A002295 -id:A002295 - cp's OEIS frontend

A251586 a(n) = 6^(n-4) * (n+1)^(n-6) * (125n^4 + 810n^3 + 2095n^2 + 2586n + 1296).

1, 1, 8, 156, 5160, 245976, 15450912, 1209613824, 113666333184, 12479546880000, 1568823886181376, 222308476014034944, 35069155573323036672, 6096327654732137496576, 1158040133351856000000000, 238674982804212474577944576, 53050036437721656891731017728, 12649916782354997981599305302016

Offset: 0

Paul D. Hanna, Dec 06 2014

nonn

			E.g.f.: A(x) = 1 + x + 8*x^2/2! + 156*x^3/3! + 5160*x^4/4! + 245976*x^5/5! +...
such that A(x) = exp( 6*x*A(x) * G(x*A(x))^5 ) / G(x*A(x))^5
where G(x) = 1 + x*G(x)^6 is the g.f. of A002295:
G(x) = 1 + x + 6*x^2 + 51*x^3 + 506*x^4 + 5481*x^5 + 62832*x^6 +...
RELATED SERIES.
Note that A(x) = F(x*A(x)) where F(x) = exp(6*x*G(x)^5)/G(x)^5,
F(x) = 1 + x + 6*x^2/2! + 96*x^3/3! + 2736*x^4/4! + 115056*x^5/5! +...
is the e.g.f. of A251576.

G. C. Greubel, Table of n, a(n) for n = 0..306

Cf. A251576, A002295.

Cf. Variants: A251583, A251584, A251585, A251587, A251588, A251589, A251590.

[6^(n - 4)*(n + 1)^(n - 6)*(125*n^4 + 810*n^3 + 2095*n^2 + 2586*n + 1296): n in [0..50]]; // G. C. Greubel, Nov 13 2017

Table[6^(n - 4)*(n + 1)^(n - 6)*(125*n^4 + 810*n^3 + 2095*n^2 + 2586*n + 1296), {n, 0, 50}] (* G. C. Greubel, Nov 13 2017 *)

{a(n) = 6^(n-4) * (n+1)^(n-6) * (125*n^4 + 810*n^3 + 2095*n^2 + 2586*n + 1296)}
for(n=0,20,print1(a(n),", "))

{a(n) = local(G=1,A=1); for(i=1,n, G=1+x*G^6 +x*O(x^n));
for(i=1,n, A = exp(6*x*A * subst(G^5,x,x*A) ) / subst(G^5,x,x*A) ); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))

Let G(x) = 1 + x*G(x)^6 be the g.f. of A002295, then the e.g.f. A(x) of this sequence satisfies:
(1) A(x) = exp( 6*x*A(x) * G(x*A(x))^5 ) / G(x*A(x))^5.
(2) A(x) = F(x*A(x)) where F(x) = exp(6*x*G(x)^5)/G(x)^5 is the e.g.f. of A251576.
(3) a(n) = [x^n/n!] F(x)^(n+1)/(n+1) where F(x) is the e.g.f. of A251576.
E.g.f.: -LambertW(-6*x) * (6 + LambertW(-6*x))^5 / (x*6^6). - Vaclav Kotesovec, Dec 07 2014

A251696 a(n) = (4n+1) (5n+1)^(n-2) 6^n.

Original entry on oeis.org

1, 5, 324, 44928, 9716112, 2870090496, 1077194894400, 490873123897344, 263285585800098048, 162505400851637010432, 113463916253636561519616, 88423664876285081860177920, 76086820231309990402228260864, 71651521268311905104861664903168, 73298071049899905319337719679434752

Offset: 0

Paul D. Hanna, Dec 07 2014

nonn

			E.g.f.: A(x) = 1 + 5*x + 324*x^2/2! + 44928*x^3/3! + 9716112*x^4/4! + 2870090496*x^5/5! +...
such that A(x) = exp( 6*x*A(x)^5 * G(x*A(x)^5)^5 ) / G(x*A(x)^5),
where G(x) = 1 + x*G(x)^6 is the g.f. A002295:
G(x) = 1 + x + 6*x^2 + 51*x^3 + 506*x^4 + 5481*x^5 + 62832*x^6 +...
Also, e.g.f. A(x) satisfies A(x) = F(x*A(x)^5) where
F(x) = 1 + 5*x + 74*x^2/2! + 2028*x^3/3! + 83352*x^4/4! + 4607496*x^5/5! +...
F(x) = exp( 6*x*G(x)^5 ) / G(x) is the e.g.f. of A251666.

G. C. Greubel, Table of n, a(n) for n = 0..256

Cf. A251666, A002295.

Cf. Variants: A127670, A251693, A251694, A251695, A251697, A251698, A251699, A251700.

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

Table[(4*n + 1)*(5*n + 1)^(n - 2)*6^n, {n, 0, 50}] (* G. C. Greubel, Nov 14 2017 *)

{a(n) = (4*n+1) * (5*n+1)^(n-2) * 6^n}
for(n=0,20,print1(a(n),", "))

{a(n)=local(G=1,A=1); for(i=0, n, G = 1 + x*G^6 +x*O(x^n));
A = ( serreverse( x*G^5 / exp(30*x*G^5) )/x )^(1/5); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))

Let G(x) = 1 + x*G(x)^6 be the g.f. of A002295, then the e.g.f. A(x) of this sequence satisfies:
(1) A(x) = exp( 6*x*A(x)^5 * G(x*A(x)^5)^5 ) / G(x*A(x)^5).
(2) A(x) = F(x*A(x)^5) where F(x) = exp(6*x*G(x)^5)/G(x) is the e.g.f. of A251666.
(3) A(x) = ( Series_Reversion( x*G(x)^5 / exp(30*x*G(x)^5) )/x )^(1/5).
E.g.f.: (-LambertW(-30*x)/(30*x))^(1/5) * (1 + LambertW(-30*x)/30). - Vaclav Kotesovec, Dec 07 2014

A346666 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * binomial(6k,k) / (5k + 1).

Original entry on oeis.org

1, 0, 5, 35, 335, 3405, 36601, 408630, 4693535, 55105970, 658390845, 7979041735, 97847884981, 1211946011450, 15139726594915, 190526268260405, 2413170608875655, 30738613968350640, 393519782671609951, 5060600804169151680, 65342131689498876095, 846781225288921612940

Offset: 0

Ilya Gutkovskiy, Jul 27 2021

nonn

Inverse binomial transform of A002295.

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

Cf. A002295, A005043, A346628, A346648, A346664, A346665, A346667, A346668.

Table[Sum[(-1)^(n - k) Binomial[n, k] Binomial[6 k, k]/(5 k + 1), {k, 0, n}], {n, 0, 21}]
nmax = 21; A[] = 0; Do[A[x] = 1/(1 + x) + x (1 + x)^4 A[x]^6 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
nmax = 21; CoefficientList[Series[Sum[(Binomial[6 k, k]/(5 k + 1)) x^k/(1 + x)^(k + 1), {k, 0, nmax}], {x, 0, nmax}], x]
Table[(-1)^n HypergeometricPFQ[{1/6, 1/3, 1/2, 2/3, 5/6, -n}, {2/5, 3/5, 4/5, 1, 6/5}, 46656/3125], {n, 0, 21}]

a(n) = sum(k=0, n, (-1)^(n-k)*binomial(n,k)*binomial(6*k,k)/(5*k + 1)); \\ Michel Marcus, Jul 28 2021

G.f. A(x) satisfies: A(x) = 1 / (1 + x) + x * (1 + x)^4 * A(x)^6.
G.f.: Sum_{k>=0} ( binomial(6*k,k) / (5*k + 1) ) * x^k / (1 + x)^(k+1).
a(n) ~ 43531^(n + 3/2) / (3359232 * sqrt(3*Pi) * n^(3/2) * 5^(5*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).

A365189 G.f. satisfies A(x) = 1 + xA(x)^5(1 + x*A(x)^5).

Original entry on oeis.org

1, 1, 6, 50, 485, 5130, 57391, 667777, 7999095, 97986680, 1221813880, 15456556791, 197887386913, 2559189842240, 33383097891135, 438714241508615, 5803049210371375, 77199163872173757, 1032215519193531310, 13864180990526161995, 186975433988014039830

Offset: 0

Seiichi Manyama, Aug 25 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..864
Jun Yan, Lattice paths enumerations weighted by ascent lengths, arXiv:2501.01152 [math.CO], 2025. See p. 13.

Cf. A002295, A365184, A365185, A365186, A365187, A365188.

Cf. A006605, A255673, A365183.

a(n) = sum(k=0, n\2, binomial(n-k, k)*binomial(5*n+1, n-k))/(5*n+1);

a(n) = (1/(5*n+1)) * Sum_{k=0..floor(n/2)} binomial(n-k,k) * binomial(5*n+1,n-k).

A212072 G.f. satisfies: A(x) = (1 + x*A(x)^2)^3.

Original entry on oeis.org

1, 3, 21, 190, 1950, 21576, 250971, 3025308, 37456650, 473498025, 6085977381, 79296104784, 1044955576496, 13903071489300, 186507160795350, 2519857658331576, 34258270557555282, 468322722628414290, 6433538749783033350, 88767899653496377050, 1229626632793564911906

Offset: 0

Paul D. Hanna, Apr 29 2012

nonn

			G.f.: A(x) = 1 + 3*x + 21*x^2 + 190*x^3 + 1950*x^4 + 21576*x^5 + ...
Related expansions:
A(x)^2 = 1 + 6*x + 51*x^2 + 506*x^3 + 5481*x^4 + ... + A002295(n+1)*x^n + ...
A(x)^(1/3) = 1 + x + 6*x^2 + 51*x^3 + 506*x^4 + ... + A002295(n)*x^n + ...

Michael De Vlieger, Table of n, a(n) for n = 0..855
Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group, arXiv:2112.11595 [math.CO], 2021.
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.

Cf. A002295, A212071, A212073, A130564.

Table[(3 Binomial[#, n])/# &[6 n + 3], {n, 0, 20}] (* Michael De Vlieger, May 13 2022 *)

{a(n)=binomial(6*n+3,n) * 3/(6*n+3)}
for(n=0, 40, print1(a(n), ", "))

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

a(n) = 3*binomial(6*n+3,n)/(6*n+3).

G.f. A(x) = G(x)^3 where G(x) = 1 + x*G(x)^6 is the g.f. of A002295.

A346065 a(n) = Sum_{k=0..n} binomial(6k,k) / (5k + 1).

Original entry on oeis.org

1, 2, 8, 59, 565, 6046, 68878, 818276, 10021910, 125629220, 1603943486, 20783993414, 272641113110, 3613484662965, 48313969712685, 650888627139801, 8826840286257595, 120398870546499685, 1650711840886884265, 22735860619151166130, 314441081323870331656

Offset: 0

Ilya Gutkovskiy, Jul 28 2021

nonn

Partial sums of A002295.

In general, for m > 1, Sum_{k=0..n} binomial(m*k,k) / ((m-1)*k + 1) ~ m^(m*(n+1) + 1/2) / (sqrt(2*Pi) * (m^m - (m-1)^(m-1)) * n^(3/2) * (m-1)^((m-1)*n + 3/2)). - Vaclav Kotesovec, Jul 28 2021

Seiichi Manyama, Table of n, a(n) for n = 0..856

Cf. A002295, A014137, A104859, A345367, A345368, A346648, A346671, A346672.

Table[Sum[Binomial[6 k, k]/(5 k + 1), {k, 0, n}], {n, 0, 20}]
nmax = 20; A[] = 0; Do[A[x] = 1/(1 - x) + x (1 - x)^5 A[x]^6 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

a(n) = sum(k=0, n, binomial(6*k, k)/(5*k+1)); \\ Michel Marcus, Jul 28 2021

G.f. A(x) satisfies: A(x) = 1 / (1 - x) + x * (1 - x)^5 * A(x)^6.

a(n) ~ 2^(6*n + 6) * 3^(6*n + 13/2) / (43531 * sqrt(Pi) * n^(3/2) * 5^(5*n + 3/2)). - Vaclav Kotesovec, Jul 28 2021

A364472 G.f. satisfies A(x) = 1 + xA(x) + x^2A(x)^6.

Original entry on oeis.org

1, 1, 2, 8, 35, 163, 808, 4162, 22041, 119325, 657384, 3673394, 20769983, 118610807, 683131766, 3963486380, 23144000681, 135911263309, 802143851323, 4755506884495, 28306896506651, 169110331570307, 1013643450123455, 6094125091837335, 36739933169338731

Offset: 0

Seiichi Manyama, Jul 26 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A000045, A000108, A001006, A182454, A186996, A364476.

Cf. A002295.

a(n) = sum(k=0, n\2, binomial(n+4*k, k)*binomial(n+3*k, n-2*k)/(5*k+1));

a(n) = Sum_{k=0..floor(n/2)} binomial(n+4*k,k) * binomial(n+3*k,n-2*k) / (5*k+1) = Sum_{k=0..floor(n/2)} binomial(n+4*k,6*k) * binomial(6*k,k) / (5*k+1).

A365186 G.f. satisfies A(x) = 1 + xA(x)^5(1 + x*A(x)^2).

Original entry on oeis.org

1, 1, 6, 47, 428, 4241, 44407, 483358, 5414618, 62014112, 722870120, 8547768832, 102284029963, 1236274747490, 15070955944288, 185089043535730, 2287843817573898, 28440852786725695, 355345599519983962, 4459821165693379625, 56200963128262312342

Offset: 0

Seiichi Manyama, Aug 25 2023

nonn

Cf. A002295, A365184, A365185, A365187, A365188, A365189.

Cf. A365192.

a(n) = sum(k=0, n, binomial(2*n+3*k+1, k)*binomial(k, n-k)/(2*n+3*k+1));

a(n) = Sum_{k=0..n} binomial(2*n+3*k+1,k) * binomial(k,n-k)/(2*n+3*k+1).

A365192 G.f. satisfies A(x) = 1 + xA(x)^5 / (1 - xA(x)^2).

Original entry on oeis.org

1, 1, 6, 48, 443, 4445, 47107, 518835, 5880223, 68130860, 803369481, 9609294542, 116310009888, 1421951861817, 17533301767624, 217796367181117, 2722942699583650, 34236790400004432, 432649744252128084, 5492060945760586212, 69998993052214823013

Offset: 0

Seiichi Manyama, Aug 25 2023

nonn

Cf. A002295, A243667, A349332, A364748, A365193, A365194.

Cf. A365186.

a(n) = sum(k=0, n, binomial(2*n+3*k+1, k)*binomial(n-1, n-k)/(2*n+3*k+1));

a(n) = Sum_{k=0..n} binomial(2*n+3*k+1,k) * binomial(n-1,n-k)/(2*n+3*k+1).

cp's OEIS Frontend

A251586 a(n) = 6^(n-4) * (n+1)^(n-6) * (125*n^4 + 810*n^3 + 2095*n^2 + 2586*n + 1296).

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A251696 a(n) = (4*n+1) * (5*n+1)^(n-2) * 6^n.

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A346666 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * binomial(6*k,k) / (5*k + 1).

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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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A365189 G.f. satisfies A(x) = 1 + x*A(x)^5*(1 + x*A(x)^5).

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A212072 G.f. satisfies: A(x) = (1 + x*A(x)^2)^3.

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A346065 a(n) = Sum_{k=0..n} binomial(6*k,k) / (5*k + 1).

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A251586 a(n) = 6^(n-4) * (n+1)^(n-6) * (125n^4 + 810n^3 + 2095n^2 + 2586n + 1296).

A251696 a(n) = (4n+1) (5n+1)^(n-2) 6^n.

A346666 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * binomial(6k,k) / (5k + 1).

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

A365189 G.f. satisfies A(x) = 1 + xA(x)^5(1 + x*A(x)^5).

A346065 a(n) = Sum_{k=0..n} binomial(6k,k) / (5k + 1).

A364472 G.f. satisfies A(x) = 1 + xA(x) + x^2A(x)^6.

A365186 G.f. satisfies A(x) = 1 + xA(x)^5(1 + x*A(x)^2).

A365192 G.f. satisfies A(x) = 1 + xA(x)^5 / (1 - xA(x)^2).