A059968 -id:A059968 - cp's OEIS frontend

A251580 E.g.f.: exp(10xG(x)^9) / G(x)^9 where G(x) = 1 + x*G(x)^10 is the g.f. of A059968.

1, 1, 10, 280, 13960, 1023760, 99935200, 12226859200, 1801725932800, 310890328768000, 61516405597830400, 13735605457885312000, 3416919943285809280000, 937247149729410729472000, 281051240591439955878400000, 91474949907165746668607488000, 32117399444469103248129863680000

Offset: 0

Paul D. Hanna, Dec 06 2014

nonn

In general, Sum_{k=0..n} m^k * n!/k! * binomial(m*n-k-m, n-k) * (k-1)/(n-1) is for m>1 asymptotic to m^(m*(n-1)-1/2) / (m-1)^((m-1)*(n-1)-1/2) * n^(n-2) / exp(n-1). - Vaclav Kotesovec, Dec 07 2014

			E.g.f.: A(x) = 1 + x + 10*x^2/2! + 280*x^3/3! + 13960*x^4/4! + 1023760*x^5/5! +...
such that A(x) = exp(10*x*G(x)^9) / G(x)^9
where G(x) = 1 + x*G(x)^10 is the g.f. of A059968:
G(x) = 1 + x + 10*x^2 + 145*x^3 + 2470*x^4 + 46060*x^5 + 910252*x^6 +...
Note that
A'(x) = exp(10*x*G(x)^9) = 1 + 10*x + 280*x^2/2! + 13960*x^3/3! +...
LOGARITHMIC DERIVATIVE.
The logarithm of the e.g.f. begins:
log(A(x)) = x + 9*x^2/2 + 252*x^3/3 + 12654*x^4/4 + 933984*x^5/5 +...
and so A'(x)/A(x) = G(x)^9.
TABLE OF POWERS OF E.G.F.
Form a table of coefficients of x^k/k! in A(x)^n as follows.
n=1: [1, 1,  10,  280,  13960,  1023760,   99935200,  12226859200, ...];
n=2: [1, 2,  22,  620,  30760,  2243120,  217911520,  26556406400, ...];
n=3: [1, 3,  36, 1026,  50760,  3683880,  356283360,  43256151360, ...];
n=4: [1, 4,  52, 1504,  74344,  5374240,  517647520,  62621962240, ...];
n=5: [1, 5,  70, 2060, 101920,  7344920,  704861200,  84980501600, ...];
n=6: [1, 6,  90, 2700, 133920,  9629280,  921060720, 110691813600, ...];
n=7: [1, 7, 112, 3430, 170800, 12263440, 1169680960, 140152067440, ...];
n=8: [1, 8, 136, 4256, 213040, 15286400, 1454475520, 173796462080, ...]; ...
in which the main diagonal begins (see A251587):
[1, 2, 36, 1504, 101920, 9629280, 1169680960, 173796462080, ...]
and is given by the formula:
[x^n/n!] A(x)^(n+1) = 10^(n-8) * (n+1)^(n-9) * (4782969*n^8 + 50309748*n^7 + 237013938*n^6 + 655232760*n^5 + 1166624361*n^4 + 1374998212*n^3 + 1051760172*n^2 + 479277840*n + 100000000) for n>=0.

Cf. A251590, A251670, A059968, A234573.

Cf. Variants: A243953, A251573, A251574, A251575, A251576, A251577, A251578, A251579.

Flatten[{1,1,Table[Sum[10^k * n!/k! * Binomial[10*n-k-10, n-k] * (k-1)/(n-1),{k,0,n}],{n,2,20}]}] (* Vaclav Kotesovec, Dec 07 2014 *)

{a(n) = local(G=1);for(i=1,n, G = 1 + x*G^10 +x*O(x^n)); n!*polcoeff( exp(10*x*G^9) / G^9, n)}
for(n=0, 20, print1(a(n), ", "))

{a(n) = if(n==0, 1, sum(k=0, n, 10^k * n!/k! * binomial(10*n-k-10,n-k)*if(n==1,1/10,(k-1)/(n-1)) ))}
for(n=0, 20, print1(a(n), ", "))

Let G(x) = 1 + x*G(x)^10 be the g.f. of A059968, then the e.g.f. A(x) of this sequence satisfies:
(1) A'(x)/A(x) = G(x)^9.
(2) A'(x) = exp(10*x*G(x)^9).
(3) A(x) = exp( Integral G(x)^9 dx ).
(4) A(x) = exp( Sum_{n>=1} A234573(n-1)*x^n/n ), where A234573(n-1) = binomial(10*n-2,n)/(9*n-1).
(5) A(x) = F(x/A(x)) where F(x) is the e.g.f. of A251590.
(6) A(x) = Sum_{n>=0} A251590(n)*(x/A(x))^n/n! and
(7) [x^n/n!] A(x)^(n+1) = (n+1)*A251590(n),
where A251590(n) = 10^(n-8) * (n+1)^(n-10) * (4782969*n^8 + 50309748*n^7 + 237013938*n^6 + 655232760*n^5 + 1166624361*n^4 + 1374998212*n^3 + 1051760172*n^2 + 479277840*n + 100000000).
a(n) = Sum_{k=0..n} 10^k * n!/k! * binomial(10*n-k-10, n-k) * (k-1)/(n-1) for n>1.
Recurrence: 81*(3*n-5)*(3*n-4)*(9*n-17)*(9*n-16)*(9*n-14)*(9*n-13)*(9*n-11)*(9*n-10)*(250000*n^8 - 5300000*n^7 + 49332500*n^6 - 263500000*n^5 + 884055975*n^4 - 1909634570*n^3 + 2596659373*n^2 - 2035277286*n + 705468040)*a(n) = 800*(3125000000000*n^17 - 111562500000000*n^16 + 1872125000000000*n^15 - 19618187500000000*n^14 + 143829395937500000*n^13 - 783195370343750000*n^12 + 3281447638218750000*n^11 - 10810863753751875000*n^10 + 28370066880833218750*n^9 - 59681174371832246875*n^8 + 100725400409628775000*n^7 - 135736802338370325750*n^6 + 144424061701272600950*n^5 - 118936947986511839915*n^4 + 73322264536912326596*n^3 - 31942069342168467356*n^2 + 8798129066413437408*n - 1156512281566561920)*a(n-1) + 10000000000*(250000*n^8 - 3300000*n^7 + 19232500*n^6 - 64805000*n^5 + 138543475*n^4 - 193260670*n^3 + 172779013*n^2 - 91243350*n + 22054032)*a(n-2). - Vaclav Kotesovec, Dec 07 2014
a(n) ~ 10^(10*(n-1)-1/2) / 9^(9*(n-1)-1/2) * n^(n-2) / exp(n-1). - Vaclav Kotesovec, Dec 07 2014

A251670 E.g.f.: exp(10xG(x)^9) / G(x) where G(x) = 1 + x*G(x)^10 is the g.f. of A059968.

Original entry on oeis.org

1, 9, 242, 11824, 856824, 82986080, 10097121280, 1481787433920, 254874712419200, 50305519571800960, 11209381628379724800, 2783746998856794752000, 762476362390276346060800, 228363072063685762536960000, 74247696727054926125971251200, 26044746725090717967744412672000

Offset: 0

Paul D. Hanna, Dec 07 2014

nonn

In general, sum_{k=0..n} m^k * n!/k! * binomial(m*n-k-2,n-k) * ((m-1)*k-1)/((m-1)*n-1), m>2, is asymptotic to (m-2) * m^(m*n-3/2) / (m-1)^((m-1)*n-1/2) * n^(n-1) / exp(n-1). - Vaclav Kotesovec, Dec 07 2014

			E.g.f.: A(x) = 1 + 9*x + 242*x^2/2! + 11824*x^3/3! + 856824*x^4/4! + 82986080*x^5/5! +...
such that A(x) = exp(10*x*G(x)^9) / G(x)
where G(x) = 1 + x*G(x)^10 is the g.f. of A059958:
G(x) = 1 + x + 10*x^2 + 145*x^3 + 2470*x^4 + 46060*x^5 + 910252*x^6 +...

Cf. A251580, A251700, A059968.

Cf. Variants: A243953, A251663, A251664, A251665, A251666, A251667, A251668, A251669.

Table[Sum[10^k * n!/k! * Binomial[10*n-k-2,n-k] * (9*k-1)/(9*n-1),{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Dec 07 2014 *)

{a(n)=local(G=1); for(i=0, n, G = 1 + x*G^10 +x*O(x^n)); n!*polcoeff(exp(10*x*G^9)/G, n)}
for(n=0, 20, print1(a(n), ", "))

{a(n) = sum(k=0, n, 9^k * n!/k! * binomial(9*n-k-2,n-k) * (8*k-1)/(8*n-1) )}
for(n=0, 20, print1(a(n), ", "))

Let G(x) = 1 + x*G(x)^10 be the g.f. of A059968, then the e.g.f. A(x) of this sequence satisfies:
(1) A'(x)/A(x) = G(x)^9 + 8*G'(x)/G(x).
(2) A(x) = F(x/A(x)^9) where F(x) is the e.g.f. of A251700.
(3) A(x) = Sum_{n>=0} A251700(n)*(x/A(x)^9)^n/n! where A251700(n) = (8*n+1) * (9*n+1)^(n-2) * 10^n.
(4) [x^n/n!] A(x)^(9*n+1) = (8*n+1) * (9*n+1)^(n-1) * 10^n.
a(n) = Sum_{k=0..n} 10^k * n!/k! * binomial(10*n-k-2,n-k) * (9*k-1)/(9*n-1) for n>=0.
Recurrence: 81*(3*n-2)*(3*n-1)*(9*n-8)*(9*n-7)*(9*n-5)*(9*n-4)*(9*n-2)*(9*n-1)*(100000000*n^9 - 1508750000*n^8 + 10158500000*n^7 - 40108637500*n^6 + 102477510000*n^5 - 175985889125*n^4 + 203494963150*n^3 - 153061617555*n^2 + 68057955478*n - 13624029912)*a(n) = 800*(1250000000000000*n^18 - 25734375000000000*n^17 + 248379687500000000*n^16 - 1494668125000000000*n^15 + 6291920187500000000*n^14 - 19707236445312500000*n^13 + 47696214907031250000*n^12 - 91443867836531250000*n^11 + 141240231848528125000*n^10 - 177729148289358906250*n^9 + 183386452781820390625*n^8 - 155416253373710737500*n^7 + 107706559814898413750*n^6 - 60246014246053412750*n^5 + 26474457002621149925*n^4 - 8675686414409435660*n^3 + 1905677176596950796*n^2 - 212632849946745072*n - 10904042717568)*a(n-1) + 10000000000*(100000000*n^9 - 608750000*n^8 + 1688500000*n^7 - 2844137500*n^6 + 3264185000*n^5 - 2692901625*n^4 + 1611256650*n^3 - 663025355*n^2 + 151278318*n + 4536)*a(n-2). - Vaclav Kotesovec, Dec 07 2014
a(n) ~ 8 * 10^(10*n-3/2) / 3^(18*n-1) * n^(n-1) / exp(n-1). - Vaclav Kotesovec, Dec 07 2014

A062744 Duplicate of A059968.

Original entry on oeis.org

1, 1, 10, 145, 2470, 46060, 910252, 18730855, 397089550, 8612835715, 190223180840, 4263421511271, 96723482198980, 2216905597676000, 51256802757808320

Offset: 0

dead

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)

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

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).

A229963 a(n) = 11binomial(10n + 11, n)/(10*n + 11) .

Original entry on oeis.org

1, 11, 165, 2860, 53900, 1072797, 22188859, 472214600, 10273141395, 227440759700, 5107663394691, 116068178638776, 2664012608972000, 61668340817988135, 1438101958237201950, 33753007927148177360, 796704536753910327114

Offset: 0

Tim Fulford, Oct 04 2013

Fuss-Catalan sequence is a(n,p,r) = r*binomial(n*p + r,n)/(n*p + r), where p = 10, r = 11.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
J-C. Aval, Multivariate Fuss-Catalan Numbers, arXiv:0711.0906 [math.CO], 2007.
J-C. Aval, Multivariate Fuss-Catalan Numbers, Discrete Math., 308 (2008), 4660-4669.
Thomas A. Dowling, Catalan Numbers Chapter 7
Wojciech Mlotkowski, Fuss-Catalan Numbers in Noncommutative Probability, Docum. Mathm. 15: 939-955.
Wikipedia, Fuss-Catalan number

Cf. A000108, A059968, A234525, A234526, A234527, A234528, A234529, A234570, A234571, A234573.

Cf. A000245 (k = 3), A006629 (k = 4), A196678 (k = 5), A233668 (k = 6), A233743 (k = 7), A233835 (k = 8), A234467 (k = 9), A232265 (k = 10).

[11*Binomial(10*n+11,n)/(10*n+11) : n in [0..20]]; // Vincenzo Librandi, Jan 10 2014

Table[11/(10 n + 11) Binomial[10 n + 11, n], {n, 0, 40}] (* Vincenzo Librandi, Jan 10 2014 *)

a(n) = 11*binomial(10*n+11,n)/(10*n+11);

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(10/11))^11+x*O(x^n)); polcoeff(B, n)}

G.f. satisfies: A(x) = {1 + x*A(x)^(p/r)}^r, where p = 10, r = 11.
From _Peter Bala, Oct 16 2015: (Start)
O.g.f. A(x) = 1/x * series reversion (x*C(-x)^11), where C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f. for the Catalan numbers A000108. See cross-references for other Fuss-Catalan sequences with o.g.f. 1/x * series reversion (x*C(-x)^k), k = 3 through 11.
A(x)^(1/11) is the o.g.f. for A059968. (End)
D-finite with recurrence: 81*n*(9*n+11)*(9*n+4)*(3*n+2)*(9*n+8)*(9*n+10)*(3*n+1)*(9*n+5)*(9*n+7)*a(n) -800*(10*n+1)*(5*n+1)*(10*n+3)*(5*n+2)*(2*n+1)*(5*n+3)*(10*n+7)*(5*n+4)*(10*n+9)*a(n-1)=0. - R. J. Mathar, Feb 21 2020

Corrected by Vincenzo Librandi, Jan 10 2014

A234571 a(n) = 4binomial(10n+8,n)/(5*n+4).

Original entry on oeis.org

1, 8, 108, 1776, 32430, 632016, 12876864, 270964320, 5843355957, 128462407840, 2868356980060, 64869895026144, 1482877843096650, 34207542810153216, 795318309360948240, 18617396126132233920, 438423206616057162258, 10379232525028947311160, 246878659984195222962220

Offset: 0

Tim Fulford, Dec 28 2013

Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), where p = 10, r = 8.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
J-C. Aval, Multivariate Fuss-Catalan Numbers, arXiv:0711.0906v1, Discrete Math., 308 (2008), 4660-4669.
Thomas A. Dowling, Catalan Numbers Chapter 7
Wojciech Mlotkowski, Fuss-Catalan Numbers in Noncommutative Probability, Docum. Mathm. 15: 939-955.
Wikipedia, Fuss-Catalan number

Cf. A059968, A234525, A234526, A234527, A234528, A234529, A234570, A234573, A059968, A069271, A118970, A212073, A233834, A234465, A234510, A235339.

[4*Binomial(10*n+8, n)/(5*n+4): n in [0..30]];

Table[4 Binomial[10 n + 8, n]/(5 n + 4), {n, 0, 30}]

PARI
```
a(n) = 4*binomial(10*n+8,n)/(5*n+4);
```

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(5/4))^8+x*O(x^n)); polcoeff(B, n)}

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, where p = 10, r = 8.

O.g.f. A(x) = 1/x * series reversion (x/C(x)^8), where C(x) is the o.g.f. for the Catalan numbers A000108. A(x)^(1/8) is the o.g.f. for A059968. - Peter Bala, Oct 14 2015

A234573 a(n) = 9binomial(10n+9,n)/(10*n+9).

Original entry on oeis.org

1, 9, 126, 2109, 38916, 763686, 15636192, 330237765, 7141879503, 157366449604, 3520256293710, 79735912636302, 1825080422272800, 42148579533938784, 980892581545169496, 22980848343194476245, 541581608172776494554, 12829884648994115426295, 305349921559399354716430

Offset: 0

Tim Fulford, Dec 28 2013

Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), where p=10, r=9.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
J-C. Aval, Multivariate Fuss-Catalan Numbers, arXiv:0711.0906 [math.CO], 2007; Discrete Math., 308 (2008), 4660-4669.
Thomas A. Dowling, Catalan Numbers Chapter 7.
Elżbieta Liszewska and Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.
Wojciech Mlotkowski, Fuss-Catalan Numbers in Noncommutative Probability, Docum. Mathm. 15: 939-955.
J. Sawada, J. Sears, A. Trautrim, and A. Williams, Demystifying our Grandparent's De Bruijn Sequences with Concatenation Trees, arXiv:2308.12405 [math.CO], 2023.

Cf. A000108, A059968, A118971, A130564, A234513, A234525, A234526, A234527, A234528, A234529, A234570, A234571, A229963.

[9*Binomial(10*n+9, n)/(10*n+9): n in [0..30]];

Table[9 Binomial[10 n + 9, n]/(10 n + 9), {n, 0, 30}]

PARI
```
a(n) = 9*binomial(10*n+9,n)/(10*n+9);
```

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(10/9))^9+x*O(x^n)); polcoeff(B, n)}

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, where p=10, r=9.
From Wolfdieter Lang, Feb 06 2020: (Start)
G.f.: hypergeom([9, 10, ..., 18]/10, [10, 11, ..., 18]/9, (10^10/9^9)*x).
E.g.f.: hypergeom([9, 11, 12, ..., 18]/10, [10, 11, ..., 18]/9, (10^10/9^9) * x). Cf. Ilya Gutkovsky in  A118971. (End)
a(n) = binomial(10*n + 8 , n+1)/(9*n + 8) which is instance k = 9 of c(k, n+1) given in a comment in A130564. x*B(x), with the above given g.f. B(x), is the compositional inverse of y*(1 - y)^9, hence  B(x)*(1 - x*B(x))^9 = 1. For another formula for B(x) involving the hypergeometric function 9F8 see the analog formula in A234513. - Wolfdieter Lang, Feb 15 2024

cp's OEIS Frontend

A251580 E.g.f.: exp(10*x*G(x)^9) / G(x)^9 where G(x) = 1 + x*G(x)^10 is the g.f. of A059968.

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A251670 E.g.f.: exp(10*x*G(x)^9) / G(x) where G(x) = 1 + x*G(x)^10 is the g.f. of A059968.

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A062744 Duplicate of A059968.

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

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A251580 E.g.f.: exp(10xG(x)^9) / G(x)^9 where G(x) = 1 + x*G(x)^10 is the g.f. of A059968.

A251670 E.g.f.: exp(10xG(x)^9) / G(x) where G(x) = 1 + x*G(x)^10 is the g.f. of A059968.

A229963 a(n) = 11binomial(10n + 11, n)/(10*n + 11) .

A234571 a(n) = 4binomial(10n+8,n)/(5*n+4).

A234573 a(n) = 9binomial(10n+9,n)/(10*n+9).