A002296 -id:A002296 - cp's OEIS frontend

A233907 9binomial(7n+9, n)/(7*n+9).

1, 9, 99, 1218, 16065, 222138, 3178140, 46656324, 698868216, 10639125640, 164128169205, 2560224004884, 40314178429707, 639948824981928, 10230035192533800, 164541833894991240, 2660919275605834701, 43239781879996449825, 705687913212419321800, 11561996402992103418000, 190100812111989146008641

Offset: 0

Tim Fulford, Dec 17 2013

nonn

Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), this is the case p=7, r=9.

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.

Cf. A000108, A002296, A233832, A233833, A143547, A233834, A130565, A233835, A233908.

[9*Binomial(7*n+9, n)/(7*n+9): n in [0..30]];

Table[9 Binomial[7 n + 9, n]/(7 n + 9), {n, 0, 30}]

PARI
```
a(n) = 9*binomial(7*n+9,n)/(7*n+9);
```

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(7/9))^9+x*O(x^n)); polcoeff(B, n)}

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, where p=7, r=9.

D-finite with recurrence 72*n*(6*n+5)*(3*n+2)*(2*n+3)*(3*n+4)*(6*n+7)*a(n) -7*(7*n+4)*(7*n+8)*(7*n+5)*(7*n+2)*(7*n+6)*(7*n+3)*a(n-1)=0. - R. J. Mathar, Nov 22 2024

A233908 10binomial(7n+10,n)/(7*n+10).

Original entry on oeis.org

1, 10, 115, 1450, 19425, 271502, 3915100, 57821940, 870238200, 13298907050, 205811513765, 3218995093860, 50802419972395, 808016193159000, 12938696992921000, 208419656266988904, 3374960506795660365, 54907659530154222000, 897060906625956765000

Offset: 0

Tim Fulford, Dec 17 2013

nonn

Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), this is the case p=7, r=10.

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 - A233835, A143547, A130565, A233907.

[10*Binomial(7*n+10, n)/(7*n+10): n in [0..30]]; // Vincenzo Librandi, Dec 23 2013

Table[10 Binomial[7 n + 10, n]/(7 n + 10), {n, 0, 40}] (* Vincenzo Librandi, Dec 23 2013 *)

PARI
```
a(n) = 10*binomial(7*n+10,n)/(7*n+10);
```

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(7/10))^10+x*O(x^n)); polcoeff(B, n)}

72*n*(6*n+5)*(3*n+5)*(2*n+3)*(3*n+4)*(6*n+7)*a(n) -7*(7*n+4)*(7*n+8)*(7*n+5)*(7*n+9)*(7*n+6)*(7*n+3)*a(n-1)=0. - R. J. Mathar, Dec 22 2013

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, where p=7, r=10.

A346768 a(n) = Sum_{k=0..n} Stirling2(n,k) * binomial(7k,k) / (6k + 1).

Original entry on oeis.org

1, 1, 8, 92, 1289, 20518, 358611, 6749268, 135095116, 2851394415, 63066764910, 1454808403309, 34869538474423, 865771965143262, 22211885496614803, 587583912259110350, 15998031596388750905, 447598845624472993496, 12850922242548662924046, 378153449033278630907275

Offset: 0

Ilya Gutkovskiy, Aug 02 2021

nonn

Stirling transform of A002296.

Michael De Vlieger, Table of n, a(n) for n = 0..493

Cf. A002296, A064856, A346764, A346765, A346766, A346767, A346769.

Table[Sum[StirlingS2[n, k] Binomial[7 k, k]/(6 k + 1), {k, 0, n}], {n, 0, 19}]
nmax = 19; CoefficientList[Series[Sum[(Binomial[7 k, k]/(6 k + 1)) x^k/Product[1 - j x, {j, 0, k}], {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 19; CoefficientList[Series[HypergeometricPFQ[{1/7, 2/7, 3/7, 4/7, 5/7, 6/7}, {1/3, 1/2, 2/3, 5/6,1, 7/6}, 823543 (Exp[x] - 1)/46656], {x, 0, nmax}], x] Range[0, nmax]!

a(n) = sum(k=0, n, stirling(n, k, 2)*binomial(7*k, k)/(6*k + 1)); \\ Michel Marcus, Aug 02 2021

G.f.: Sum_{k>=0} ( binomial(7*k,k) / (6*k + 1) ) * x^k / Product_{j=0..k} (1 - j*x).

A349302 G.f. A(x) satisfies: A(x) = 1 / ((1 + x) * (1 - x * A(x)^6)).

Original entry on oeis.org

1, 0, 1, 6, 43, 321, 2500, 20096, 165621, 1392397, 11896823, 103014141, 902035660, 7974080834, 71070247438, 637937825112, 5761970031357, 52329993278856, 477588786637264, 4377832437503643, 40288077072190109, 372086539388626537, 3447632819399550915

Offset: 0

Ilya Gutkovskiy, Nov 13 2021

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

Cf. A002296, A005043, A346627, A346667, A349292, A349299, A349300, A349301, A349303.

nmax = 22; 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[(-1)^(n - k) Binomial[n + 5 k, 6 k] Binomial[7 k, k]/(6 k + 1), {k, 0, n}], {n, 0, 22}]

a(n) = Sum_{k=0..n} (-1)^(n-k) * 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 = 0.1008057775745727124639860500770912830001828593281202101426766... is the root of the equation 7^7 * r = 6^6 * (1+r)^6. - Vaclav Kotesovec, Nov 14 2021
From Peter Bala, Jun 02 2024: (Start)
A(x) = 1/(1 + x)*F(x/(1 + x)^6), where F(x) = Sum_{n >= 0} A002296(n)*x^n.
A(x) = 1/(1 + x) + x*A(x)^7. (End)

A349363 G.f. A(x) satisfies: A(x) = 1 + x * A(x)^7 / (1 + x).

Original entry on oeis.org

1, 1, 6, 57, 629, 7589, 96942, 1288729, 17643920, 247089010, 3522891561, 50964747400, 746241617226, 11038241689188, 164696773030055, 2475832560808858, 37462189433509758, 570112127356828846, 8720472842436039280, 133997057207982607092, 2067402314984991892461

Offset: 0

Ilya Gutkovskiy, Nov 15 2021

nonn

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

Cf. A001006, A002296, A127897, A317133, A346667, A346671 (binomial transform), A349334, A349361, A349362, A349364.

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 = 20; 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[(-1)^(n - k) Binomial[n - 1, k - 1] Binomial[7 k, k]/(6 k + 1), {k, 0, n}], {n, 0, 20}]

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n-1,k-1) * binomial(7*k,k) / (6*k+1).
a(n) = (-1)^(n+1)* F([8/7, 9/7, 10/7, 11/7, 12/7, 13/7, 1-n], [4/3, 3/2, 5/3, 11/6, 2, 13/6], 7^7/6^6), where F is the generalized hypergeometric function. - Stefano Spezia, Nov 15 2021
a(n) ~ 776887^(n + 1/2) / (343 * sqrt(Pi) * n^(3/2) * 2^(6*n + 2) * 3^(6*n + 3/2)). - Vaclav Kotesovec, Nov 17 2021

A365218 G.f. satisfies A(x) = 1 + xA(x)^6 / (1 + xA(x)^6).

Original entry on oeis.org

1, 1, 5, 34, 265, 2232, 19766, 181300, 1706737, 16392049, 159959240, 1581278838, 15800619070, 159321921844, 1618981274136, 16562211506496, 170426473666497, 1762771226922775, 18316562635133813, 191104193378725552, 2001224271292820200

Offset: 0

Seiichi Manyama, Aug 26 2023

nonn

Conjecture: Is a(n)>0 correct? It is correct up to the first 10000 terms.

Seiichi Manyama, Table of n, a(n) for n = 0..956

Cf. A291534, A364864, A364865, A364866.

Cf. A002296.

a(n) = sum(k=0, n, (-1)^(n-k)*binomial(6*n+1, k)*binomial(n-1, n-k))/(6*n+1);

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

A346581 a(n) = (1/(7n)) Sum_{d|n} mu(n/d) * binomial(7*d,d).

Original entry on oeis.org

1, 6, 63, 728, 9275, 124866, 1753073, 25365600, 375677595, 5667202850, 86775157139, 1345153422600, 21069043965983, 332927798516614, 5301031234076325, 84967018610221440, 1369846562874360886, 22199151535757655354, 361411377745122110421, 5908312923789590118600

Offset: 1

Ilya Gutkovskiy, Jul 24 2021

nonn

Inverse Euler transform of A002296.

Moebius transform of A261500.

Cf. A002296, A004368, A008683, A022553, A261500, A346577, A346578, A346579, A346580, A346582.

Table[(1/(7 n)) Sum[MoebiusMu[n/d] Binomial[7 d, d], {d, Divisors[n]}], {n, 20}]

a(n) = sumdiv(n, d, moebius(n/d)*binomial(7*d,d))/(7*n); \\ Michel Marcus, Jul 24 2021

A346683 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(7k,k) / (6k + 1).

Original entry on oeis.org

1, 0, 7, 63, 756, 9716, 132062, 1865626, 27124049, 403197584, 6100155272, 93626517858, 1454221328232, 22815183746508, 361030984965596, 5755543515895284, 92350704790963431, 1490287557170676816, 24171116970619575559, 393808998160695560841, 6442255541764422795759

Offset: 0

Ilya Gutkovskiy, Jul 29 2021

nonn

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

Cf. A002296, A032357, A188678, A346667, A346671.

Cf. A346680, A346681, A346682, A346684.

Table[Sum[(-1)^(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)^6 A[x]^7 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

a(n) = sum(k=0, n, (-1)^(n-k)*binomial(7*k, k)/(6*k + 1)); \\ Michel Marcus, Jul 29 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) / (870199 * sqrt(Pi) * n^(3/2) * 2^(6*n + 2) * 3^(6*n + 3/2)). - Vaclav Kotesovec, Jul 30 2021

A206290 G.f.: Sum_{n>=0} Product_{k=1..n} Series_Reversion( x/(1 + x^k) ).

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 12, 17, 29, 44, 77, 114, 218, 330, 617, 987, 1913, 2968, 6068, 9500, 19263, 31399, 64268, 101702, 218891, 348559, 735823, 1205239, 2576727, 4119884, 9100854, 14588992, 31841260, 52163378, 114485092, 183947681, 414704366, 667453931, 1487920000

Offset: 0

Paul D. Hanna, Feb 05 2012

nonn

Compare to the g.f. of partition numbers (A000041): Sum_{n>=0} Product_{k=1..n} x/(1 - x^k).

			G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 12*x^6 + 17*x^7 +...
such that, by definition,
A(x) = 1 + G_1(x) + G_1(x)*G_2(x) + G_1(x)*G_2(x)*G_3(x) + G_1(x)*G_2(x)*G_3(x)*G_4(x) +...
where G_n( x/(1 + x^n) ) = x.
The first few expansions of G_n(x) begin:
G_1(x) = x + x^2 + x^3 + x^4 + x^5 + x^6 +...+ x^(n+1) +...
G_2(x) = x + x^3 + 2*x^5 + 5*x^7 + 14*x^9 +...+ A000108(n)*x^(2*n+1) +...
G_3(x) = x + x^4 + 3*x^7 + 12*x^10 + 55*x^13 +...+ A001764(n)*x^(3*n+1) +...
G_4(x) = x + x^5 + 4*x^9 + 22*x^13 + 140*x^17 +...+ A002293(n)*x^(4*n+1) +...
G_5(x) = x + x^6 + 5*x^11 + 35*x^16 + 285*x^21 +...+ A002294(n)*x^(5*n+1) +...
G_6(x) = x + x^7 + 6*x^13 + 51*x^19 + 506*x^25 +...+ A002295(n)*x^(6*n+1) +...
G_7(x) = x + x^8 + 7*x^15 + 70*x^22 + 819*x^29 +...+ A002296(n)*x^(7*n+1) +...
Note that G_n(x) = x + x*G_n(x)^n.

Vaclav Kotesovec, Table of n, a(n) for n = 0..250

Cf. A206289, A194560, A110448.

{a(n)=polcoeff(sum(m=0,n,prod(k=1,m,serreverse(x/(1+x^k+x*O(x^n))))),n)}
for(n=0,45,print1(a(n),", "))

G.f.: Sum_{n>=0} Product_{k=1..n} G_k(x), where G_n(x) is defined by:
(1) G_n(x) = Series_Reversion( x/(1 + x^n) ),
(2) G_n(x) = x + x*G_n(x)^n,
(3) G_n(x) = Sum_{k>=0} binomial(n*k+1, k) * x^(n*k+1) / (n*k+1).

A235534 a(n) = binomial(6n, 2n) / (4*n + 1).

Original entry on oeis.org

1, 3, 55, 1428, 43263, 1430715, 50067108, 1822766520, 68328754959, 2619631042665, 102240109897695, 4048514844039120, 162250238001816900, 6568517413771094628, 268225186597703313816, 11034966795189838872624, 456949965738717944767791

Offset: 0

Bruno Berselli, Jan 12 2014

This is the case l=4, k=2 of binomial((l+k)*n,k*n)/((l*n+1)/gcd(k,l*n+1)), see Theorem 1.1 in Zhi-Wei Sun's paper.

First bisection of A001764.

Zhi-Wei Sun, On Divisibility Of Binomial Coefficients, Journal of the Australian Mathematical Society 93 (2012), p. 189-201.

Cf. similar sequences generated by binomial((l+k)*n,k*n)/(l*n+1), where l is divisible by all the factors of k: A000108 (l=1, k=1), A001764 (l=2, k=1), A002293 (l=3, k=1), A002294 (l=4, k=1), A002295 (l=5, k=1), A002296 (l=6, k=1), A007556 (l=7, k=1), A062994 (l=8, k=1), A059968 (l=9, k=1), A230388 (l=10, k=1), A048990 (l=2, k=2), this sequence (l=4, k=2), A235536 (l=6, k=2), A187357 (l=3, k=3), A235535 (l=6, k=3).

l:=4; k:=2; [Binomial((l+k)*n,k*n)/(l*n+1): n in [0..20]]; /* where l is divisible by all the prime factors of k */

Table[Binomial[6 n, 2 n]/(4 n + 1), {n, 0, 20}]

a(n) = A047749(4*n-2) for n>0.
From Ilya Gutkovskiy, Jun 21 2018: (Start)
G.f.: 4F3(1/6,1/3,2/3,5/6; 1/2,3/4,5/4; 729*x/16).
a(n) ~ 3^(6*n+1/2)/(sqrt(Pi)*2^(4*n+7/2)*n^(3/2)). (End)

cp's OEIS Frontend

A233907 9*binomial(7*n+9, n)/(7*n+9).

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A233908 10*binomial(7*n+10,n)/(7*n+10).

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A346768 a(n) = Sum_{k=0..n} Stirling2(n,k) * binomial(7*k,k) / (6*k + 1).

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A349302 G.f. A(x) satisfies: A(x) = 1 / ((1 + x) * (1 - x * A(x)^6)).

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A349363 G.f. A(x) satisfies: A(x) = 1 + x * A(x)^7 / (1 + x).

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A365218 G.f. satisfies A(x) = 1 + x*A(x)^6 / (1 + x*A(x)^6).

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A346581 a(n) = (1/(7*n)) * Sum_{d|n} mu(n/d) * binomial(7*d,d).

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A346683 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(7*k,k) / (6*k + 1).

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A233907 9binomial(7n+9, n)/(7*n+9).

A233908 10binomial(7n+10,n)/(7*n+10).

A346768 a(n) = Sum_{k=0..n} Stirling2(n,k) * binomial(7k,k) / (6k + 1).

A365218 G.f. satisfies A(x) = 1 + xA(x)^6 / (1 + xA(x)^6).

A346581 a(n) = (1/(7n)) Sum_{d|n} mu(n/d) * binomial(7*d,d).

A346683 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(7k,k) / (6k + 1).

A235534 a(n) = binomial(6n, 2n) / (4*n + 1).