A002296 -id:A002296 - cp's OEIS frontend

A349292 G.f. A(x) satisfies A(x) = 1 / ((1 - x) * (1 - x * A(x)^6)).

1, 2, 15, 190, 2871, 47643, 838888, 15389452, 290951545, 5629024955, 110908062511, 2217739684483, 44891645810124, 918086053852234, 18941156419798530, 393742848618632760, 8239112912485293357, 173406208518520952066, 3668419587671991125142

Offset: 0

Ilya Gutkovskiy, Nov 13 2021

nonn

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

Cf. A002296, A007317, A199475, A346649, A349289, A349290, A349291, A349293.

nmax = 18; A[] = 0; Do[A[x] = 1/((1 - x) (1 - x A[x]^6)) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[Binomial[n + 5 k, 6 k] Binomial[7 k, k]/(6 k + 1), {k, 0, n}], {n, 0, 18}]

a(n) = Sum_{k=0..n} binomial(n+5*k,6*k) * binomial(7*k,k) / (6*k+1).
a(n) ~ sqrt(1 + 5*r) / (2 * 7^(2/3) * sqrt(3*Pi*(1-r)) * n^(3/2) * r^(n + 1/6)), where r = 0.043408935906208378827553096713877784793679356... is the root of the equation 7^7 * r = 6^6 * (1-r)^6. - Vaclav Kotesovec, Nov 14 2021
a(n) = 1 + Sum_{x_1, x_2, ..., x_7>=0 and x_1+x_2+...+x_7=n-1} Product_{k=1..7} a(x_k). - Seiichi Manyama, Jul 11 2025

A349313 G.f. A(x) satisfies: A(x) = (1 + x * A(x)^7) / (1 - x).

Original entry on oeis.org

1, 2, 16, 212, 3320, 57024, 1038928, 19718512, 385668448, 7718866880, 157326086656, 3254310606208, 68142850580480, 1441588339943168, 30765576147680000, 661561298256228096, 14319744815795062272, 311756656998135770112, 6822215641015820419072

Offset: 0

Ilya Gutkovskiy, Nov 14 2021

nonn

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

Cf. A002296, A006318, A346626, A346649, A349310, A349311, A349312, A349314.

nmax = 18; A[] = 0; Do[A[x] = (1 + x A[x]^7)/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[Binomial[n + 6 k, 7 k] Binomial[7 k, k]/(6 k + 1), {k, 0, n}], {n, 0, 18}]

a(n) = Sum_{k=0..n} binomial(n+6*k,7*k) * binomial(7*k,k) / (6*k+1).
a(n) = F([(1+n)/6, (2+n)/6, (3+n)/6, (4+n)/6, (5+n)/6, 1+n/6, -n], [1/3, 1/2, 2/3, 5/6, 1, 7/6], -1), where F is the generalized hypergeometric function. - Stefano Spezia, Nov 14 2021
a(n) ~ sqrt(1 + 6*r) / (2 * 7^(2/3) * sqrt(3*Pi) * (1-r)^(1/3) * n^(3/2) * r^(n + 1/6)), where r = 0.04196526794785323647696104132939153750367778616407409162... is the real root of the equation 6^6 * (1-r)^7 = 7^7 * r. - Vaclav Kotesovec, Nov 15 2021

A251697 a(n) = (5n+1) (6n+1)^(n-2) 7^n.

Original entry on oeis.org

1, 6, 539, 104272, 31513125, 13018130762, 6835288192159, 4358439870247764, 3271482918202092041, 2826044644022395468750, 2761781119675422226696419, 3012587650584028093856586776, 3628565076873134344787430377389, 4783177086109789054912470697687698, 6849486554475843842876951982177734375

Offset: 0

Paul D. Hanna, Dec 07 2014

nonn

			E.g.f.: A(x) = 1 + 6*x + 539*x^2/2! + 104272*x^3/3! + 31513125*x^4/4! + 13018130762*x^5/5! +...
such that A(x) = exp( 7*x*A(x)^6 * G(x*A(x)^6)^6 ) / G(x*A(x)^6),
where G(x) = 1 + x*G(x)^7 is the g.f. A002296:
G(x) = 1 + x + 7*x^2 + 70*x^3 + 819*x^4 + 10472*x^5 + 141778*x^6 +...
Also, e.g.f. A(x) satisfies A(x) = F(x*A(x)^6) where
F(x) = 1 + 6*x + 107*x^2/2! + 3508*x^3/3! + 171741*x^4/4! + 11280842*x^5/5! +...
F(x) = exp( 7*x*G(x)^6 ) / G(x) is the e.g.f. of A251667.

G. C. Greubel, Table of n, a(n) for n = 0..248

Cf. A251667, A002296.

Cf. Variants: A127670, A251693, A251694, A251695, A251696, A251698, A251699, A251700.

[(5*n + 1)*(6*n + 1)^(n - 2)*7^n: n in [0..50]]; // G. C. Greubel, Nov 14 2017

Table[(5*n + 1)*(6*n + 1)^(n - 2)*7^n, {n, 0, 50}] (* G. C. Greubel, Nov 14 2017 *)

{a(n) = (5*n+1) * (6*n+1)^(n-2) * 7^n}
for(n=0,20,print1(a(n),", "))

{a(n)=local(G=1,A=1); for(i=0, n, G = 1 + x*G^7 +x*O(x^n));
A = ( serreverse( x*G^6 / exp(42*x*G^6) )/x )^(1/6); 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)^6 * G(x*A(x)^6)^6 ) / G(x*A(x)^6).
(2) A(x) = F(x*A(x)^6) where F(x) = exp(7*x*G(x)^6)/G(x) is the e.g.f. of A251667.
(3) A(x) = ( Series_Reversion( x*G(x)^6 / exp(42*x*G(x)^6) )/x )^(1/6).
E.g.f.: (-LambertW(-42*x)/(42*x))^(1/6) * (1 + LambertW(-42*x)/42). - Vaclav Kotesovec, Dec 07 2014

A346667 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * binomial(7k,k) / (6k + 1).

Original entry on oeis.org

1, 0, 6, 51, 578, 7011, 89931, 1198798, 16445122, 230643888, 3292247673, 47672499727, 698569117499, 10339672571689, 154357100458366, 2321475460350492, 35140713973159266, 534971413383669580, 8185501429052369700, 125811555778930237392, 1941590759206061655069

Offset: 0

Ilya Gutkovskiy, Jul 27 2021

nonn

Inverse binomial transform of A002296.

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

Cf. A002296, A005043, A346628, A346649, A346664, A346665, A346666, A346668.

Table[Sum[(-1)^(n - k) Binomial[n, k] Binomial[7 k, k]/(6 k + 1), {k, 0, n}], {n, 0, 20}]
nmax = 20; 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 = 20; CoefficientList[Series[Sum[(Binomial[7 k, k]/(6 k + 1)) x^k/(1 + x)^(k + 1), {k, 0, nmax}], {x, 0, nmax}], x]
Table[(-1)^n 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, 20}]

a(n) = sum(k=0, n, (-1)^(n-k)*binomial(n,k)*binomial(7*k,k)/(6*k + 1)); \\ Michel Marcus, Jul 28 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) ~ 776887^(n + 3/2) / (282475249 * sqrt(Pi) * n^(3/2) * 2^(6*n + 2) * 3^(6*n + 3/2)). - Vaclav Kotesovec, Jul 30 2021

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

Original entry on oeis.org

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

Offset: 0

Peter Luschny, Jun 26 2022

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

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

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

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

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

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

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

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

A233832 a(n) = 2binomial(7n+2,n)/(7*n+2).

Original entry on oeis.org

1, 2, 15, 154, 1827, 23562, 320866, 4540200, 66096459, 983592304, 14894775896, 228784720710, 3555866673450, 55819631671902, 883738853546472, 14094715154157680, 226245021605612955, 3652242142988400570, 59254515909624764575, 965678197027521177200

Offset: 0

Tim Fulford, Dec 16 2013

nonn

Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), this is the case p=7, r=2.

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
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.
Wojciech Mlotkowski, Fuss-Catalan Numbers in Noncommutative Probability, Docum. Mathm. 15: 939-955.
Wikipedia, Fuss-Catalan number
Jun Yan, Lattice paths enumerations weighted by ascent lengths, arXiv:2501.01152 [math.CO], 2025. See p. 7.
Sheng-liang Yang and Mei-yang Jiang, Pattern avoiding problems on the hybrid d-trees, J. Lanzhou Univ. Tech., (China, 2023) Vol. 49, No. 2, 144-150. (in Mandarin)

Cf. A000108, A002296, A233833 - A233835, A143547, A130565, A233907, A233908.

[2*Binomial(7*n+2, n)/(7*n+2): n in [0..30]];

Table[2 Binomial[7 n + 2, n]/(7 n + 2), {n, 0, 30}]

PARI
```
a(n) = 2*binomial(7*n+2,n)/(7*n+2);
```

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(7/2))^2+x*O(x^n)); polcoeff(B, n)}

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, where p=7, r=2.
a(n) = 2*binomial(7n+1, n-1)/n for n>0, a(0)=1. [Bruno Berselli, Jan 19 2014]
From Ilya Gutkovskiy, Sep 14 2018: (Start)
E.g.f.: 6F6(2/7,3/7,4/7,5/7,6/7,8/7; 1/2,2/3,5/6,1,7/6,4/3; 823543*x/46656).
a(n) ~ 7^(7*n+3/2)/(sqrt(Pi)*3^(6*n+5/2)*4^(3*n+1)*n^(3/2)). (End)

A349334 G.f. A(x) satisfies A(x) = 1 + x * A(x)^7 / (1 - x).

Original entry on oeis.org

1, 1, 8, 85, 1051, 14197, 203064, 3022909, 46347534, 726894786, 11606936525, 188060979332, 3084087347910, 51094209834068, 853859480938095, 14376597494941454, 243649099741045190, 4153091242153905838, 71152973167920086796, 1224593757045581062444

Offset: 0

Ilya Gutkovskiy, Nov 15 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..791

Cf. A002212, A307678, A349331, A349332, A349333, A349335.

Cf. A002296, A346649 (partial sums), A349363.

a:= n-> coeff(series(RootOf(1+x*A^7/(1-x)-A, A), x, n+1), x, n):
seq(a(n), n=0..20);  # Alois P. Heinz, Nov 15 2021

nmax = 19; A[] = 0; Do[A[x] = 1 + x A[x]^7/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[Binomial[n - 1, k - 1] Binomial[7 k, k]/(6 k + 1), {k, 0, n}], {n, 0, 19}]

{a(n) = my(A=[1]); for(m=1, n, A=concat(A, 0);
A[#A] = 1 + sum(k=1, m-1, (polcoeff(Ser(A)^7, k)) )); A[n+1]}
for(n=0, 30, print1(a(n), ", ")) \\ Vaclav Kotesovec, Nov 23 2024, after Paul D. Hanna

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

a(n) ~ 870199^(n + 1/2) / (343 * sqrt(Pi) * n^(3/2) * 2^(6*n + 2) * 3^(6*n + 3/2)). - Vaclav Kotesovec, Nov 15 2021

A346671 a(n) = Sum_{k=0..n} binomial(7k,k) / (6k + 1).

Original entry on oeis.org

1, 2, 9, 79, 898, 11370, 153148, 2150836, 31140511, 461462144, 6964815000, 106691488130, 1654539334220, 25923944408960, 409770113121064, 6526344613981944, 104632592920840659, 1687270854882480906, 27348675382672733281, 445328790513987869681, 7281393330439106226281

Offset: 0

Ilya Gutkovskiy, Jul 28 2021

nonn

Partial sums of A002296.

Seiichi Manyama, Table of n, a(n) for n = 0..806

Cf. A002296, A014137, A104859, A345367, A345368, A346065, A346649, A346672.

Table[Sum[Binomial[7 k, k]/(6 k + 1), {k, 0, n}], {n, 0, 20}]
nmax = 20; A[] = 0; Do[A[x] = 1/(1 - x) + x (1 - x)^6 A[x]^7 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

a(n) = sum(k=0, n, binomial(7*k, k)/(6*k+1)); \\ Michel Marcus, Jul 28 2021

G.f. A(x) satisfies: A(x) = 1 / (1 - x) + x * (1 - x)^6 * A(x)^7.

a(n) ~ 7^(7*n + 15/2) / (776887 * sqrt(Pi) * n^(3/2) * 2^(6*n + 2) * 3^(6*n + 3/2)). - Vaclav Kotesovec, Jul 30 2021

A079563 a(n) = a(n,m) = Sum_{k=0..n} binomial(mk,k)binomial(m*(n-k),n-k) for m=7.

Original entry on oeis.org

1, 14, 231, 3934, 67851, 1177974, 20531770, 358788696, 6281076123, 110103674128, 1931983053056, 33926800240578, 596145343139514, 10480467311987778, 184327560283768776, 3243034966775972144, 57074433199551436347

Offset: 0

Benoit Cloitre, Jan 26 2003

nonn

More generally, for m>=2, a(n,m) = Sum_{k=0..n} binomial(m*k,k)*binomial(m*(n-k),n-k) is asymptotic to 1/2*m/(m-1)*(m^m/(m-1)^(m-1))^n * (1 + (2*m-4)/(3*sqrt(Pi*n*m*(m-1)/2))), extended by Vaclav Kotesovec, May 25 2020

See A000302, A006256, A078995 for cases m=2,3 and 4.

Seiichi Manyama, Table of n, a(n) for n = 0..802
Rui Duarte and António Guedes de Oliveira, Short note on the convolution of binomial coefficients, arXiv:1302.2100 [math.CO], 2013.
D. Merlini, R. Sprugnoli, and M. C. Verri, The tennis ball problem, J. Combin. Theory, A 99 (2002), 307-344.

Cf. A006256, A078995, A079678, A079679.

Cf. A002296.

a(n) = (7/12)*(823543/46656)^n*(1+c/sqrt(n)+o(n^-1/2)) where c=0.41...
c = 10/(3*sqrt(21*Pi)) = 0.410387535383... - Vaclav Kotesovec, May 25 2020
From Rui Duarte and António Guedes de Oliveira, Feb 17 2013: (Start)
a(n) = Sum_{k=0..n} binomial(7*k+x,k)*binomial(7*(n-k)-x,n-k) for any real x.
a(n) = Sum_{k=0..n} 6^(n-k)*binomial(7*n+1,k).
a(n) = Sum_{k=0..n} 7^(n-k)*binomial(6*n+k,k). (End)
a(n) = [x^n] 1/((1-7*x) * (1-x)^(6*n+1)). - Seiichi Manyama, Aug 03 2025
From Seiichi Manyama, Aug 14 2025: (Start)
a(n) = Sum_{k=0..n} 7^k * (-6)^(n-k) * binomial(7*n+1,k) * binomial(7*n-k,n-k).
G.f.: g^2/(7-6*g)^2 where g = 1+x*g^7 is the g.f. of A002296. (End)

A233833 a(n) = 3binomial(7n+3, n)/(7*n+3).

Original entry on oeis.org

1, 3, 24, 253, 3045, 39627, 543004, 7718340, 112752783, 1682460520, 25533901536, 392912889915, 6116090678334, 96133810101609, 1523687678528400, 24324750346691480, 390786855500604195, 6313161418594235271, 102494297789621214400, 1671366110239940499000

Offset: 0

Tim Fulford, Dec 16 2013

nonn

Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), this is the case p=7, r=3.

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
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.
Wojciech Mlotkowski, Fuss-Catalan Numbers in Noncommutative Probability, Docum. Mathm. 15: 939-955.

Cf. A000108, A002296, A233832, A143547, A233834, A130565, A233835, A233907, A233908.

[3*Binomial(7*n+3, n)/(7*n+3): n in [0..30]];

Table[3 Binomial[7 n + 3, n]/(7 n + 3), {n, 0, 30}]

PARI
```
a(n)=3*binomial(7*n+3,n)/(7*n+3);
```

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(7/3))^3+x*O(x^n)); polcoeff(B, n)}

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, where p=7, r=3.
From Ilya Gutkovskiy, Sep 14 2018: (Start)
E.g.f.: 6F6(3/7,4/7,5/7,6/7,8/7,9/7; 2/3,5/6,1,7/6,4/3,3/2; 823543*x/46656).
a(n) ~ 7^(7*n+5/2)/(sqrt(Pi)*3^(6*n+5/2)*4^(3*n+2)*n^(3/2)). (End)

cp's OEIS Frontend

A349292 G.f. A(x) satisfies A(x) = 1 / ((1 - x) * (1 - x * A(x)^6)).

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A349313 G.f. A(x) satisfies: A(x) = (1 + x * A(x)^7) / (1 - x).

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Mathematica

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A251697 a(n) = (5*n+1) * (6*n+1)^(n-2) * 7^n.

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Magma

Mathematica

PARI

PARI

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A346667 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * binomial(7*k,k) / (6*k + 1).

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Mathematica

PARI

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

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Maple

Mathematica

Formula

A233832 a(n) = 2*binomial(7*n+2,n)/(7*n+2).

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PARI

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A349334 G.f. A(x) satisfies A(x) = 1 + x * A(x)^7 / (1 - x).

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Maple

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PARI

Formula

A251697 a(n) = (5n+1) (6n+1)^(n-2) 7^n.

A346667 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * binomial(7k,k) / (6k + 1).

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

A233832 a(n) = 2binomial(7n+2,n)/(7*n+2).

A346671 a(n) = Sum_{k=0..n} binomial(7k,k) / (6k + 1).

A079563 a(n) = a(n,m) = Sum_{k=0..n} binomial(mk,k)binomial(m*(n-k),n-k) for m=7.

A233833 a(n) = 3binomial(7n+3, n)/(7*n+3).