A002295 -id:A002295 - cp's OEIS frontend

A349312 G.f. A(x) satisfies: A(x) = (1 + x * A(x)^6) / (1 - x).

1, 2, 14, 158, 2106, 30762, 476406, 7683926, 127692530, 2171184146, 37592376734, 660522703886, 11747865153962, 211093333172282, 3826315983647366, 69880933123237958, 1284661783610775010, 23753502514840942882, 441458929706855144494, 8242097867816771820926

Offset: 0

Ilya Gutkovskiy, Nov 14 2021

nonn

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

Cf. A002295, A006318, A346626, A346648, A349310, A349311, A349313, A349314.

nmax = 19; A[] = 0; Do[A[x] = (1 + x A[x]^6)/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[Binomial[n + 5 k, 6 k] Binomial[6 k, k]/(5 k + 1), {k, 0, n}], {n, 0, 19}]

a(n) = Sum_{k=0..n} binomial(n+5*k,6*k) * binomial(6*k,k) / (5*k+1).
a(n) = F([(1+n)/5, (2+n)/5, (3+n)/5, (4+n)/5, 1+n/5, -n], [2/5, 3/5, 4/5, 1, 6/5], -1), where F is the generalized hypergeometric function. - Stefano Spezia, Nov 14 2021
a(n) ~ sqrt(1 + 5*r) / (2^(6/5) * 3^(7/10) * sqrt(5*Pi) * (1-r)^(3/10) * n^(3/2) * r^(n + 1/5)), where r = 0.04941755525635041337247049893940451999923592381716... is the smallest real root of the equation 5^5 * (1-r)^6 = 6^6 * r. - Vaclav Kotesovec, Nov 15 2021

A349333 G.f. A(x) satisfies A(x) = 1 + x * A(x)^6 / (1 - x).

Original entry on oeis.org

1, 1, 7, 64, 678, 7836, 95838, 1219527, 15979551, 214151601, 2921712145, 40444378948, 566634504256, 8019501351103, 114484746457075, 1646614155398872, 23837794992712680, 347081039681365623, 5079306905986689309, 74670702678690897079, 1102218694940440851877

Offset: 0

Ilya Gutkovskiy, Nov 15 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..836

Cf. A002212, A307678, A349331, A349332, A349334, A349335.

Cf. A002295, A346648 (partial sums), A349362.

a:= n-> coeff(series(RootOf(1+x*A^6/(1-x)-A, A), x, n+1), x, n):
seq(a(n), n=0..20);  # Alois P. Heinz, Nov 15 2021

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

{a(n) = my(A=[1]); for(m=1, n, A=concat(A, 0);
A[#A] = 1 + sum(k=1, m-1, (polcoeff(Ser(A)^6, 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(6*k,k) / (5*k+1).

a(n) ~ 49781^(n + 1/2) / (72 * sqrt(3*Pi) * n^(3/2) * 5^(5*n + 3/2)). - Vaclav Kotesovec, Nov 15 2021

A349362 G.f. A(x) satisfies: A(x) = 1 + x * A(x)^6 / (1 + x).

Original entry on oeis.org

1, 1, 5, 40, 370, 3740, 40006, 445231, 5102165, 59799505, 713496815, 8637432580, 105826926716, 1309793896431, 16351672606365, 205665994855320, 2603696877136060, 33151784577226295, 424258396639960591, 5454120586840761631, 70402732493668027775

Offset: 0

Ilya Gutkovskiy, Nov 15 2021

nonn

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

Cf. A001006, A002295, A127897, A317133, A346065 (binomial transform), A346666, A349333, A349361, A349363, A349364.

a:= n-> coeff(series(RootOf(1+x*A^6/(1+x)-A, A), x, n+1), x, n):
seq(a(n), n=0..20);  # Alois P. Heinz, Nov 15 2021

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

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n-1,k-1) * binomial(6*k,k) / (5*k+1).
a(n) = (-1)^(n+1)* F([7/6, 4/3, 3/2, 5/3, 11/6, 1-n], [7/5, 8/5, 9/5, 2, 11/5], 6^6/5^5), where F is the generalized hypergeometric function. - Stefano Spezia, Nov 15 2021
a(n) ~ 43531^(n + 1/2) / (72 * sqrt(3*Pi) * n^(3/2) * 5^(5*n + 3/2)). - Vaclav Kotesovec, Nov 17 2021

A079679 a(n) = a(n,m) = Sum_{k=0..n} binomial(mk,k)binomial(m*(n-k),n-k) for m=6.

Original entry on oeis.org

1, 12, 168, 2424, 35400, 520236, 7674144, 113482584, 1681028136, 24932533800, 370144424376, 5499182587416, 81748907485248, 1215834858032820, 18090048027643200, 269246037610828656, 4008495234662771688, 59692297399976544120, 889090275714779739120, 13245013739104555683600

Offset: 0

Benoit Cloitre, Jan 26 2003

nonn

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

Seiichi Manyama, Table of n, a(n) for n = 0..851
D. Merlini, R. Sprugnoli and M. C. Verri, The tennis ball problem, J. Combin. Theory, A 99 (2002), 307-344.
Rui Duarte and António Guedes de Oliveira, Short note on the convolution of binomial coefficients, arXiv:1302.2100 [math.CO], 2013.

Cf. A000302, A006256, A078995, A079563, A079678.

Cf. A002295, A226705.

a(n) = sum(k=0,n,5^(n-k)*binomial(6*n+1,k));
vector(30, n, a(n-1)) \\  Altug Alkan, Sep 30 2015

a(n) = 3/5*(46656/3125)^n*(1+c/sqrt(n)+o(n^-1/2)) where c=0.388...
c = 8/(3*sqrt(15*Pi)) = 0.388461664210517... - Vaclav Kotesovec, May 25 2020
a(n) = Sum_{k=0..n} binomial(6*k+l,k)*binomial(6*(n-k)-l,n-k) for every real number l. - Rui Duarte and António Guedes de Oliveira, Feb 16 2013
From Rui Duarte and António Guedes de Oliveira, Feb 17 2013: (Start)
a(n) = Sum_{k=0..n} 5^(n-k) * binomial(6*n+1,k).
a(n) = Sum_{k=0..n} 6^(n-k) * binomial(5*n+k,k). (End)
G.f.: hypergeom([1/6, 1/3, 1/2, 2/3, 5/6],[1/5, 2/5, 3/5, 4/5],46656*x/3125)^2.  - Mark van Hoeij, Apr 19 2013
a(n) = [x^n] 1/((1-6*x) * (1-x)^(5*n+1)). - Seiichi Manyama, Aug 03 2025
From Seiichi Manyama, Aug 14 2025: (Start)
a(n) = Sum_{k=0..n} 6^k * (-5)^(n-k) * binomial(6*n+1,k) * binomial(6*n-k,n-k).
G.f.: g^2/(6-5*g)^2 where g = 1+x*g^6 is the g.f. of A002295. (End)

A346648 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(6k,k) / (5k + 1).

Original entry on oeis.org

1, 2, 9, 73, 751, 8587, 104425, 1323952, 17303503, 231455104, 3153167249, 43597546197, 610232050453, 8629733401556, 123114479858631, 1769728635257503, 25607523627970183, 372688563309335806, 5451995469296025115, 80122698147986922194, 1182341393088427774071

Offset: 0

Ilya Gutkovskiy, Jul 26 2021

nonn

Binomial transform of A002295.

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

Cf. A002295, A007317, A188687, A226910, A346646, A346647, A346649, A346650.

Table[Sum[Binomial[n, k] 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)^4 A[x]^6 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
nmax = 20; CoefficientList[Series[Sum[(Binomial[6 k, k]/(5 k + 1)) x^k/(1 - x)^(k + 1), {k, 0, nmax}], {x, 0, nmax}], x]
Table[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, 20}]

a(n) = sum(k=0, n, binomial(n,k)*binomial(6*k,k)/(5*k + 1)); \\ Michel Marcus, Jul 26 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) ~ 49781^(n + 3/2) / (3359232 * sqrt(3*Pi) * n^(3/2) * 5^(5*n + 3/2)). - Vaclav Kotesovec, Jul 30 2021

A364748 G.f. A(x) satisfies A(x) = 1 + xA(x)^5 / (1 - xA(x)).

Original entry on oeis.org

1, 1, 6, 47, 424, 4159, 43097, 464197, 5145475, 58313310, 672598269, 7869856070, 93183973405, 1114471042413, 13443614108307, 163372291277764, 1998239045199623, 24580340878055298, 303893356012560280, 3774099648814193998, 47061518776483143441

Offset: 0

Seiichi Manyama, Aug 05 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..890

Cf. A000108, A001003, A108447, A364747, A378691, A378692.

Cf. A002295, A243667, A365192, A365193, A365194.

a(n) = if(n==0, 1, sum(k=0, n-1, binomial(n, k)*binomial(5*n-4*k, n-1-k))/n);

a(n, r=1, s=1, t=5, u=1) = r*sum(k=0, n, binomial(t*k+u*(n-k)+r, k)*binomial(n+(s-1)*k-1, n-k)/(t*k+u*(n-k)+r)); \\ Seiichi Manyama, Dec 05 2024

a(n) = (1/n) * Sum_{k=0..n-1} binomial(n,k) * binomial(5*n-4*k,n-1-k) for n > 0.
From Seiichi Manyama, Dec 05 2024: (Start)
G.f. A(x) satisfies A(x) = 1/(1 - x*A(x)^4/(1 - x*A(x))).
If g.f. satisfies A(x) = ( 1 + x*A(x)^(t/r) / (1 - x*A(x)^(u/r))^s )^r, then a(n) = r * Sum_{k=0..n} binomial(t*k+u*(n-k)+r,k) * binomial(n+(s-1)*k-1,n-k)/(t*k+u*(n-k)+r). (End)

A349291 G.f. A(x) satisfies A(x) = 1 / ((1 - x) * (1 - x * A(x)^5)).

Original entry on oeis.org

1, 2, 13, 139, 1775, 24886, 370099, 5733304, 91518691, 1494815215, 24862931821, 419674102147, 7170713484877, 123783319369420, 2155542171446485, 37820343323942566, 667957770644685811, 11865421405897931581, 211856917750711562695, 3800040255017879663415

Offset: 0

Ilya Gutkovskiy, Nov 13 2021

nonn

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

Cf. A002295, A007317, A199475, A346648, A349289, A349290, A349292, A349293.

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

a(n) = Sum_{k=0..n} binomial(n+4*k,5*k) * binomial(6*k,k) / (5*k+1).
a(n) ~ sqrt(1 + 4*r) / (2^(6/5) * 3^(7/10) * sqrt(5*Pi*(1-r)) * n^(3/2) * r^(n + 1/5)), where r = 0.051436794119208432185504972091697516647... is the real root of the equation 6^6 * r = 5^5 * (1-r)^5. - Vaclav Kotesovec, Nov 14 2021
a(n) = 1 + Sum_{x_1, x_2, ..., x_6>=0 and x_1+x_2+...+x_6=n-1} Product_{k=1..6} a(x_k). - Seiichi Manyama, Jul 10 2025

A233743 a(n) = 7binomial(6n + 7, n)/(6*n + 7).

Original entry on oeis.org

1, 7, 63, 644, 7105, 82467, 992446, 12271512, 154962990, 1990038435, 25909892008, 341225775072, 4537563627415, 60842326873230, 821692714673340, 11167153485624304, 152610018401940330, 2095863415900961490, 28910564819681953485, 400379714692751795820

Offset: 0

Tim Fulford, Dec 15 2013

Fuss-Catalan sequence is a(n,p,r) = r*binomial(n*p + r, n)/(n*p + r); this is the case p = 6, r = 7.

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, A002295, A212071, A212072, A212073, A130564, A233827, A233829, A233830.

Cf. A000245 (k = 3), A006629 (k = 4), A196678 (k = 5), A233668 (k = 6), A233835 (k = 8), A234467 (k = 9), A232265 (k = 10), A229963 (k = 11).

[7*Binomial(6*n+7, n)/(6*n+7): n in [0..30]]; // Vincenzo Librandi, Dec 16 2013

Table[7 Binomial[6 n + 7, n]/(6 n + 7), {n, 0, 40}] (* Vincenzo Librandi, Dec 16 2013 *)

PARI
```
a(n) = 7*binomial(6*n+7,n)/(6*n+7);
```

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(6/7))^7+x*O(x^n)); polcoeff(B, n)}

G.f. satisfies: A(x) = {1 + x*A(x)^(p/r)}^r, where p = 6, r = 7.
From _Peter Bala, Oct 16 2015: (Start)
O.g.f. A(x) = 1/x * series reversion (x*C(-x)^7), 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)^(1/7) is the o.g.f. for A002295. (End)

More terms from Vincenzo Librandi, Dec 16 2013

A251667 E.g.f.: exp(7xG(x)^6) / G(x) where G(x) = 1 + x*G(x)^7 is the g.f. of A002296.

Original entry on oeis.org

1, 6, 107, 3508, 171741, 11280842, 933014767, 93212094024, 10925496633401, 1470493880790382, 223555405538724819, 37892802280129883324, 7086076189702624109653, 1449303152891376476830962, 321848482510755456019058519, 77124029495405859198280522768

Offset: 0

Paul D. Hanna, Dec 07 2014

nonn

			E.g.f.: A(x) = 1 + 6*x + 107*x^2/2! + 3508*x^3/3! + 171741*x^4/4! + 11280842*x^5/5! +...
such that A(x) = exp(7*x*G(x)^6) / G(x)
where G(x) = 1 + x*G(x)^7 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 +...

Cf. A251577, A251697, A002295.

Cf. Variants: A243953, A251663, A251664, A251665, A251666, A251668, A251669, A251670.

Table[Sum[7^k * n!/k! * Binomial[7*n-k-2,n-k] * (6*k-1)/(6*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^7 +x*O(x^n)); n!*polcoeff(exp(7*x*G^6)/G, n)}
for(n=0, 20, print1(a(n), ", "))

{a(n) = sum(k=0, n, 7^k * n!/k! * binomial(7*n-k-2,n-k) * (6*k-1)/(6*n-1) )}
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)/A(x) = G(x)^6 + 5*G'(x)/G(x).
(2) A(x) = F(x/A(x)^6) where F(x) is the e.g.f. of A251697.
(3) A(x) = Sum_{n>=0} A251697(n)*(x/A(x)^6)^n/n! where A251697(n) = (5*n+1) * (6*n+1)^(n-2) * 7^n .
(4) [x^n/n!] A(x)^(6*n+1) = (5*n+1) * (6*n+1)^(n-1) * 7^n .
a(n) = Sum_{k=0..n} 7^k * n!/k! * binomial(7*n-k-2,n-k) * (6*k-1)/(6*n-1) for n>=0.
Recurrence: 72*(2*n-1)*(3*n-2)*(3*n-1)*(6*n-5)*(6*n-1)*(588245*n^6 - 6117748*n^5 + 26651100*n^4 - 62321728*n^3 + 82554122*n^2 - 58646294*n + 17291583)*a(n) = 7*(69206436005*n^12 - 996572678472*n^11 + 6516703994430*n^10 - 25624338676965*n^9 + 67604945463195*n^8 - 126360374558838*n^7 + 171960790012102*n^6 - 171911061779835*n^5 + 125050872537045*n^4 - 63802357502870*n^3 + 20814954345360*n^2 - 3329274812661*n - 3763584000)*a(n-1) - 823543*(588245*n^6 - 2588278*n^5 + 4886035*n^4 - 5129908*n^3 + 3141733*n^2 - 958104*n - 720)*a(n-2). - Vaclav Kotesovec, Dec 07 2014
a(n) ~ 5 * 7^(7*n-3/2) / 6^(6*n-1/2) * n^(n-1) / exp(n-1). - Vaclav Kotesovec, Dec 07 2014

A365184 G.f. satisfies A(x) = 1 + xA(x)^5(1 + x).

Original entry on oeis.org

1, 1, 6, 45, 395, 3775, 38146, 400826, 4335455, 47951065, 539823620, 6165377836, 71261299056, 831990025420, 9797505040130, 116235417614900, 1387958781395535, 16668362761081560, 201190667288072005, 2439418470063468505, 29698136499328762445

Offset: 0

Seiichi Manyama, Aug 25 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..898

Cf. A002295, A365185, A365186, A365187, A365188, A365189.
Cf. A364475, A365178.
Cf. A002294, A349332.

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

a(n) = Sum_{k=0..n} binomial(5*k+1,k) * binomial(k,n-k)/(5*k+1) = Sum_{k=0..n} binomial(k,n-k) * A002294(k).

cp's OEIS Frontend

A349312 G.f. A(x) satisfies: A(x) = (1 + x * A(x)^6) / (1 - x).

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A349333 G.f. A(x) satisfies A(x) = 1 + x * A(x)^6 / (1 - x).

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A349362 G.f. A(x) satisfies: A(x) = 1 + x * A(x)^6 / (1 + x).

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A079679 a(n) = a(n,m) = Sum_{k=0..n} binomial(m*k,k)*binomial(m*(n-k),n-k) for m=6.

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A346648 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(6*k,k) / (5*k + 1).

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A364748 G.f. A(x) satisfies A(x) = 1 + x*A(x)^5 / (1 - x*A(x)).

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A349291 G.f. A(x) satisfies A(x) = 1 / ((1 - x) * (1 - x * A(x)^5)).

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A233743 a(n) = 7*binomial(6*n + 7, n)/(6*n + 7).

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Magma

A079679 a(n) = a(n,m) = Sum_{k=0..n} binomial(mk,k)binomial(m*(n-k),n-k) for m=6.

A346648 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(6k,k) / (5k + 1).

A364748 G.f. A(x) satisfies A(x) = 1 + xA(x)^5 / (1 - xA(x)).

A233743 a(n) = 7binomial(6n + 7, n)/(6*n + 7).

A251667 E.g.f.: exp(7xG(x)^6) / G(x) where G(x) = 1 + x*G(x)^7 is the g.f. of A002296.

A365184 G.f. satisfies A(x) = 1 + xA(x)^5(1 + x).