A002295 -id:A002295 - cp's OEIS frontend

A059968 Number of 10-ary trees.

1, 1, 10, 145, 2470, 46060, 910252, 18730855, 397089550, 8612835715, 190223180840, 4263421511271, 96723482198980, 2216905597676000, 51256802757808320, 1194060413809070710, 27999654303202465310, 660370070571422998410, 15654733143626084944150

Offset: 0

Claude Lenormand (claude.lenormand(AT)free.fr), Mar 05 2001

From Wolfdieter Lang, Feb 06 2020: (Start)
Ninth column of triangle A062993 (without leading zeros). A Pfaff-Fuss or 10-Raney sequence.
a(n), n>=1, enumerates 10-ary trees (rooted, ordered, incomplete) with n vertices (including the root).
See Graham et al., Hilton and Pedersen, Hoggat and Bicknell, Frey and Sellers references given in A062993. (End)
This is instance k = 10 of the generalized Catalan family {C(k, n)}A130564%20-%20_Wolfdieter%20Lang">{n>=0} given in a comment of A130564 - _Wolfdieter Lang, Feb 05 2024

			There are a(2)=10 10-ary trees (vertex degree <=10 and 10 possible branchings) with 2 vertices (one of them the root). Adding one more branch (one more vertex) to these 10 trees yields 10*10+binomial(10,2)=145=a(3) such trees. - _Wolfdieter Lang_, Sep 14 2007.

G. Pólya and G. Szegő, Problems and Theorems in Analysis, Springer-Verlag, Heidelberg, New York, 2 vols., 1972, Vol. 1, problem 211, p. 146 with solution on p. 348.

S. Heubach, N. Y. Li and T. Mansour, Staircase tilings and k-Catalan structures, Discrete Math., 308 (2008), 5954-5964.

Related algebraic sequences concerning trees: strictly k-ary trees (A000108: s=x+s^2, A001263: s=(x, y)+(x, s)+(s, y)+(s, s))), (A001764: s=x+s^3), (A002293: s=x+s^4), (A002294: s=x+s^5), (A002295: s=x+s^6), (A002296: s=x+s^7), (A007556: s=x+s^8), at most k-ary trees (A001006: s=x+xs+xs^2), (A036765-A036769, s=x+xs^2....+xs^k, k=3, 4, 5, 6, 7).
Pfaff-Fuss A062993 column members: A000108, A001764, A002293, A002294, A002295, A002296, A007556, A062994, A230388.
Cf. A130564.

seq(binomial(10*k+1, k)/(9*k+1), k=0..30);
n:=30:G:=series(RootOf(g = 1+x*g^10, g), x=0, n+1):seq(coeff(G, x, k), k=0..n); # Robert FERREOL, Apr 01 2015

a[n_] := Binomial[10n, n]/(9n+1);
a /@ Range[0, 25] (* Jean-François Alcover, Jan 17 2020 *)

G.f. A(x) satisfies: A = x + A^10.
a(n) = binomial(k*n, n)/((k-1)*n+1), for k=10.
Recurrence: a(0) = 1; a(n) = Sum_{i1+i2+..i10=n-1} a(i1)*a(i2)*...*a(i10) for n>=1. - Robert FERREOL, Apr 01 2015
From Wolfdieter Lang, Feb 06 2020: (Start)
a(n) = A062993(n+8, 8). [Corrected by Robert FERREOL, Apr 01 2015]
G.f.: RootOf((_Z^10)*x-_Z+1) (Maple notation, from ECS, see links for A007556).
G.f.: hypergeometric([1, 2, 3, 4, 5, 6, 7, 8, 9]/10, [2, 3, 4, 5, 6, 7, 8, 10]/9, (10^10/9^9)*x),
E.g.f.: hypergeometric([1, 2, 3, 4, 5, 6, 7, 8, 9]/10, [2, 3, 4, 5, 6, 7, 8, 9, 10]/9, (10^10/9^9)*x).
For other family members see the crossreferences.
(End)
D-finite with recurrence 81*n*(9*n-7)*(9*n-5)*(3*n-1)*(9*n-1)*(9*n+1)*(3*n-2)*(9*n-4)*(9*n-2)*a(n) -800*(10*n-9)*(5*n-4)*(10*n-7)*(5*n-3)*(2*n-1)*(5*n-2)*(10*n-3)*(5*n-1)*(10*n-1)*a(n-1)=0. - R. J. Mathar, Mar 21 2022
a(n) ~ (10^10/9^9)^n*sqrt(10/(2*Pi*(9*n)^3)). - Robert A. Russell, Jul 15 2024
G.f. A(x) satisfies A(x) = 1/A(-x*A(x)^19). - Seiichi Manyama, Jun 16 2025

More terms from James Sellers, Mar 15 2001
a(0)=1 inserted by Alois P. Heinz, Jan 17 2020
A062744 merged into this sequence by Wolfdieter Lang, Feb 06 2020

A062994 Eighth column of triangle A062993 (without leading zeros). A Pfaff-Fuss or 9-Raney sequence.

Original entry on oeis.org

1, 1, 9, 117, 1785, 29799, 527085, 9706503, 184138713, 3573805950, 70625252863, 1416298046436, 28748759731965, 589546754316126, 12195537924351375, 254184908607118800, 5332692942907262361

Offset: 0

Wolfdieter Lang, Jul 12 2001

See Graham et al., Hilton and Pedersen, Hoggat and Bicknell, Frey and Sellers references given in A062993.
Essentially the same as A059967. a(n), n>=1, enumerates 9-ary trees (rooted, ordered, incomplete) with n vertices (including the root).
These numbers appear in a formula on p. 24 of Gross et al. for b = -2 or 4. For b = -1 or 3, see A002293.- Tom Copeland, Dec 24 2019
This is instance k = 9 of the generalized Catalan family {C(k, n)}_{n>=0} given in a comment of A130564. - Wolfdieter Lang, Feb 05 2024

			There are a(2)=9 9-ary trees (vertex degree <=9 and 9 possible branchings) with 2 vertices (one of them the root). Adding one more branch (one more vertex) to these 9 trees yields 9*9 + binomial(9,2) = 117 = a(3) such trees.

G. Pólya and G. Szegő, Problems and Theorems in Analysis, Springer-Verlag, Heidelberg, New York, 2 vols., 1972, Vol. 1, problem. 211, p. 146 with solution on p. 348.

Harry J. Smith, Table of n, a(n) for n = 0..100
M. Gross, P. Hacking, S. Keel, and M. Kontsevich, Canonical bases for cluster algebras, arXiv preprint arXiv:1411.1394 [math.AG], 2016.

Column k=8 of A070914.

Cf. A000108, A001764, A002293, A002294, A002295, A002296, A007556, A059968, A130564.

seq(binomial(9*k+1,k)/(8*k+1),k=0..30);
n:=30: G:=series(RootOf(g = 1+x*g^9, g),x=0,n+1): seq(coeff(G,x,k),k=0..n); # Robert FERREOL, Apr 01 2015

Table[Binomial[9n,n]/(8n+1),{n,0,30}] (* Harvey P. Dale, Oct 28 2012 *)

{ for (n=0, 100, write("b062994.txt", n, " ", binomial(9*n, n)/(8*n + 1)) ) } \\ Harry J. Smith, Aug 15 2009

a(n) = A062993(n+7, 7) = binomial(9*n, n)/(8*n+1).
G.f.: RootOf((_Z^9)*x-_Z+1) (Maple notation, from ECS, see links for A007556).
Recurrence: a(0) = 1; a(n) = Sum_{i1+i2+..+i9=n-1} a(i1)*a(i2)*...*a(i9) for n>=1. - Robert FERREOL, Apr 01 2015
From Ilya Gutkovskiy, Jan 16 2017: (Start)
O.g.f.: 8F7(1/9,2/9,1/3,4/9,5/9,2/3,7/9,8/9; 1/4,3/8,1/2,5/8,3/4,7/8,9/8; 387420489*x/16777216).
E.g.f.: 8F8(1/9,2/9,1/3,4/9,5/9,2/3,7/9,8/9; 1/4,3/8,1/2,5/8,3/4,7/8,1,9/8; 387420489*x/16777216).
a(n) ~ 3^(18*n+1)/(sqrt(Pi)*2^(24*n+5)*n^(3/2)). (End)
D-finite with recurrence: 128*n*(8*n-5)*(4*n-1)*(8*n+1)*(2*n-1)*(8*n-1)*(4*n-3)*(8*n-3)*a(n) -81*(9*n-7)*(9*n-5)*(3*n-1)*(9*n-1)*(9*n-8)*(3*n-2)*(9*n-4)*(9*n-2)*a(n-1)=0. - R. J. Mathar, Feb 20 2020
G.f. A(x) satisfies A(x) = 1/A(-x*A(x)^17). - Seiichi Manyama, Jun 16 2025

9-ary tree comments and Pólya and G. Szegő reference from Wolfdieter Lang, Sep 14 2007

A251592 Triangle of coefficients of polynomials P(n,t) related to the Mittag-Leffler function, where P(n,t) = Product_{k=0..n-2} n*t-k.

Original entry on oeis.org

1, 0, 2, 0, -3, 9, 0, 8, -48, 64, 0, -30, 275, -750, 625, 0, 144, -1800, 7560, -12960, 7776, 0, -840, 13426, -77175, 204085, -252105, 117649, 0, 5760, -112896, 831488, -3010560, 5734400, -5505024, 2097152, 0, -45360, 1058508, -9573228

Offset: 1

Jean-François Alcover, Dec 05 2014

Second column (unsigned) 2, 3, 8, 30, 144, ... is A001048.

Diagonal 1, 2, 9, 64, 625, 7776, ... is A000169.

			Triangle begins :
  1;
  0,   2;
  0,  -3,     9;
  0,   8,   -48,   64;
  0, -30,   275, -750,    625;
  0, 144, -1800, 7560, -12960, 7776;
  ...

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, Reading, MA, 2nd ed. 1998

Peter Bala, Fractional iteration of a series inversion operator
MathOverflow, Properties and name of some polynomials
Eric Weisstein's MathWorld, Mittag-Leffler Function
Eric Weisstein's MathWorld, Wright Function

Cf. A000169, A001048, A156136, A000108 (B_2(x)), A001764 (B_3(x)), A002293 (B_4(x)), A002294 (B_5(x)), A002295 (B_6(x)), A002296 (B_7(x)), A007556 (B_8(x)), A062994 (B_9(x)), A059968 (B_10(x)), A230388 (B_11(x)), A139526, A260687.

P[n_, t_] := Product[n*t - k, {k, 0, n-2}]; row[n_] := CoefficientList[P[n, t], t]; Table[row[n], {n, 1, 10}] // Flatten

P(n,t) = (n-1)!*binomial(n*t, n-1).
From Peter Bala, Nov 15 2015: (Start)
E.g.f. (with constant term 1): B_t(x) = Sum_{n >= 0} 1/(n*t + 1)*binomial(n*t + 1,n)*x^n = 1 + x + 2*t*x^2/2! + 3*t(3*t - 1)*x^3/3! + 4*t*(4*t - 1)*(4*t - 2)*x^4/4! + ... is the generalized binomial series of Lambert. See Graham et al., Section 5.4 and Section 7.5.
In the notation of the Bala link, B_t(x) = I^t(1 + x) where I^t is a fractional inversion operator. B_(1+t)(x) is the e.g.f. for A260687.
B_t(x) = 1 + x*B_t(x)^t.
For complex r, B_t(x)^r = Sum_{n >= 0} r/(n*t + r)*binomial(n*t + r,n)*x^n.
log (B_t(x)) = Sum_{n >= 1} 1/(n*t)*binomial(n*t,n)*x^n.
B_2(x) is the o.g.f. for the Catalan numbers A000108. B_t(x) for t = 3,4,5,... gives the o.g.f. for various Fuss-Catalan sequences. See the cross references. (End)

A004355 Binomial coefficient C(6n,n).

Original entry on oeis.org

1, 6, 66, 816, 10626, 142506, 1947792, 26978328, 377348994, 5317936260, 75394027566, 1074082795968, 15363284301456, 220495674290430, 3173734438530120, 45795673964460816, 662252084388541314

Offset: 0

N. J. A. Sloane

a(n) is asymptotic to c*(46656/3125)^n/sqrt(n), with c = sqrt(3/(5*Pi)) = 0.437019372236831628217354... - Benoit Cloitre, Jan 23 2008

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.

T. D. Noe, Table of n, a(n) for n = 0..100
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

binomial(k*n,n): A000984 (k = 2), A005809 (k = 3), A005810 (k = 4), A001449 (k = 5), A004368 (k = 7), A004381 (k = 8), A169958 - A169961 (k = 9 thru 12).

Cf. A002295

[Binomial(6*n,n): n in [0..100]]; // Vincenzo Librandi, Apr 13 2011

Table[Binomial[6 n, n], {n, 0, 16}] (* Michael De Vlieger, Oct 05 2015 *)

B(x):=sum(binomial(6*n,n-1)/n*x^n,n,1,30);
taylor(x*diff(B(x),x)/B(x),x,0,10); /* Vladimir Kruchinin, Oct 05 2015 */

a(n) = binomial(6*n,n) \\ Altug Alkan, Oct 05 2015

a(n) = C(6*n-1,n-1)*C(36*n^2,2)/(3*n*C(6*n+1,3)), n>0. - Gary Detlefs, Jan 02 2014
G.f.: A(x) = x*B'(x)/B(x), where B(x)+1 is g.f. of A002295. - Vladimir Kruchinin, Oct 05 2015
a(n) = GegenbauerC(n, -3*n, -1). - Peter Luschny, May 07 2016
From Ilya Gutkovskiy, Jan 16 2017: (Start)
O.g.f.: 5F4(1/6,1/3,1/2,2/3,5/6; 1/5,2/5,3/5,4/5; 46656*x/3125).
E.g.f.: 5F5(1/6,1/3,1/2,2/3,5/6; 1/5,2/5,3/5,4/5,1; 46656*x/3125). (End)
RHS of identities Sum_{k = 0..n} binomial(3*n, k)*binomial(3*n, n-k) =
Sum_{k = 0..2*n} (-1)^(n+k)*binomial(6*n, k)*binomial(6*n, 2*n-k) = binomial(6*n,n). - Peter Bala, Oct 07 2021
From Peter Bala, Feb 20 2022: (Start)
5*n*(5*n-1)*(5*n-2)*(5*n-3)*(5*n-4)*a(n) = 6*(6*n-1)*(6*n-2)*(6*n-3)(6*n-4)*(6*n-5)*a(n-1).
The o.g.f. A(x) is algebraic: (1 - A(x))*(1 + 5*A(x))^5 +  (6^6)*x*A(x)^6 = 0.
Sum_{n >= 1} a(n)*( x*(5*x + 6)^5/(6^6*(1 + x)^6) )^n = x. (End)

A070914 Array read by antidiagonals giving number of paths up and left from (0,0) to (n,kn) where x/y <= k for all intermediate points.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 5, 1, 1, 1, 4, 12, 14, 1, 1, 1, 5, 22, 55, 42, 1, 1, 1, 6, 35, 140, 273, 132, 1, 1, 1, 7, 51, 285, 969, 1428, 429, 1, 1, 1, 8, 70, 506, 2530, 7084, 7752, 1430, 1, 1, 1, 9, 92, 819, 5481, 23751, 53820, 43263, 4862, 1, 1, 1, 10, 117, 1240

Offset: 0

Henry Bottomley, May 20 2002

Also related to dissections of polygons and enumeration of trees.
Number of dissections of a polygon into n (k+2)-gons by nonintersecting diagonals. All dissections are counted separately. See A295260 for nonequivalent solutions up to rotation and reflection. - Andrew Howroyd, Nov 20 2017
Number of rooted polyominoes composed of n (k+2)-gonal cells of the hyperbolic (Euclidean for k=0) regular tiling with Schläfli symbol {k+2,oo}. A rooted polyomino has one external edge identified, and chiral pairs are counted as two. For k>0, a stereographic projection of the {k+2,oo} tiling on the Poincaré disk can be obtained via the Christensson link. - Robert A. Russell, Jan 27 2024

			Rows start:
===========================================================
n\k| 0     1      2       3        4        5         6
---|-------------------------------------------------------
0  | 1,    1,     1,      1,       1,       1,        1 ...
1  | 1,    1,     1,      1,       1,       1,        1 ...
2  | 1,    2,     3,      4,       5,       6,        7 ...
3  | 1,    5,    12,     22,      35,      51,       70 ...
4  | 1,   14,    55,    140,     285,     506,      819 ...
5  | 1,   42,   273,    969,    2530,    5481,    10472 ...
6  | 1,  132,  1428,   7084,   23751,   62832,   141778 ...
7  | 1,  429,  7752,  53820,  231880,  749398,  1997688 ...
8  | 1, 1430, 43263, 420732, 2330445, 9203634, 28989675 ...
...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
Malin Christensson, Make hyperbolic tilings of images, web page, 2019.
Peter Hilton and Jean Pedersen, Catalan Numbers, Their Generalization, and Their Uses, The Mathematical Intelligencer, March 1991, Volume 13, Issue 2, pp. 64-75.
V. E. Hoggatt, Jr. and M. Bicknell, Catalan and related sequences arising from inverses of Pascal's triangle matrices, Fib. Quart., 14 (1976), 395-405.

Rows include A000012 (twice), A000027, A000326.
Columns include A000012, A000108 (Catalan), A001764, A002293, A002294, A002295, A002296, A007556, A062994, A059968.
Reflected version of A062993 (which is the main entry).
Cf. A295260.
Polyominoes: A295224 (oriented), A295260 (unoriented).

A:= (n, k)-> binomial((k+1)*n, n)/(k*n+1):
seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Mar 25 2015

T[n_, k_] = Binomial[n(k+1), n]/(k*n+1); Flatten[Table[T[n-k, k], {n, 0, 9}, {k, n, 0, -1}]] (* Jean-François Alcover, Apr 08 2016 *)

T(n, k) = binomial(n*(k+1), n)/(n*k+1); \\ Andrew Howroyd, Nov 20 2017

T(n, k) = binomial(n*(k+1), n)/(n*k+1) = A071201(n, k*n) = A071201(n, k*n+1) = A071202(n, k*n+1) = A062993(n+k-1, k-1).

If P(k,x) = Sum_{n>=0} T(n,k)*x^n is the g.f. of column k (k>=0), then P(k,x) = exp(1/(k+1)*(Sum_{j>0} (1/j)*binomial((k+1)*j,j)*x^j)). - Werner Schulte, Oct 13 2015

A213104 G.f. satisfies: A(x) = 1 + x/A(-x*A(x)^10)^5.

Original entry on oeis.org

1, 1, 5, 40, 360, 3820, 43651, 543240, 7146185, 98885725, 1420274645, 21037156031, 319127602075, 4935547265370, 77525696636995, 1233356748777015, 19829269320322346, 321631227310756885, 5255920261950786655, 86436636022328320125, 1429253483704685851315

Offset: 0

Paul D. Hanna, Jun 05 2012

sign

Compare definition of g.f. to:
(1) B(x) = 1 + x/B(-x*B(x)) when B(x) = 1/(1-x).
(2) C(x) = 1 + x/C(-x*C(x)^3)^2 when C(x) = 1 + x*C(x)^2 (A000108).
(3) D(x) = 1 + x/D(-x*D(x)^5)^3 when D(x) = 1 + x*D(x)^3 (A001764).
(4) E(x) = 1 + x/E(-x*E(x)^7)^4 when E(x) = 1 + x*E(x)^4 (A002293).
(5) F(x) = 1 + x/F(-x*F(x)^9)^5 when F(x) = 1 + x*F(x)^5 (A002294).
(6) G(x) = 1 + x/G(-x*G(x)^11)^6 when G(x) = 1 + x*G(x)^6 (A002295).
The first negative term is a(306). - Georg Fischer, Feb 16 2019

			G.f.: A(x) = 1 + x + 5*x^2 + 40*x^3 + 360*x^4 + 3820*x^5 + 43651*x^6 +...
Related expansions:
A(x)^10 = 1 + 10*x + 95*x^2 + 970*x^3 + 10335*x^4 + 116452*x^5 +...
A(-x*A(x)^10)^5 = 1 - 5*x - 15*x^2 - 85*x^3 - 995*x^4 - 10776*x^5 -...

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

Cf. A000108, A001764, A002293, A002294, A002295, A213091, A213092, A213093, A213094, A213095, A213096, A213098, A213099, A213100, A213101, A213102, A213103, A213105.

m = 21; A[] = 1; Do[A[x] = 1 + x/A[-x A[x]^10]^5 + O[x]^m, {m}];
CoefficientList[A[x], x] (* Jean-François Alcover, Nov 06 2019 *)

{a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=1+x/subst(A^5,x,-x*subst(A^10,x,x+x*O(x^n))) );polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

A062993 A triangle (lower triangular matrix) composed of Pfaff-Fuss (or Raney) sequences.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 5, 3, 1, 1, 14, 12, 4, 1, 1, 42, 55, 22, 5, 1, 1, 132, 273, 140, 35, 6, 1, 1, 429, 1428, 969, 285, 51, 7, 1, 1, 1430, 7752, 7084, 2530, 506, 70, 8, 1, 1, 4862, 43263, 53820, 23751, 5481, 819, 92, 9

Offset: 0

Wolfdieter Lang, Jul 12 2001

The column sequences (without leading zeros) appear in eq.(7.66), p. 347 of the Graham et al. reference, in Th. 0.3, p. 66, of Hilton and Pedersen reference, as first columns of the S-triangles in the Hoggatt and Bicknell reference and in eq. 5 of the Frey and Sellers reference. They are also called m-Raney (here m=k+2) or Fuss-Catalan sequences (see Graham et al. for reference). For the history and the name Pfaff-Fuss see Brown reference, p. 975. PF(n,m) := binomial(m*n+1,n)/(m*n+1), m >= 2.

Also called generalized Catalan numbers.

			The triangle a(n, k) begins:
n\k     0      1      2      3     4     5    6   7  8  9 10 ...
0:      1
1:      1      1
2:      2      1      1
3:      5      3      1      1
4:     14     12      4      1     1
5:     42     55     22      5     1     1
6:    132    273    140     35     6     1    1
7:    429   1428    969    285    51     7    1   1
8:   1430   7752   7084   2530   506    70    8   1  1
9:   4862  43263  53820  23751  5481   819   92   9  1  1
10: 16796 246675 420732 231880 62832 10472 1240 117 10  1  1
... Reformatted by _Wolfdieter Lang_, Feb 06 2020

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd. ed., 1994.

O. Aichholzer, A. Asinowski, and T. Miltzow, Disjoint compatibility graph of non-crossing matchings of points in convex position, arXiv preprint arXiv:1403.5546 [math.CO], 2014.
Jean-Christophe Aval, Multivariate Fuss-Catalan numbers, arXiv:0711.0906 [math.CO], 2007.
C. Banderier and D. Merlini, Lattice paths with an infinite set of jumps
W. G. Brown, Historical note on a recurrent combinatorial problem, Am. Math. Monthly 72 (1965) 973-977.
Sergio Caracciolo and Andrea Sportiello, Spanning forests on random planar lattices, J. Stat. Phys. 135, No. 5-6, 1063-1104 (2009).
CombOS - Combinatorial Object Server, Generate k-ary trees and dissections
M. Dziemianczuk, Enumerations of plane trees with multiple edges and Raney lattice paths, Discrete Mathematics 337 (2014): 9-24.
D. D. Frey and J. A. Sellers, Generalizing Bailey's generalization of the Catalan numbers, The Fibonacci Quarterly, 39 (2001) 142-148.
P. Hilton and J. Pedersen, Catalan Numbers, their generalization and their uses, The Mathematical Intelligencer 13 (1991) 64-75.
V. E. Hoggatt, Jr. and M. Bicknell, Catalan and related sequences arising from inverses of Pascal's triangle matrices, Fib. Quart., 14 (1976), 395-405.
V. U. Pierce, Combinatoric results for graphical enumeration and the higher Catalan numbers, arXiv:math/0703160 [math.CO], 2007.
J. H. Przytycki and A. S. Sikora, Polygon dissections and Euler, Fuss, Kirkman and Cayley numbers, arXiv:math/9811086 [math.CO], 1998.
H. S. Snevily and D. B. West, The Bricklayer Problem and the Strong Cycle Lemma, arXiv:math/9802026 [math.CO], 1998.
Donovan Young, Linear k-Chord Diagrams, arXiv:2004.06921 [math.CO], 2020. See also J. Int. Seq., Vol. 23 (2020), Article 20.9.1.

Reflected version of A070914.

Columns k=0..9 (without leading zeros) give sequences A000108 (Catalan), A001764, A002293, A002294, A002295, A002296, A007556, A062994, A059968, A230388.

a[n_, k_] = Binomial[(k+2)*(n-k), n-k]/((k+1)*(n-k) + 1);
Flatten[Table[a[n, k], {n, 0, 9}, {k, 0, n}]][[1 ;; 53]]
(* Jean-François Alcover, May 27 2011, after formula *)

a(n, k) = binomial((k+2)*(n-k), n-k)/((k+1)*(n-k)+1) = PF(n-k, k+2) if n-k >= 0, otherwise 0.

G.f. for column k: A(k, x) := x^k*RootOf(_Z^(k+2)*x-_Z+1) (Maple notation, from ECS, see links for column sequences and Graham et al. reference eq.(5.59) p. 200 and p. 349).

A213105 G.f. satisfies: A(x) = 1 + x/A(-x*A(x)^12)^6.

Original entry on oeis.org

1, 1, 6, 57, 614, 7716, 104322, 1529385, 23689968, 385885521, 6531397090, 114147452526, 2045979734964, 37435147640010, 696431496524796, 13134442980269397, 250527556214516892, 4824098879117797749, 93639919777995946446, 1830133457257882605430

Offset: 0

Paul D. Hanna, Jun 05 2012

nonn

Compare definition of g.f. to:
(1) B(x) = 1 + x/B(-x*B(x)) when B(x) = 1/(1-x).
(2) C(x) = 1 + x/C(-x*C(x)^3)^2 when C(x) = 1 + x*C(x)^2 (A000108).
(3) D(x) = 1 + x/D(-x*D(x)^5)^3 when D(x) = 1 + x*D(x)^3 (A001764).
(4) E(x) = 1 + x/E(-x*E(x)^7)^4 when E(x) = 1 + x*E(x)^4 (A002293).
(5) F(x) = 1 + x/F(-x*F(x)^9)^5 when F(x) = 1 + x*F(x)^5 (A002294).
(6) G(x) = 1 + x/G(-x*G(x)^11)^6 when G(x) = 1 + x*G(x)^6 (A002295).

			G.f.: A(x) = 1 + x + 6*x^2 + 57*x^3 + 614*x^4 + 7716*x^5 + 104322*x^6 +...
Related expansions:
A(x)^12 = 1 + 12*x + 138*x^2 + 1696*x^3 + 21723*x^4 + 292836*x^5 +...
A(-x*A(x)^12)^6 = 1 - 6*x - 21*x^2 - 146*x^3 - 1959*x^4 - 25056*x^5 -...

Cf. A213091, A213092, A213093, A213094, A213095, A213096.

Cf. A213098, A213099, A213100, A213101, A213102, A213103, A213104.

m = 20; A[] = 1; Do[A[x] = 1 + x/A[-x A[x]^12]^6 + O[x]^m, {m}];
CoefficientList[A[x], x] (* Jean-François Alcover, Nov 06 2019 *)

{a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=1+x/subst(A^6,x,-x*subst(A^12,x,x+x*O(x^n))) );polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

A212073 G.f. satisfies: A(x) = (1 + x*A(x)^(3/2))^4.

Original entry on oeis.org

1, 4, 30, 280, 2925, 32736, 383838, 4654320, 57887550, 734405100, 9467075926, 123648163392, 1632743088275, 21761329287600, 292362576381900, 3955219615609056, 53834425161872586, 736687428853685400, 10129401435828605700, 139876690363085200200

Offset: 0

Paul D. Hanna, Apr 29 2012

Fuss-Catalan sequence is a(n,p,r) = r*binomial(p*n + r, n)/(p*n + r); this is the case p = 6, r = 4. The o.g.f. B(x) of the Fuss_catalan sequence a(n,p,r) satisfies B(x) = {1 + x*B(x)^(p/r)}^r. - Peter Bala, Oct 14 2015

			G.f.: A(x) = 1 + 4*x + 30*x^2 + 280*x^3 + 2925*x^4 + 32736*x^5 +...
Related expansions:
A(x)^(3/2) = 1 + 6*x + 51*x^2 + 506*x^3 + 5481*x^4 +...+ A002295(n+1)*x^n +...
A(x)^(1/4) = 1 + x + 6*x^2 + 51*x^3 + 506*x^4 +...+ A002295(n)*x^n +...

Wikipedia, Fuss-Catalan number

Cf. A002295, A212071, A212072, A130564, A069271, A118970, A233834, A234465, A234510, A234571, A235339.

m = 20; A[_] = 0;
Do[A[x_] = (1 + x*A[x]^(3/2))^4 + O[x]^m, {m}];
CoefficientList[A[x], x] (* Jean-François Alcover, Oct 20 2019 *)

{a(n)=binomial(6*n+4,n) * 4/(6*n+4)}
for(n=0, 40, print1(a(n), ", "))

{a(n)=local(A=1+4*x); for(i=1, n, A=(1+x*A^(3/2))^4+x*O(x^n)); polcoeff(A, n)}

a(n) = 4*binomial(6*n+4,n)/(6*n+4).
G.f. A(x) = G(x)^4 where G(x) = 1 + x*G(x)^6 is the g.f. of A002295.
O.g.f. A(x) = 1/x * series reversion (x/C(x)^4), where C(x) is the o.g.f. for the Catalan numbers A000108. - Peter Bala, Oct 14 2015
D-finite with recurrence 5*n*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)*a(n) -72*(6*n-1)*(3*n-1)*(2*n+1)*(3*n+1)*(6*n+1)*a(n-1)=0. - R. J. Mathar, Nov 22 2024

A212071 G.f. satisfies: A(x) = (1 + x*A(x)^3)^2.

Original entry on oeis.org

1, 2, 13, 114, 1150, 12586, 145299, 1741844, 21475146, 270570300, 3468352701, 45089941936, 593082894768, 7878407177270, 105542811922950, 1424267372100456, 19343105144742098, 264182048662182420, 3626176386241346070, 49995713597946235350, 692084935397470961346

Offset: 0

Paul D. Hanna, Apr 29 2012

The two parameter Fuss-Catalan sequence is A(n,p,r) := r*binomial(n*p + r, n)/(n*p + r), with o.g.f. G(p,r,x) = G(x) satisfying G(x) = {1 + x*G(x)^(p/r)}^r ; this is the case p = 6, r = 2. - Peter Bala, Oct 14 2015

			G.f.: A(x) = 1 + 2*x + 13*x^2 + 114*x^3 + 1150*x^4 + 12586*x^5 +...
Related expansions:
A(x)^3 = 1 + 6*x + 51*x^2 + 506*x^3 + 5481*x^4 +...+ A002295(n+1)*x^n +...
A(x)^(1/2) = 1 + x + 6*x^2 + 51*x^3 + 506*x^4 +...+ A002295(n)*x^n +...

Michael De Vlieger, Table of n, a(n) for n = 0..856
Gi-Sang Cheon, S.-T. Jin, and L. W. Shapiro, A combinatorial equivalence relation for formal power series, Linear Algebra and its Applications, Volume 491, 15 February 2016, Pages 123-137.
Clemens Heuberger, Sarah J. Selkirk, and Stephan Wagner, Enumeration of Generalized Dyck Paths Based on the Height of Down-Steps Modulo k, arXiv:2204.14023 [math.CO], 2022.
Wikipedia, Fuss-Catalan number

Cf. A002295, A212072, A212073, A130564.

Table[c=6n+2;(2*Binomial[c,n])/c,{n,0,20}] (* Harvey P. Dale, Oct 14 2013 *)

{a(n)=binomial(6*n+2,n) * 2/(6*n+2)}
for(n=0, 40, print1(a(n), ", "))

{a(n)=local(A=1+2*x); for(i=1, n, A=(1+x*A^3)^2+x*O(x^n)); polcoeff(A, n)}

a(n) = 2*binomial(6*n+2,n)/(6*n+2).
G.f.: A(x) = G(x)^2 where G(x) = 1 + x*G(x)^6 is the g.f. of A002295.
a(n) = 2*binomial(6n+1, n-1)/n for n>0, a(0)=1. [Bruno Berselli, Jan 19 2014]
A(x^2) = 1/x * series reversion (x/C(x^2)^2), where C(x) = (1 - sqrt( 1 - 4*x))/(2*x) is the o.g.f. for the Catalan numbers A000108. - Peter Bala, Oct 14 2015
D-finite with recurrence 5*n*(5*n+1)*(5*n+2)*(5*n-2)*(5*n-1)*a(n) -72*(6*n-1)*(3*n-1)*(2*n-1)*(3*n-2)*(6*n+1)*a(n-1)=0. - R. J. Mathar, Nov 22 2024

cp's OEIS Frontend

A059968 Number of 10-ary trees.

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A062994 Eighth column of triangle A062993 (without leading zeros). A Pfaff-Fuss or 9-Raney sequence.

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A251592 Triangle of coefficients of polynomials P(n,t) related to the Mittag-Leffler function, where P(n,t) = Product_{k=0..n-2} n*t-k.

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A004355 Binomial coefficient C(6n,n).

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A070914 Array read by antidiagonals giving number of paths up and left from (0,0) to (n,kn) where x/y <= k for all intermediate points.

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A213104 G.f. satisfies: A(x) = 1 + x/A(-x*A(x)^10)^5.

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