A062994 -id:A062994 - cp's OEIS frontend

A251579 E.g.f.: exp(9xG(x)^8) / G(x)^8 where G(x) = 1 + x*G(x)^9 is the g.f. of A062994.

1, 1, 9, 225, 10017, 656289, 57255849, 6262226721, 825067217025, 127305462542913, 22527254639457801, 4498536675388410081, 1000890043482114644769, 245556248365681036646625, 65862976584851401437170217, 19174678419336874098038167329, 6022064808176665662053835550209

Offset: 0

Paul D. Hanna, Dec 06 2014

nonn

			E.g.f.: A(x) = 1 + x + 9*x^2/2! + 225*x^3/3! + 10017*x^4/4! + 656289*x^5/5! +...
such that A(x) = exp(9*x*G(x)^8) / G(x)^8
where G(x) = 1 + x*G(x)^9 is the g.f. of A062994:
G(x) = 1 + x + 9*x^2 + 117*x^3 + 1785*x^4 + 29799*x^5 + 527085*x^6 +...
Note that
A'(x) = exp(9*x*G(x)^8) = 1 + 9*x + 225*x^2/2! + 10017*x^3/3! +...
LOGARITHMIC DERIVATIVE.
The logarithm of the e.g.f. begins:
log(A(x)) = x + 8*x^2/2 + 200*x^3/3 + 8976*x^4/4 + 592368*x^5/5 +...
and so A'(x)/A(x) = G(x)^8.
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,   9,  225,  10017,   656289,  57255849,  6262226721, ...];
n=2: [1, 2,  20,  504,  22320,  1453248, 126104256, 13731880320, ...];
n=3: [1, 3,  33,  843,  37233,  2411667, 208241361, 22581193851, ...];
n=4: [1, 4,  48, 1248,  55104,  3554496, 305558784, 33002857728, ...];
n=5: [1, 5,  65, 1725,  76305,  4906965, 420159825, 45211985325, ...];
n=6: [1, 6,  84, 2280, 101232,  6496704, 554376384, 59448214656, ...];
n=7: [1, 7, 105, 2919, 130305,  8353863, 710786601, 75977951175, ...];
n=8: [1, 8, 128, 3648, 163968, 10511232, 892233216, 95096756736, ...]; ...
in which the main diagonal begins (see A251587):
[1, 2, 33, 1248, 76305, 6496704, 710786601, 95096756736, ...]
and is given by the formula:
[x^n/n!] A(x)^(n+1) = 9^(n-7) * (n+1)^(n-8) * (262144*n^7 + 2494464*n^6 + 10470208*n^5 + 25229505*n^4 + 37857568*n^3 + 35537670*n^2 + 19414368*n + 4782969) for n>=0.

Cf. A251589, A251669, A062994, A234513.

Cf. Variants: A243953, A251573, A251574, A251575, A251576, A251577, A251578, A251580.

Flatten[{1,1,Table[Sum[9^k * n!/k! * Binomial[9*n-k-9, 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^9 +x*O(x^n)); n!*polcoeff(exp(9*x*G^8)/G^8, n)}
for(n=0, 20, print1(a(n), ", "))

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

Let G(x) = 1 + x*G(x)^9 be the g.f. of A062994, then the e.g.f. A(x) of this sequence satisfies:
(1) A'(x)/A(x) = G(x)^8.
(2) A'(x) = exp(8*x*G(x)^8).
(3) A(x) = exp( Integral G(x)^8 dx ).
(4) A(x) = exp( Sum_{n>=1} A234513(n-1)*x^n/n ), where A234513(n-1) = binomial(9*n-2,n)/(8*n-1).
(5) A(x) = F(x/A(x)) where F(x) is the e.g.f. of A251589.
(6) A(x) = Sum_{n>=0} A251589(n)*(x/A(x))^n/n! and
(7) [x^n/n!] A(x)^(n+1) = (n+1)*A251589(n),
where A251589(n) = 9^(n-7) * (n+1)^(n-9) * (262144*n^7 + 2494464*n^6 + 10470208*n^5 + 25229505*n^4 + 37857568*n^3 + 35537670*n^2 + 19414368*n + 4782969).
a(n) = Sum_{k=0..n} 9^k * n!/k! * binomial(9*n-k-9, n-k) * (k-1)/(n-1) for n>1.
Recurrence: 128*(2*n-3)*(4*n-7)*(4*n-5)*(8*n-15)*(8*n-13)*(8*n-11)*(8*n-9)*(59049*n^7 - 1102248*n^6 + 8858079*n^5 - 39764115*n^4 + 107806473*n^3 - 176772075*n^2 + 162618742*n - 64907105)*a(n) = 81*(282429536481*n^15 - 8943601988565*n^14 + 132044525265870*n^13 - 1206188364304287*n^12 + 7627178203628841*n^11 - 35382975568258428*n^10 + 124478964551078775*n^9 - 338415281830783431*n^8 + 717436315214480025*n^7 - 1187215577095780764*n^6 + 1522794566607803919*n^5 - 1488866286016780047*n^4 + 1075889068341959448*n^3 - 543536112365518695*n^2 + 172059320987344825*n - 25799292366848000)*a(n-1) - 387420489*(59049*n^7 - 688905*n^6 + 3484620*n^5 - 9940725*n^4 + 17352558*n^3 - 18650247*n^2 + 11527801*n - 3203200)*a(n-2). - Vaclav Kotesovec, Dec 07 2014
a(n) ~ 9^(9*(n-1)-1/2) / 8^(8*(n-1)-1/2) * n^(n-2) / exp(n-1). - Vaclav Kotesovec, Dec 07 2014

A251669 E.g.f.: exp(9xG(x)^8) / G(x) where G(x) = 1 + x*G(x)^9 is the g.f. of A062994.

Original entry on oeis.org

1, 8, 191, 8310, 537117, 46444164, 5047987707, 662002733394, 101779688986425, 17959176833948928, 3578033935192224951, 794559576204365478318, 194620831940208238831701, 52129134740350115227721340, 15158273263608217360939225587, 4755712518628181890216523759754

Offset: 0

Paul D. Hanna, Dec 07 2014

nonn

			E.g.f.: A(x) = 1 + 8*x + 191*x^2/2! + 8310*x^3/3! + 537117*x^4/4! + 46444164*x^5/5! +...
such that A(x) = exp(9*x*G(x)^8) / G(x)
where G(x) = 1 + x*G(x)^9 is the g.f. of A062994:
G(x) = 1 + x + 9*x^2 + 117*x^3 + 1785*x^4 + 29799*x^5 + 527085*x^6 +...

Cf. A251579, A251699, A062994.

Cf. Variants: A243953, A251663, A251664, A251665, A251666, A251667, A251668, A251670.

Table[Sum[9^k * n!/k! * Binomial[9*n-k-2,n-k] * (8*k-1)/(8*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^9 +x*O(x^n)); n!*polcoeff(exp(9*x*G^8)/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)^9 be the g.f. of A062994, then the e.g.f. A(x) of this sequence satisfies:
(1) A'(x)/A(x) = G(x)^8 + 7*G'(x)/G(x).
(2) A(x) = F(x/A(x)^8) where F(x) is the e.g.f. of A251699.
(3) A(x) = Sum_{n>=0} A251699(n)*(x/A(x)^8)^n/n! where A251699(n) = (7*n+1) * (8*n+1)^(n-2) * 9^n.
(4) [x^n/n!] A(x)^(8*n+1) = (7*n+1) * (8*n+1)^(n-1) * 9^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.
Recurrence: 128*(2*n-1)*(4*n-3)*(4*n-1)*(8*n-7)*(8*n-5)*(8*n-3)*(8*n-1)*(33480783*n^8 - 453319173*n^7 + 2697889761*n^6 - 9230277240*n^5 + 19886167926*n^4 - 27672715746*n^3 + 24328423881*n^2 - 12365760717*n + 2776106045)*a(n) = 81*(160137547184727*n^16 - 2968899287488272*n^15 + 25604779347830979*n^14 - 136506824772659775*n^13 + 504285657127489314*n^12 - 1371500076773316825*n^11 + 2847804013092225933*n^10 - 4619534029925962572*n^9 + 5937710241656343834*n^8 - 6090889132598477481*n^7 + 4986522977501530773*n^6 - 3228624422259256476*n^5 + 1615386846720554091*n^4 - 595058403096826425*n^3 + 145565831993332122*n^2 - 17972427186502245*n - 2554359808000)*a(n-1) - 387420489*(33480783*n^8 - 185472909*n^7 + 462117474*n^6 - 687717459*n^5 + 680611896*n^4 - 464268429*n^3 + 210617505*n^2 - 51824070*n - 4480)*a(n-2). - Vaclav Kotesovec, Dec 07 2014
a(n) ~ 7 * 3^(18*n-3) / 8^(8*n-1/2) * n^(n-1) / exp(n-1). - Vaclav Kotesovec, Dec 07 2014

A002293 Number of dissections of a polygon: binomial(4n, n)/(3n + 1).

Original entry on oeis.org

1, 1, 4, 22, 140, 969, 7084, 53820, 420732, 3362260, 27343888, 225568798, 1882933364, 15875338990, 134993766600, 1156393243320, 9969937491420, 86445222719724, 753310723010608, 6594154339031800, 57956002331347120, 511238042454541545

Offset: 0

N. J. A. Sloane

The number of rooted loopless n-edge maps in the plane (planar with a distinguished outside face). - Valery A. Liskovets, Mar 17 2005
Number of lattice paths from (1,0) to (3*n+1,n) which, starting from (1,0), only utilize the steps +(1,0) and +(0,1) and additionally, the paths lie completely below the line y = (1/3)*x (i.e., if (a,b) is in the path, then b < a/3). - Joseph Cooper (jecooper(AT)mit.edu), Feb 07 2006
Number of length-n restricted growth strings (RGS) [s(0), s(1), ..., s(n-1)] where s(0) = 0 and s(k) <= s(k-1) + 3, see fxtbook link below. - Joerg Arndt, Apr 08 2011
From Wolfdieter Lang, Sep 14 2007: (Start)
a(n), n >= 1, enumerates quartic trees (rooted, ordered, incomplete) with n vertices (including the root).
Pfaff-Fuss-Catalan sequence C^{m}_n for m = 4. See the Graham et al. reference, p. 347. eq. 7.66. (Second edition, p. 361, eq. 7.67.) See also the Pólya-Szegő reference.
Also 4-Raney sequence. See the Graham et al. reference, pp. 346-347.
(End)
Bacher: "We describe the statistics of checkerboard triangulations obtained by coloring black every other triangle in triangulations of convex polygons." The current sequence (A002293) occurs on p. 12 as one of two "extremal sequences" of an array of coefficients of polynomials, whose generating functions are given in terms of hypergeometric functions. - Jonathan Vos Post, Oct 05 2007
A generating function in terms of a (labyrinthine) solution to a depressed quartic equation is given in the Copeland link for signed A005810. With D(z,t) that g.f., a g.f. for signed A002293 is {[-1+1/D(z,t)]/(4t)}^(1/3). - Tom Copeland, Oct 10 2012
For a relation to the inviscid Burgers's equation, see A001764. - Tom Copeland, Feb 15 2014
For relations to compositional inversion, the Legendre transform, and convex geometry, see the Copeland, the Schuetz and Whieldon, and the Gross (p. 58) links. - Tom Copeland, Feb 21 2017 (See also Gross et al. in A062994. - Tom Copeland, Dec 24 2019)
This is the number of A'Campo bicolored forests of degree n and co-dimension 0. This can be shown using generating functions or a combinatorial approach. See Combe and Jugé link below. - Noemie Combe, Feb 28 2017
Conjecturally, a(n) is the number of 3-uniform words over the alphabet [n] that avoid the patterns 231 and 221 (see the Defant and Kravitz link). - Colin Defant, Sep 26 2018
The compositional inverse o.g.f. pair in Copeland's comment above are related to a pair of quantum fields in Balduf's thesis by Theorem 4.2 on p. 92. Cf. A001764. - Tom Copeland, Dec 13 2019
a(n) is the total number of down steps before the first up step in all 3_1-Dyck paths of length 4*n. A 3_1-Dyck path is a lattice path with steps (1, 3), (1, -1) that starts and ends at y = 0 and stays above the line y = -1. - Sarah Selkirk, May 10 2020
a(n) is the number of pairs (A<=B) of noncrossing partitions of [2n] such that every block of A has exactly two elements. In fact, it is proved that a(n) is the number of planar tied arc diagrams with n arcs (see Aicardi link below). A planar diagram with n arcs represents a noncrossing partition A of [2n] with n blocks, each block containing the endpoints of one arc; each tie connects two arcs, so that the ties define a partition B >= A: the endpoints of two arcs connected by a tie belong to the same block of B. Ties do not cross arcs nor other ties iff B has a planar diagram, i.e., B is a noncrossing partition. - Francesca Aicardi, Nov 07 2022
Dropping the initial 1 (starting 1, 4, 22 with offset 1) yields the REVERT transformation 1, -4 ,10, -20, 35.. essentially A000292 without leading 0. - R. J. Mathar, Aug 17 2023
Number of rooted polyominoes composed of n pentagonal cells of the hyperbolic regular tiling with Schläfli symbol {5,oo}. A rooted polyomino has one external edge identified, and chiral pairs are counted as two. A stereographic projection of the {5,oo} tiling on the Poincaré disk can be obtained via the Christensson link. - Robert A. Russell, Jan 27 2024
This is instance k = 4 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
a(n) is the cardinality of the planar ramified Jones monoid PR(J_n). - Diego Arcis, Nov 21 2024

			There are a(2) = 4 quartic trees (vertex degree <= 4 and 4 possible branchings) with 2 vertices (one of them the root). Adding one more branch (one more vertex) to these four trees yields 4*4 + 6 = 22 = a(3) such trees.

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Column k=3 of triangle A062993 and A070914.
Cf. A000260, A002295, A002296, A027836, A062994, A346646 (binomial transform), A346664 (inverse binomial transform).
Cf. A006632, A006633, A006634, A025174, A069271, A196678, A224274, A233658, A233666, A233667, A277877, A283049, A283101, A283102, A283103.
Polyominoes: A005038 (oriented), A005040 (unoriented), A369471 (chiral), A369472 (achiral), A001764 {4,oo}, A002294 {6,oo}.
Cf. A130564 (for generalized Catalan C(k, n), for = 4).

List([0..22],n->Binomial(4*n,n)/(3*n+1)); # Muniru A Asiru, Nov 01 2018

[ Binomial(4*n,n)/(3*n+1): n in [0..50]]; // Vincenzo Librandi, Apr 19 2011

series(RootOf(g = 1+x*g^4, g),x=0,20); # Mark van Hoeij, Nov 10 2011
seq(binomial(4*n, n)/(3*n+1),n=0..20); # Robert FERREOL, Apr 02 2015
# Using the integral representation above:
Digits:=6;
R:=proc(x)((I + sqrt(3))*(4*sqrt(256 - 27*x) - 12*I*sqrt(3)*sqrt(x))^(1/3))/16 - ((I - sqrt(3))*(4*sqrt(256 - 27*x) + 12*I*sqrt(3)*sqrt(x))^(1/3))/16;end;
W:=proc(x) x^(-3/4) * sqrt(4*R(x) - 3^(3/4)*x^(1/4)/sqrt(R(x)))/(2*3^(1/4)*Pi);end;
# Attention: W(x) is singular at x = 0. Integration is done from  a very small positive x to x = 256/27.
# For a(8):  ... gives 420732
evalf(int(x^8*W(x),x=0.000001..256/27));
# Karol A. Penson, Jul 05 2024

CoefficientList[InverseSeries[ Series[ y - y^4, {y, 0, 60}], x], x][[Range[2, 60, 3]]]
Table[Binomial[4n,n]/(3n+1),{n,0,25}] (* Harvey P. Dale, Apr 18 2011 *)
CoefficientList[1 + InverseSeries[Series[x/(1 + x)^4, {x, 0, 60}]], x] (* Gheorghe Coserea, Aug 12 2015 *)
terms = 22; A[] = 0; Do[A[x] = 1 + x*A[x]^4 + O[x]^terms, terms];
CoefficientList[A[x], x] (* Jean-François Alcover, Jan 13 2018 *)

a(n)=binomial(4*n,n)/(3*n+1) /* Charles R Greathouse IV, Jun 16 2011 */

my(x='x+O('x^33)); Vec(1 + serreverse(x/(1+x)^4)) \\ Gheorghe Coserea, Aug 12 2015

A002293_list, x = [1], 1
for n in range(100):
    x = x*4*(4*n+3)*(4*n+2)*(4*n+1)//((3*n+2)*(3*n+3)*(3*n+4))
    A002293_list.append(x) # Chai Wah Wu, Feb 19 2016

O.g.f. satisfies: A(x) = 1 + x*A(x)^4 = 1/(1 - x*A(x)^3).
a(n) = binomial(4*n,n-1)/n, n >= 1, a(0) = 1. From the Lagrange series of the o.g.f. A(x) with its above given implicit equation.
From Karol A. Penson, Apr 02 2010: (Start)
Integral representation as n-th Hausdorff power moment of a positive function on the interval [0, 256/27]:
a(n) = Integral_{x=0..256/27}(x^n((3/256) * sqrt(2) * sqrt(3) * ((2/27) * 3^(3/4) * 27^(1/4) * 256^(/4) * hypergeom([-1/12, 1/4, 7/12], [1/2, 3/4], (27/256)*x)/(sqrt(Pi) * x^(3/4)) - (2/27) * sqrt(2) * sqrt(27) * sqrt(256) * hypergeom([1/6, 1/2, 5/6], [3/4, 5/4], (27/256)*x)/ (sqrt(Pi) * sqrt(x)) - (1/81) * 3^(1/4) * 27^(3/4) * 256^(1/4) * hypergeom([5/12, 3/4, 13/12], [5/4, 3/2], (27/256)*x/(sqrt(Pi)*x^(1/4)))/sqrt(Pi))).
This representation is unique as it represents the solution of the Hausdorff moment problem.
O.g.f.: hypergeom([1/4, 1/2, 3/4], [2/3, 4/3], (256/27)*x);
E.g.f.: hypergeom([1/4, 1/2, 3/4], [2/3, 1, 4/3], (256/27)*x). (End)
a(n) = upper left term in M^n, M = the production matrix:
  1, 1
  3, 3, 1
  6, 6, 3, 1
  ...
(where 1, 3, 6, 10, ...) is the triangular series. - Gary W. Adamson, Jul 08 2011
O.g.f. satisfies g = 1+x*g^4. If h is the series reversion of x*g, so h(x*g)=x, then (x-h(x))/x^2 is the o.g.f. of A006013. - Mark van Hoeij, Nov 10 2011
a(n) = binomial(4*n+1, n)/(4*n+1) = A062993(n+2,2). - Robert FERREOL, Apr 02 2015
a(n) = Sum_{i=0..n-1} Sum_{j=0..n-1-i} Sum_{k=0..n-1-i-j} a(i)*a(j)*a(k)*a(n-1-i-j-k) for n>=1; and a(0) = 1. - Robert FERREOL, Apr 02 2015
a(n) ~ 2^(8*n+1/2) / (sqrt(Pi) * n^(3/2) * 3^(3*n+3/2)). - Vaclav Kotesovec, Jun 03 2015
From Peter Bala, Oct 16 2015: (Start)
A(x)^2 is o.g.f. for A069271; A(x)^3 is o.g.f. for A006632;
A(x)^5 is o.g.f. for A196678; A(x)^6 is o.g.f. for A006633;
A(x)^7 is o.g.f. for A233658; A(x)^8 is o.g.f. for A233666;
A(x)^9 is o.g.f. for A006634; A(x)^10 is o.g.f. for A233667. (End)
D-finite with recurrence: a(n+1) = a(n)*4*(4*n + 3)*(4*n + 2)*(4*n + 1)/((3*n + 2)*(3*n + 3)*(3*n + 4)). - Chai Wah Wu, Feb 19 2016
E.g.f.: F([1/4, 1/2, 3/4], [2/3, 1, 4/3], 256*x/27), where F is the generalized hypergeometric function. - Stefano Spezia, Dec 27 2019
x*A'(x)/A(x) = (A(x) - 1)/(- 3*A(x) + 4) = x + 7*x^2 + 55*x^3 + 455*x^4 + ... is the o.g.f. of A224274. Cf. A001764 and A002294 - A002296. - Peter Bala, Feb 04 2022
a(n) = hypergeom([1 - n, -3*n], [2], 1). Row sums of A173020. - Peter Bala, Aug 31 2023
G.f.: t*exp(4*t*hypergeom([1, 1, 5/4, 3/2, 7/4], [4/3, 5/3, 2, 2], (256*t)/27))+1. - Karol A. Penson, Dec 20 2023
From Karol A. Penson, Jul 03 2024: (Start)
a(n) = Integral_{x=0..256/27} x^(n)*W(x)dx, n>=0, where W(x) = x^(-3/4) * sqrt(4*R(x) - 3^(3/4)*x^(1/4)/sqrt(R(x)))/(2*3^(1/4)*Pi), with R(x) = ((i + sqrt(3))*(4*sqrt(256 - 27*x) -12*i*sqrt(3*x))^(1/3))/16 - ((i - sqrt(3))*(4*sqrt(256 - 27*x) + 12*i* sqrt(3*x))^(1/3))/16, where i is the imaginary unit.
The elementary function W(x) is positive on the interval x = (0, 256/27) and is equal to the combination of hypergeometric functions in my formula from 2010; see above.
(Pi*W(x))^6 satisfies an algebraic equation of order 6, with integer polynomials as coefficients. (End)
G.f.: (Sum_{n >= 0} binomial(4*n+1, n)*x^n) / (Sum_{n >= 0} binomial(4*n, n)*x^n). - Peter Bala, Dec 14 2024
G.f. A(x) satisfies A(x) = 1/A(-x*A(x)^7). - Seiichi Manyama, Jun 16 2025

A130564 Member k=5 of a family of generalized Catalan numbers.

Original entry on oeis.org

1, 5, 40, 385, 4095, 46376, 548340, 6690585, 83615350, 1064887395, 13770292256, 180320238280, 2386316821325, 31864803599700, 428798445360120, 5809228810425801, 79168272296871450, 1084567603590147950

Offset: 1

Wolfdieter Lang, Jul 13 2007

The generalized Catalan numbers C(k,n):= binomial(k*n+1,n)/(k*n+1) become for negative k=-|k|, with |k|>=2, ((-1)^(n-1))*binomial((|k|+1)*n-2,n)/(|k|*n-1), n>=0.
The family c(k,n):=binomial((k+1)*n-2,n)/(k*n-1), n>=1, has the members A000108, A006013, A006632, A118971 for k=1,2,3,4, respectively (but the offset there is 0).
The members of the C(k,n) family for positive k are: A000012 (powers of 1), A000108, A001764, A002293, A002294, A002295, A002296, A007556, A062994, for k=1..9.
The ordinary generating functions for the k-family {c(k, n+1)}{n>=0}, k >= 1, are G(k, x) = hypergeometric(Aseq(k+1), Bseq(k), ((k+1)^(k+1)/k^k)*x), with Aseq(k+1) = [a(k)_1,..., a(k){k+1}], where a(k)j = (2*k - (j-1))/(k+1), and Bseq(k) = [b(k)_1, ..., b(k)_k], where b(k)_j = (2*k - (j-1))/k. The e.g.f. has an extra 1 in the B-section, which leads to a cancellation with the A-section 1 term. Thanks to Dixon J. Jones for asking for the general formulas. - _Wolfdieter Lang, Feb 04 2024
The o.g.f. of {C(k, n)}{n>=0} is in the Graham-Knuth-Patashnik book denoted as B_k(z) on pp. 200, 349 (2nd ed. 1994, pp. 200, 363). - _Wolfdieter Lang, Mar 07 2024

Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Mathematics, Addison-Wesley, Reading, MA, 2nd ed. 1994, pp. 200, 363.

Michael De Vlieger, Table of n, a(n) for n = 1..856
K. Kobayashi, H. Morita and M. Hoshi, Coding of ordered trees, Proceedings, IEEE International Symposium on Information Theory, ISIT 2000, Sorrento, Italy, Jun 25 2000.
Elżbieta Liszewska, Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.

Cf. A000012, A000108, A001764, A002293, A002294, A002295, A002296, A006013, A062994, A006632, A007556, A118971, A130565, A234466, A234513, A234573, A235340.

Rest@ CoefficientList[InverseSeries[Series[y (1 - y)^5, {y, 0, 18}], x], x] (* Michael De Vlieger, Oct 13 2019 *)

a(n) = binomial((k+1)*n-2,n)/(k*n-1), with k=5.
G.f.: inverse series of y*(1-y)^5.
a(n) = (5/6)*binomial(6*n,n)/(6*n-1). [Bruno Berselli, Jan 17 2014]
From Wolfdieter Lang, Feb 06 2020: (Start)
G.f.: (5/6)*(1 - hypergeom([-1, 1, 2, 3, 4]/6, [1, 2, 3, 4]/5,(6^6/5^5)*x)).
E.g.f.: (5/6)*(1 - hypergeom([-1, 1, 2, 3, 4]/6, [1, 2, 3, 4, 5]/5,(6^6/5^5)*x)). (End)
D-finite with recurrence 5*n*(5*n-4)*(5*n-3)*(5*n-2)*(5*n-1)*a(n) -72*(6*n-7)*(3*n-1)*(2*n-1)*(3*n-2)*(6*n-5)*a(n-1)=0. - R. J. Mathar, May 07 2021

A059968 Number of 10-ary trees.

Original entry on oeis.org

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

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

A143554 G.f. A(x) satisfies A(x) = 1 + xA(x)^5A(-x)^4.

Original entry on oeis.org

1, 1, 1, 5, 9, 55, 117, 775, 1785, 12350, 29799, 211876, 527085, 3818430, 9706503, 71282640, 184138713, 1366368375, 3573805950, 26735839650, 70625252863, 531838637759, 1416298046436, 10723307329700, 28748759731965, 218658647805780, 589546754316126

Offset: 0

Paul D. Hanna, Aug 24 2008

nonn

Number of achiral noncrossing partitions composed of n blocks of size 9. - Andrew Howroyd, Feb 08 2024

			G.f.: A(x) = 1 + x + x^2 + 5*x^3 + 9*x^4 + 55*x^5 + 117*x^6 + 775*x^7 +...
Let G(x) = 1 + x*G(x)^9 be the g.f. of A062994, then
G(x^2) = A(x)*A(-x) and A(x) = G(x^2) + x*G(x^2)^5 where
G(x) = 1 + x + 9*x^2 + 117*x^3 + 1785*x^4 + 29799*x^5 + 527085*x^6 +...
G(x)^5 = 1 + 5*x + 55*x^2 + 775*x^3 + 12350*x^4 + 211876*x^5 +...

Andrew Howroyd, Table of n, a(n) for n = 0..500
Michel Bousquet and Cédric Lamathe, On symmetric structures of order two, Discrete Math. Theor. Comput. Sci. 10 (2008), 153-176. See Table 1. - From _N. J. A. Sloane_, Jul 12 2011

Column k=9 of A369929 and k=10 of A370062.
Cf. A143338, A143546, A143547, A143550, A062994 (bisection).
Cf. A143551, A143552, A143553.
Cf. A143047.

terms = 25;
A[] = 1; Do[A[x] = 1 + x A[x]^5 A[-x]^4 + O[x]^terms // Normal, {terms}];
CoefficientList[A[x], x] (* Jean-François Alcover, Jul 24 2018 *)

{a(n)=my(A=1+x*O(x^n));for(i=0,n,A=1+x*A^5*subst(A^4,x,-x));polcoef(A,n)}

{a(n)=my(m=n\2,p=4*(n%2)+1);binomial(9*m+p-1,m)*p/(8*m+p)}

G.f. satisfies: A(x) = [A(x)*A(-x)] + x*[A(x)*A(-x)]^5.
G.f. satisfies: A(x)*A(-x) = (A(x) + A(-x))/2 = G(x^2) where G(x) = 1 + x*G(x)^9 is the g.f. of A062994.
a(2n) = binomial(9*n,n)/(8*n+1); a(2n+1) = binomial(9*n+4,n)*5/(8*n+5).
a(0) = 1; a(n) = Sum_{i, j, k, l, m>=0 and i+2*j+2*k+2*l+2*m=n-1} a(i) * a(2*j) * a(2*k) * a(2*l) * a(2*m). - Seiichi Manyama, Jul 07 2025
a(0) = 1; a(n) = Sum_{x_1, x_2, ..., x_9>=0 and x_1+x_2+...+x_9=n-1} (-1)^(x_1+x_2+x_3+x_4) * Product_{k=1..9} a(x_k). - Seiichi Manyama, Jul 09 2025

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).

cp's OEIS Frontend

A251579 E.g.f.: exp(9*x*G(x)^8) / G(x)^8 where G(x) = 1 + x*G(x)^9 is the g.f. of A062994.

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A251669 E.g.f.: exp(9*x*G(x)^8) / G(x) where G(x) = 1 + x*G(x)^9 is the g.f. of A062994.

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A002293 Number of dissections of a polygon: binomial(4*n, n)/(3*n + 1).

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A130564 Member k=5 of a family of generalized Catalan numbers.

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A059968 Number of 10-ary trees.

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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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A251579 E.g.f.: exp(9xG(x)^8) / G(x)^8 where G(x) = 1 + x*G(x)^9 is the g.f. of A062994.

A251669 E.g.f.: exp(9xG(x)^8) / G(x) where G(x) = 1 + x*G(x)^9 is the g.f. of A062994.

A002293 Number of dissections of a polygon: binomial(4n, n)/(3n + 1).

A143554 G.f. A(x) satisfies A(x) = 1 + xA(x)^5A(-x)^4.

A232265 a(n) = 10binomial(9n + 10, n)/(9*n + 10).