A002296 -id:A002296 - cp's OEIS frontend

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.

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).

A258708 Triangle read by rows: T(i,j) = integer part of binomial(i+j, i-j)/(2*j+1) for i >= 1 and j = 0..i-1.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 5, 7, 4, 1, 1, 7, 14, 12, 5, 1, 1, 9, 25, 30, 18, 6, 1, 1, 12, 42, 66, 55, 26, 7, 1, 1, 15, 66, 132, 143, 91, 35, 8, 1, 1, 18, 99, 245, 334, 273, 140, 45, 9, 1, 1, 22, 143, 429, 715, 728, 476, 204, 57, 10, 1

Offset: 1

N. J. A. Sloane, Jun 12 2015

In the Loh-Shannon-Horadam paper, Table 3 contains a typo (see Extensions lines).
T(n,k) = round(A258993(n,k)/(2*k+1)). - Reinhard Zumkeller, Jun 22 2015
From Reinhard Zumkeller, Jun 23 2015: (Start)
(using tables 4 and 5 of the Loh-Shannon-Horadam paper, p. 8f).
T(n, n-1)  = 1;
T(n, n-2)  = n for n > 1;
T(n, n-3)  = A000969(n-3) for n > 2;
T(n, n-4)  = A000330(n-3) for n > 3;
T(n, n-5)  = T(2*n-7, 2) = A000970(n) for n > 4;
T(n, n-6)  = A000971(n) for n > 5;
T(n, n-7)  = A000972(n) for n > 6;
T(n, n-8)  = A000973(n) for n > 7;
T(n, 1)    = A001840(n-1) for n > 1;
T(2*n, n)  = A001764(n);
T(3*n-1, 1) = A000326(n);
T(3*n, 2*n) = A002294(n);
T(4*n, 3*n) = A002296(n). (End)

			Triangle T(i, j) (with rows i >= 1 and columns j >= 0) begins as follows:
  1;
  1,  1;
  1,  2,  1;
  1,  3,  3,   1;
  1,  5,  7,   4,   1;
  1,  7, 14,  12,   5,   1;
  1,  9, 25,  30,  18,   6,   1;
  1, 12, 42,  66,  55,  26,   7,  1;
  1, 15, 66, 132, 143,  91,  35,  8, 1;
  1, 18, 99, 245, 334, 273, 140, 45, 9, 1;
  ...

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened
R. P. Loh, A. G. Shannon, and A. F. Horadam, Divisibility Criteria and Sequence Generators Associated with Fermat Coefficients, Preprint, 1980.

Cf. A011793.
Cf. A007318, A258993, A158405, A005408, A006013 (central terms).
Cf. A000326, A000330, A000969, A000970, A000971, A000972, A000973, A000976, A001764, A001840, A002294, A002296.

a258708 n k = a258708_tabl !! (n-1) !! k
a258708_row n = a258708_tabl !! (n-1)
a258708_tabl = zipWith (zipWith ((round .) . ((/) `on` fromIntegral)))
                       a258993_tabl a158405_tabl
-- Reinhard Zumkeller, Jun 22 2015, Jun 16 2015

Corrected T(8,5) = 26 from Reinhard Zumkeller, Jun 13 2015

A143547 G.f. A(x) satisfies A(x) = 1 + xA(x)^4A(-x)^3.

Original entry on oeis.org

1, 1, 1, 4, 7, 34, 70, 368, 819, 4495, 10472, 59052, 141778, 814506, 1997688, 11633440, 28989675, 170574723, 430321633, 2552698720, 6503352856, 38832808586, 99726673130, 598724403680, 1547847846090, 9335085772194, 24269405074740, 146936230074004, 383846168712104

Offset: 0

Paul D. Hanna, Aug 23 2008

nonn

Number of achiral noncrossing partitions composed of n blocks of size 7. - Andrew Howroyd, Feb 08 2024

			G.f.: A(x) = 1 + x + x^2 + 4*x^3 + 7*x^4 + 34*x^5 + 70*x^6 + 368*x^7 + ...
Let G(x) = 1 + x*G(x)^7 be the g.f. of A002296, then
A(x)*A(-x) = G(x^2) and A(x) = G(x^2) + x*G(x^2)^4 where
G(x) = 1 + x + 7*x^2 + 70*x^3 + 819*x^4 + 10472*x^5 + 141778*x^6 + ...
G(x)^4 = 1 + 4*x + 34*x^2 + 368*x^3 + 4495*x^4 + 59052*x^5 + ...
form the bisections of A(x).
By definition, A(x) = 1 + x*A(x)^4*A(-x)^3 where
A(x)^4 = 1 + 4*x + 10*x^2 + 32*x^3 + 95*x^4 + 332*x^5 + 1074*x^6 + ...
A(-x)^3 = 1 - 3*x + 6*x^2 - 19*x^3 + 51*x^4 - 183*x^5 + 550*x^6 -+ ...

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=7 of A369929 and k=8 of A370062.
Cf. A002296 (bisection), A143546.
Cf. A143549, A143550.
Cf. A143046, A143553.

terms = 26;
A[] = 1; Do[A[x] = 1 + x A[x]^4 A[-x]^3 + O[x]^terms // Normal, {terms}];
CoefficientList[A[x], x] (* Jean-François Alcover, Jul 24 2018 *)

{a(n)=my(A=1+O(x^(n+1)));for(i=0,n,A=1+x*A^4*subst(A^3,x,-x));polcoef(A,n)}

{a(n)=my(m=n\2,p=3*(n%2)+1);binomial(7*m+p-1,m)*p/(6*m+p)}

G.f.: A(x) = G(x^2) + x*G(x^2)^4 where G(x^2) = A(x)*A(-x) and G(x) = 1 + x*G(x)^7 is the g.f. of A002296.
a(2n) = binomial(7*n,n)/(6*n+1); a(2n+1) = binomial(7*n+3,n)*4/(6*n+4).
G.f. satisfies: A(x)*A(-x) = (A(x) + A(-x))/2.
a(0) = 1; a(n) = Sum_{i, j, k, l>=0 and i+2*j+2*k+2*l=n-1} a(i) * a(2*j) * a(2*k) * a(2*l). - Seiichi Manyama, Jul 07 2025
a(0) = 1; a(n) = Sum_{x_1, x_2, ..., x_7>=0 and x_1+x_2+...+x_7=n-1} (-1)^(x_1+x_2+x_3) * Product_{k=1..7} a(x_k). - Seiichi Manyama, Jul 08 2025

a(26) onwards from Andrew Howroyd, Feb 08 2024

A004368 Binomial coefficient C(7n,n).

Original entry on oeis.org

1, 7, 91, 1330, 20475, 324632, 5245786, 85900584, 1420494075, 23667689815, 396704524216, 6681687099710, 112992892764570, 1917283000904460, 32626924340528840, 556608279578340080, 9516306085765295355, 163011740982048945441

Offset: 0

N. J. A. Sloane

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), A004355 (k = 6), A004381 (k = 8), A169958 - A169961 (k = 9 thru 12).

Cf. A002296.

[Binomial(7*n,n): n in [0..20]]; // Vincenzo Librandi, Oct 06 2015

Table[Binomial[7n,n],{n,0,20}] (* Harvey P. Dale, Apr 05 2014 *)

B(x):=sum(binomial(7*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(7*n,n) \\ Altug Alkan, Oct 05 2015

a(n) = C(7*n-1,n-1)*C(49*n^2,2)/(3*n*C(7*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 A002296. - Vladimir Kruchinin, Oct 05 2015
From Ilya Gutkovskiy, Jan 16 2017: (Start)
O.g.f.: 6F5(1/7,2/7,3/7,4/7,5/7,6/7; 1/6,1/3,1/2,2/3,5/6; 823543*x/46656).
E.g.f.: 6F6(1/7,2/7,3/7,4/7,5/7,6/7; 1/6,1/3,1/2,2/3,5/6,1; 823543*x/46656).
a(n) ~ sqrt(7/3)*7^(7*n)/(2*sqrt(Pi*n)*6^(6*n)). (End)
From Peter Bala, Feb 20 2022: (Start)
The o.g.f. A(x) is algebraic: (1 - A(x))*(1 + 6*A(x))^6 +  (7^7)*x*A(x)^7 = 0.
Sum_{n >= 1} a(n)*( x*(6*x + 7)^6/(7^7*(1 + x)^7) )^n = x. (End)

A233835 a(n) = 8binomial(7n + 8, n)/(7*n + 8).

Original entry on oeis.org

1, 8, 84, 1008, 13090, 179088, 2542512, 37106784, 553270671, 8391423040, 129058047580, 2008018827360, 31550226597162, 499892684834368, 7978140653296800, 128138773298754240, 2069603881026760323, 33593111381834512200, 547698081896206040800, 8965330544164089648000, 147285313888568167177866

Offset: 0

Tim Fulford, Dec 16 2013

Fuss-Catalan sequence is a(n,p,r) = r*binomial(n*p + r, n)/(n*p + r); this is the case p = 7, r = 8.

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, A002296, A233832, A233833, A143547, A233834, A233835, A233907, A233908.

Cf. A000245 (k = 3), A006629 (k = 4), A196678 (k = 5), A233668 (k = 6), A233743 (k = 7), A234467 (k = 9), A232265 (k = 10), A229963 (k = 11).

[8*Binomial(7*n+8, n)/(7*n+8): n in [0..30]];

Table[8 Binomial[7 n + 8, n]/(7 n + 8), {n, 0, 30}]

PARI
```
a(n) = 8*binomial(7*n+8,n)/(7*n+8);
```

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(7/8))^8+x*O(x^n)); polcoeff(B, n)}

G.f. satisfies: A(x) = {1 + x*A(x)^(p/r)}^r, where p = 7, r = 8.
From _Peter Bala, Oct 16 2015: (Start)
O.g.f. A(x) = 1/x * series reversion (x*C(-x)^8), 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)^(1/8) is the o.g.f. for A002296. (End)

A251587 a(n) = 7^(n-5) * (n+1)^(n-7) * (1296n^5 + 9720n^4 + 30555n^3 + 50665n^2 + 44621*n + 16807).

Original entry on oeis.org

1, 1, 9, 202, 7745, 429192, 31383169, 2862173104, 313456891041, 40120056928000, 5880757402778489, 971776774765633536, 178783183332534538849, 36248462186643418851328, 8031317081954231731640625, 1930762515644331053063077888, 500587184017640118192794723009, 139240118930461640299714951839744

Offset: 0

Paul D. Hanna, Dec 06 2014

			E.g.f.: A(x) = 1 + x + 9*x^2/2! + 202*x^3/3! + 7745*x^4/4! + 429192*x^5/5! +...
such that A(x) = exp( 7*x*A(x) * G(x*A(x))^6 ) / G(x*A(x))^6
where G(x) = 1 + x*G(x)^5 is the g.f. of A002296:
G(x) = 1 + x + 7*x^2 + 70*x^3 + 819*x^4 + 10472*x^5 + 141778*x^6 +...
RELATED SERIES.
Note that A(x) = F(x*A(x)) where F(x) = exp(7*x*G(x)^6)/G(x)^6,
F(x) = 1 + x + 7*x^2/2! + 133*x^3/3! + 4501*x^4/4! + 224497*x^5/5! +...
is the e.g.f. of A251577.

G. C. Greubel, Table of n, a(n) for n = 0..300

Cf. A251577, A002296.

Cf. Variants: A251583, A251584, A251585, A251586, A251588, A251589, A251590.

Table[7^(n - 5)*(n + 1)^(n - 7)*(1296*n^5 + 9720*n^4 + 30555*n^3 + 50665*n^2 + 44621*n + 16807), {n, 0, 20}] (* G. C. Greubel, Nov 10 2017 *)

{a(n) = 7^(n-5) * (n+1)^(n-7) * (1296*n^5 + 9720*n^4 + 30555*n^3 + 50665*n^2 + 44621*n + 16807)}
for(n=0,20,print1(a(n),", "))

{a(n) = local(G=1,A=1); for(i=1,n, G=1+x*G^7 +x*O(x^n));
for(i=1,n, A = exp(7*x*A * subst(G^6,x,x*A) ) / subst(G^6,x,x*A) ); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))

Let G(x) = 1 + x*G(x)^7 be the g.f. of A002296, then the e.g.f. A(x) of this sequence satisfies:
(1) A(x) = exp( 7*x*A(x) * G(x*A(x))^6 ) / G(x*A(x))^6.
(2) A(x) = F(x*A(x)) where F(x) = exp(7*x*G(x)^6)/G(x)^6 is the e.g.f. of A251577.
(3) a(n) = [x^n/n!] F(x)^(n+1)/(n+1) where F(x) is the e.g.f. of A251577.
E.g.f.: -LambertW(-7*x) * (7 + LambertW(-7*x))^6 / (x*7^7). - Vaclav Kotesovec, Dec 07 2014

A346649 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(7k,k) / (6k + 1).

Original entry on oeis.org

1, 2, 10, 95, 1146, 15343, 218407, 3241316, 49588850, 776483636, 12383420161, 200444399493, 3284531747403, 54378741581471, 908238222519566, 15284835717461020, 258933935458506210, 4412025177612412048, 75564998345532498844, 1300158755391113561288

Offset: 0

Ilya Gutkovskiy, Jul 26 2021

nonn

Binomial transform of A002296.

Seiichi Manyama, Table of n, a(n) for n = 0..500

Cf. A002296, A007317, A188687, A346646, A346647, A346648, A346650.

Table[Sum[Binomial[n, k] Binomial[7 k, k]/(6 k + 1), {k, 0, n}], {n, 0, 19}]
nmax = 19; A[] = 0; Do[A[x] = 1/(1 - x) + x (1 - x)^5 A[x]^7 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
nmax = 19; CoefficientList[Series[Sum[(Binomial[7 k, k]/(6 k + 1)) x^k/(1 - x)^(k + 1), {k, 0, nmax}], {x, 0, nmax}], x]
Table[HypergeometricPFQ[{1/7, 2/7, 3/7, 4/7, 5/7, 6/7, -n}, {1/3, 1/2, 2/3, 5/6, 1, 7/6}, -823543/46656], {n, 0, 19}]

a(n) = sum(k=0, n, binomial(n,k)*binomial(7*k,k)/(6*k + 1)); \\ Michel Marcus, Jul 26 2021

G.f. A(x) satisfies: A(x) = 1 / (1 - x) + x * (1 - x)^5 * A(x)^7.
G.f.: Sum_{k>=0} ( binomial(7*k,k) / (6*k + 1) ) * x^k / (1 - x)^(k+1).
a(n) ~ 870199^(n + 3/2) / (282475249 * sqrt(Pi) * n^(3/2) * 2^(6*n + 2) * 3^(6*n + 3/2)). - Vaclav Kotesovec, Jul 30 2021

A233834 a(n) = 5binomial(7n+5,n)/(7*n+5).

Original entry on oeis.org

1, 5, 45, 500, 6200, 82251, 1142295, 16398200, 241379325, 3623534200, 55262073757, 853814730600, 13335836817420, 210225027967325, 3340362288091500, 53443628421286320, 860246972339613855, 13921016318025200505, 226352372251889455000, 3696160728052814340000

Offset: 0

Tim Fulford, Dec 16 2013

Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), this is the case p = 7, r = 5.

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. A000108, A002296, A233832, A233833, A143547, A130565, A233835, A233907, A233908, A002296, A069271, A118970, A212073, A234465, A234510, A234571, A235339.

[5*Binomial(7*n+5, n)/(7*n+5): n in [0..30]];

Table[5 Binomial[7 n + 5, n]/(7 n + 5), {n, 0, 30}]

PARI
```
a(n) = 5*binomial(7*n+5,n)/(7*n+5);
```

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(7/5))^5+x*O(x^n)); polcoeff(B, n)}

G.f. satisfies: A(x) = {1 + x*A(x)^(p/r)}^r, where p = 7, r = 5.

O.g.f. A(x) = 1/x * series reversion (x/C(x)^5), where C(x) is the o.g.f. for the Catalan numbers A000108. A(x)^(1/5) is the o.g.f. for A002296. - Peter Bala, Oct 14 2015

cp's OEIS Frontend

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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A258708 Triangle read by rows: T(i,j) = integer part of binomial(i+j, i-j)/(2*j+1) for i >= 1 and j = 0..i-1.

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A143547 G.f. A(x) satisfies A(x) = 1 + x*A(x)^4*A(-x)^3.

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A004368 Binomial coefficient C(7n,n).

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Magma

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Maxima

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A233835 a(n) = 8*binomial(7*n + 8, n)/(7*n + 8).

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A143547 G.f. A(x) satisfies A(x) = 1 + xA(x)^4A(-x)^3.

A233835 a(n) = 8binomial(7n + 8, n)/(7*n + 8).

A251587 a(n) = 7^(n-5) * (n+1)^(n-7) * (1296n^5 + 9720n^4 + 30555n^3 + 50665n^2 + 44621*n + 16807).

A346649 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(7k,k) / (6k + 1).

A233834 a(n) = 5binomial(7n+5,n)/(7*n+5).