185000000000, 333992093359375, 39470976000000000, 5192072114658203125, 752537122540000000000, 119176291179656982421875, 20476256583680000000000000, 3793880513498167242431640625, 754086862404270000000000000000

Offset: 0

Paul D. Hanna, Dec 06 2014

nonn

			E.g.f.: A(x) = 1 + x + 7*x^2/2! + 116*x^3/3! + 3229*x^4/4! + 129000*x^5/5! + ...
such that A(x) = exp( 5*x*A(x) * G(x*A(x))^4 ) / G(x*A(x))^4
where G(x) = 1 + x*G(x)^5 is the g.f. of A002294:
G(x) = 1 + x + 5*x^2 + 35*x^3 + 285*x^4 + 2530*x^5 + 23751*x^6 + ...
RELATED SERIES.
Note that A(x) = F(x*A(x)) where F(x) = exp(5*x*G(x)^4)/G(x)^4,
F(x) = 1 + x + 5*x^2/2! + 65*x^3/3! + 1505*x^4/4! + 51505*x^5/5! + ...
is the e.g.f. of A251575.

G. C. Greubel, Table of n, a(n) for n = 0..312

Cf. A251575, A002294.

Cf. Variants: A251583, A251584, A251586, A251587, A251588, A251589, A251590.

[5^(n - 3)*(n + 1)^(n - 5)*(16*n^3 + 87*n^2 + 172*n + 125): n in [0..50]]; // G. C. Greubel, Nov 13 2017

Table[5^(n - 3)*(n + 1)^(n - 5)*(16*n^3 + 87*n^2 + 172*n + 125), {n, 0, 50}] (* G. C. Greubel, Nov 13 2017 *)

{a(n) = 5^(n-3) * (n+1)^(n-5) * (16*n^3 + 87*n^2 + 172*n + 125)}
for(n=0,20,print1(a(n),", "))

{a(n) = local(G=1,A=1); for(i=1,n, G=1+x*G^5 +x*O(x^n));
for(i=1,n, A = exp(5*x*A * subst(G^4,x,x*A) ) / subst(G^4,x,x*A) ); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))

Let G(x) = 1 + x*G(x)^5 be the g.f. of A002294, then the e.g.f. A(x) of this sequence satisfies:
(1) A(x) = exp( 5*x*A(x) * G(x*A(x))^4 ) / G(x*A(x))^4.
(2) A(x) = F(x*A(x)) where F(x) = exp(5*x*G(x)^4)/G(x)^4 is the e.g.f. of A251575.
(3) a(n) = [x^n/n!] F(x)^(n+1)/(n+1) where F(x) is the e.g.f. of A251575.
E.g.f.: -LambertW(-5*x) * (5 + LambertW(-5*x))^4 / (x*5^5). - Vaclav Kotesovec, Dec 07 2014

A251695 a(n) = (3n+1) (4n+1)^(n-2) 5^n.

Original entry on oeis.org

1, 4, 175, 16250, 2348125, 463050000, 115966796875, 35253537343750, 12611991884765625, 5191587030710937500, 2417311348659677734375, 1256208098030090332031250, 720779749270420907470703125, 452589644988876542822265625000, 308707218248583408960223388671875

Offset: 0

Paul D. Hanna, Dec 07 2014

nonn

			E.g.f.: A(x) = 1 + 4*x + 175*x^2/2! + 16250*x^3/3! + 2348125*x^4/4! + 463050000*x^5/5! +...
such that A(x) = exp( 5*x*A(x)^4 * G(x*A(x)^4)^4 ) / G(x*A(x)^4),
where G(x) = 1 + x*G(x)^5 is the g.f. A002294:
G(x) = 1 + x + 5*x^2 + 35*x^3 + 285*x^4 + 2530*x^5 + 23751*x^6 +...
Also, e.g.f. A(x) satisfies A(x) = F(x*A(x)^4) where
F(x) = 1 + 4*x + 47*x^2/2! + 1034*x^3/3! + 34349*x^4/4! + 1540480*x^5/5! +...
F(x) = exp( 5*x*G(x)^4 ) / G(x) is the e.g.f. of A251665.

G. C. Greubel, Table of n, a(n) for n = 0..268

Cf. A251665, A002294.

Cf. Variants: A127670, A251693, A251694, A251696, A251697, A251698, A251699, A251700.

[(3*n + 1)*(4*n + 1)^(n - 2)*5^n: n in [0..50]]; // G. C. Greubel, Nov 13 2017

Table[(3*n + 1)*(4*n + 1)^(n - 2)*5^n, {n, 0, 50}] (* G. C. Greubel, Nov 13 2017 *)

{a(n) = (3*n+1) * (4*n+1)^(n-2) * 5^n}
for(n=0,20,print1(a(n),", "))

{a(n)=local(G=1,A=1); for(i=0, n, G = 1 + x*G^5 +x*O(x^n));
A = ( serreverse( x*G^4 / exp(20*x*G^4) )/x )^(1/4); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))

Let G(x) = 1 + x*G(x)^5 be the g.f. of A002294, then the e.g.f. A(x) of this sequence satisfies:
(1) A(x) = exp( 5*x*A(x)^4 * G(x*A(x)^4)^4 ) / G(x*A(x)^4).
(2) A(x) = F(x*A(x)^4) where F(x) = exp(5*x*G(x)^4)/G(x) is the e.g.f. of A251665.
(3) A(x) = ( Series_Reversion( x*G(x)^4 / exp(20*x*G(x)^4) )/x )^(1/4).
E.g.f.: (-LambertW(-20*x)/(20*x))^(1/4) * (1 + LambertW(-20*x)/20). - Vaclav Kotesovec, Dec 07 2014

A355262 Array of Fuss-Catalan numbers read by ascending antidiagonals, A(n, k) = binomial(kn + 1, k)/(kn + 1).

Original entry on oeis.org

1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 3, 5, 1, 0, 1, 1, 4, 12, 14, 1, 0, 1, 1, 5, 22, 55, 42, 1, 0, 1, 1, 6, 35, 140, 273, 132, 1, 0, 1, 1, 7, 51, 285, 969, 1428, 429, 1, 0, 1, 1, 8, 70, 506, 2530, 7084, 7752, 1430, 1, 0

Offset: 0

Peter Luschny, Jun 26 2022

An alternative definition is: the Fuss-Catalan sequences (A(n, k), k >= 0 ) are the main diagonals of the Fuss-Catalan triangles of order n - 1. See A355173 for the definition of a Fuss-Catalan triangle.

			Array A(n, k) begins:
[0] 1, 1, 0,   0,    0,     0,      0,       0,         0, ...  A019590
[1] 1, 1, 1,   1,    1,     1,      1,       1,         1, ...  A000012
[2] 1, 1, 2,   5,   14,    42,    132,     429,      1430, ...  A000108
[3] 1, 1, 3,  12,   55,   273,   1428,    7752,     43263, ...  A001764
[4] 1, 1, 4,  22,  140,   969,   7084,   53820,    420732, ...  A002293
[5] 1, 1, 5,  35,  285,  2530,  23751,  231880,   2330445, ...  A002294
[6] 1, 1, 6,  51,  506,  5481,  62832,  749398,   9203634, ...  A002295
[7] 1, 1, 7,  70,  819, 10472, 141778, 1997688,  28989675, ...  A002296
[8] 1, 1, 8,  92, 1240, 18278, 285384, 4638348,  77652024, ...  A007556
[9] 1, 1, 9, 117, 1785, 29799, 527085, 9706503, 184138713, ...  A062994

N. I. Fuss, Solutio quaestionis, quot modis polygonum n laterum in polygona m laterum, per diagonales resolvi queat, Nova Acta Academiae Scientiarum Imperialis Petropolitanae, vol.9 (1791), 243-251.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, Reading, MA, 1990, (Eqs. 5.70, 7.66, and sec. 7.5, example 5).

Per Alexandersson, Frether Getachew Kebede, Samuel Asefa Fufa, and Dun Qiu, Pattern-Avoidance and Fuss-Catalan Numbers, J. Int. Seq. (2023) Vol. 26, Art. 23.4.2.
Joerg Arndt, Matters Computational (The Fxtbook), pp. 337-338.
Jean-Luc Baril, Mireille Bousquet-Mélou, Sergey Kirgizov, and Mehdi Naima, The ascent lattice on Dyck paths, arXiv:2409.15982 [math.CO], 2024. See p. 6.
Jean-Christophe Aval, Multivariate Fuss-Catalan Numbers, arXiv:0711.0906v1, Discrete Math., 308 (2008), 4660-4669.
Thomas A. Dowling, Catalan Numbers, Chapter 7.
D. E. Knuth, Convolution polynomials, arXiv:math/9207221 [math.CA]; Mathematica J. 2.1 (1992), no. 4, 67-78.
Donald Knuth's 20th Annual Christmas Tree Lecture, (3/2)-ary Trees, Stanford Online, Video 2014.
Wojciech Młotkowski, Fuss-Catalan Numbers in Noncommutative Probability, Docum. Math. 15:939-955, (2010).
Wikipedia, Fuss-Catalan number.

Cf. A091144 (main diagonal), A019590, A000012, A000108, A001764, A002293, A002294, A002295, A002296, A007556, A062994.
Variants: A062993, A070914.
Fuss-Catalan triangles: A123110 (order 0), A355173 (order 1), A355172 (order 2), A355174 (order 3).

A := (n, k) -> binomial(k*n + 1, k)/(k*n + 1):
for n from 0 to 9 do seq(A(n, k), k = 0..8) od;

(* See the Knuth references. In the christmas lecture Knuth has fun calculating the Fuss-Catalan development of Pi and i. *)
B[t_, n_] := Sum[Binomial[t k+1, k] z^k / (t k+1), {k, 0, n-1}] + O[z]^n
Table[CoefficientList[B[n, 9], z], {n, 0, 9}] // TableForm

A(n, k) = (1/k!) * Product_{j=1..k-1} (k*n + 1 - j).
A(n, k) = (binomial(k*n, k) + binomial(k*n, k-1)) / (k*n + 1).
Let B(t, z) = Sum_{k>=0} binomial(k*t + 1, k)*z^k / (k*t + 1), then
A(n, k) = [z^k] B(n, z).

A365180 G.f. satisfies A(x) = 1 + xA(x)^4(1 + x*A(x)).

Original entry on oeis.org

1, 1, 5, 31, 223, 1740, 14328, 122549, 1078197, 9695359, 88710199, 823247686, 7730244098, 73310150097, 701163085849, 6755544043969, 65506554804129, 638794412442172, 6260571309256152, 61632794482411367, 609197871548209907, 6043456939539775056

Offset: 0

Seiichi Manyama, Aug 25 2023

nonn

Cf. A002294, A365178, A365181, A365182, A365183.

Cf. A364747.

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

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

A365182 G.f. satisfies A(x) = 1 + xA(x)^4(1 + x*A(x)^3).

Original entry on oeis.org

1, 1, 5, 33, 252, 2091, 18319, 166750, 1561599, 14948572, 145615404, 1438752770, 14384289530, 145248707646, 1479212551278, 15175516654760, 156691764630780, 1627069871618145, 16980373299730925, 178006989972532900, 1873607777794186000

Offset: 0

Seiichi Manyama, Aug 25 2023

nonn

Cf. A002294, A365178, A365180, A365181, A365183.

Cf. A365177.

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

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

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A213229 G.f. satisfies: A(x) = 1/(1 - x/A(-x*A(x)^7)^2).

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A213230 G.f. satisfies: A(x) = 1/(1 - x/A(-x*A(x)^8)^2).

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A213231 G.f. satisfies: A(x) = 1/(1 - x/A(-x*A(x)^8)^3).

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A213232 G.f. satisfies: A(x) = 1/(1 - x/A(-x*A(x)^9)^3).

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A213233 G.f. satisfies: A(x) = 1/(1 - x/A(-x*A(x)^10)^4).

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A251585 a(n) = 5^(n-3) * (n+1)^(n-5) * (16*n^3 + 87*n^2 + 172*n + 125).

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A251695 a(n) = (3*n+1) * (4*n+1)^(n-2) * 5^n.

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A355262 Array of Fuss-Catalan numbers read by ascending antidiagonals, A(n, k) = binomial(k*n + 1, k)/(k*n + 1).

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A251585 a(n) = 5^(n-3) * (n+1)^(n-5) * (16n^3 + 87n^2 + 172*n + 125).

A251695 a(n) = (3n+1) (4n+1)^(n-2) 5^n.

A355262 Array of Fuss-Catalan numbers read by ascending antidiagonals, A(n, k) = binomial(kn + 1, k)/(kn + 1).

A365180 G.f. satisfies A(x) = 1 + xA(x)^4(1 + x*A(x)).

A365182 G.f. satisfies A(x) = 1 + xA(x)^4(1 + x*A(x)^3).