A059968 -id:A059968 - cp's OEIS frontend

A234525 Binomial(10n+2,n)/(5n+1).

1, 2, 21, 310, 5330, 99960, 1983049, 40919714, 869304150, 18885977110, 417663940540, 9371084905962, 212791660837756, 4880918206648000, 112925143575796455, 2632162372046272660, 61752662230350642670, 1457074607325333325524

Offset: 0

Tim Fulford, Dec 27 2013

nonn

Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), where p=10, 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
Wojciech Mlotkowski, Fuss-Catalan Numbers in Noncommutative Probability, Docum. Mathm. 15: 939-955.

Cf. A000108, A059968, A234526, A234527, A234528, A234529, A234570, A234571, A234573, A229963.

[Binomial(10*n+2, n)/(5*n+1): n in [0..30]]; // Vincenzo Librandi, Dec 27 2013

Table[Binomial[10 n + 2, n]/(5 n + 1), {n, 0, 40}] (* Vincenzo Librandi, Dec 27 2013 *)

PARI
```
a(n) = binomial(10*n+2,n)/(5*n+1);
```

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^5)^2+x*O(x^n)); polcoeff(B, n)}

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, where p=10, r=2.

a(n) = 2*binomial(10n+1,n-1)/n for n>0, a(0)=1. [Bruno Berselli, Jan 19 2014]

A234527 2binomial(10n+4,n)/(5*n+2).

Original entry on oeis.org

1, 4, 46, 704, 12341, 234260, 4685898, 97274544, 2075959314, 45262862788, 1003884090440, 22577660493024, 513698787408521, 11802947663348800, 273471432969603198, 6382396843322710560, 149902629054480517590, 3540479504783000035464

Offset: 0

Tim Fulford, Dec 27 2013

nonn

Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), where p=10, r=4.

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, A059968, A234525, A234526, A234528, A234529, A234570, A234571, A234573, A229963.

[2*Binomial(10*n+4, n)/(5*n+2): n in [0..30]]; // Vincenzo Librandi, Dec 28 2013

Table[2 Binomial[10 n + 4, n]/(5 n + 2), {n, 0, 30}] (* Vincenzo Librandi, Dec 28 2013 *)

PARI
```
a(n) = 2*binomial(10*n+4,n)/(5*n+2);
```

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(5/2))^4+x*O(x^n)); polcoeff(B, n)}

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, where p=10, r=4.

A234526 3binomial(10n+3,n)/(10*n+3).

Original entry on oeis.org

1, 3, 33, 496, 8610, 162435, 3235501, 66959532, 1425658806, 31026962395, 687124547340, 15434728080408, 350818684083868, 8053515040969200, 186457795206547635, 4348790005989493960, 102080931442008205230, 2409777235191897422982

Offset: 0

Tim Fulford, Dec 27 2013

nonn

Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), where p=10, 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
Wojciech Mlotkowski, Fuss-Catalan Numbers in Noncommutative Probability, Docum. Mathm. 15: 939-955.

Cf. A000108, A059968, A234525, A234527, A234528, A234529, A234570, A234571, A234573, A229963.

[3*Binomial(10*n+3, n)/(10*n+3): n in [0..30]]; // Vincenzo Librandi, Dec 28 2013

Table[3 Binomial[10 n + 3, n]/(10 n + 3), {n, 0, 30}] (* Vincenzo Librandi, Dec 28 2013 *)

PARI
```
a(n) = 3*binomial(10*n+3,n)/(10*n+3);
```

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(10/3))^3+x*O(x^n)); polcoeff(B, n)}

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, where p=10, r=3.

A234528 Binomial(10n+5,n)/(2n+1).

Original entry on oeis.org

1, 5, 60, 935, 16555, 316251, 6353760, 132321990, 2830853610, 61841702065, 1373736123760, 30935736733230, 704631080073635, 16204866668942000, 375762274309378440, 8775795659568727020, 206241872189050376550, 4873761343609509542490

Offset: 0

Tim Fulford, Dec 27 2013

nonn

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

Cf. A000108, A059968, A234525, A234526, A234527, A234529, A234570, A234571, A234573, A229963.

[Binomial(10*n+5, n)/(2*n+1): n in [0..30]]; // Vincenzo Librandi, Dec 28 2013

Table[Binomial[10 n + 5, n]/(2 n + 1), {n, 0, 30}] (* Vincenzo Librandi, Dec 28 2013 *)

PARI
```
a(n) = binomial(10*n+5,n)/(2*n+1);
```

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^2)^5+x*O(x^n)); polcoeff(B, n)}

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, where p=10, r=5.

A234529 3binomial(10n+6,n)/(5*n+3).

Original entry on oeis.org

1, 6, 75, 1190, 21285, 409266, 8259888, 172593900, 3701885490, 81033954430, 1803028662435, 40658396849388, 927146157991625, 21342995124948000, 495322997953271580, 11576581508367256920, 272239271289546497046, 6437043284012559773100

Offset: 0

Tim Fulford, Dec 27 2013

nonn

Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), where p=10, r=6.

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, A059968, A234525, A234526, A234527, A234528, A234570, A234571, A234573, A229963.

[3*Binomial(10*n+6, n)/(5*n+3): n in [0..30]]; // Vincenzo Librandi, Dec 27 2013

Table[3 Binomial[10 n + 6, n]/(5 n + 3), {n, 0, 30}] (* Vincenzo Librandi, Dec 27 2013 *)

PARI
```
a(n) = 3*binomial(10*n+6,n)/(5*n+3);
```

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(5/3))^6+x*O(x^n)); polcoeff(B, n)}

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, where p=10, r=6.

A234570 7binomial(10n+7,n)/(10*n+7).

Original entry on oeis.org

1, 7, 91, 1470, 26565, 514206, 10426416, 218618940, 4701550770, 103134123820, 2298706645235, 51909777109596, 1185134654128425, 27309853977084000, 634361032466470620, 14837590383963667320, 349163392095422769942, 8260872214482785042145, 196380752260155290992675

Offset: 0

Tim Fulford, Dec 28 2013

nonn

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

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, A059968, A234525, A234526, A234527, A234528, A234529, A234571, A234573, A229963.

[7*Binomial(10*n+7, n)/(10*n+7): n in [0..30]];

Table[7 Binomial[10 n + 7, n]/(10 n + 7), {n, 0, 30}]

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

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

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

A251590 a(n) = 10^(n-8) * (n+1)^(n-10) * (4782969n^8 + 50309748n^7 + 237013938n^6 + 655232760n^5 + 1166624361n^4 + 1374998212n^3 + 1051760172n^2 + 479277840n + 100000000).

Original entry on oeis.org

1, 1, 12, 376, 20384, 1604880, 167097280, 21724557760, 3393929393280, 619917062849920, 129708290461760000, 30601444681382400000, 8038933665661600000000, 2327544788978773504000000, 736481767572932400000000000, 252867957872989831168000000000, 93638020040848371127040000000000

Offset: 0

Paul D. Hanna, Dec 06 2014

nonn

			E.g.f.: A(x) = 1 + x + 12*x^2/2! + 376*x^3/3! + 20384*x^4/4! + 1604880*x^5/5! +...
such that A(x) = exp( 10*x*A(x) * G(x*A(x))^9 ) / G(x*A(x))^9
where G(x) = 1 + x*G(x)^10 is the g.f. of A059968:
G(x) = 1 + x + 10*x^2 + 145*x^3 + 2470*x^4 + 46060*x^5 + 910252*x^6 +...
RELATED SERIES.
Note that A(x) = F(x*A(x)) where F(x) = exp(9*x*G(x)^8)/G(x)^8,
F(x) = 1 + x + 10*x^2/2! + 280*x^3/3! + 13960*x^4/4! + 1023760*x^5/5! +...
is the e.g.f. of A251580.

G. C. Greubel, Table of n, a(n) for n = 0..289

Cf. A251580, A059968.

Cf. Variants: A251583, A251584, A251585, A251586, A251587, A251588, A251589.

[10^(n - 8)*(n + 1)^(n - 10)*(4782969*n^8 + 50309748*n^7 + 237013938*n^6 + 655232760*n^5 + 1166624361*n^4 + 1374998212*n^3 + 1051760172*n^2 + 479277840*n + 100000000): n in [0..50]]; // G. C. Greubel, Nov 13 2017

Table[10^(n - 8)*(n + 1)^(n - 10)*(4782969*n^8 + 50309748*n^7 + 237013938*n^6 + 655232760*n^5 + 1166624361*n^4 + 1374998212*n^3 + 1051760172*n^2 + 479277840*n + 100000000), {n, 0, 50}] (* G. C. Greubel, Nov 13 2017 *)

{a(n) = 10^(n-8) * (n+1)^(n-10) * (4782969*n^8 + 50309748*n^7 + 237013938*n^6 + 655232760*n^5 + 1166624361*n^4 + 1374998212*n^3 + 1051760172*n^2 + 479277840*n + 100000000)}
for(n=0,20,print1(a(n),", "))

{a(n) = local(G=1,A=1); for(i=1,n, G=1+x*G^10 +x*O(x^n));
for(i=1,n, A = exp(10*x*A * subst(G^9,x,x*A) ) / subst(G^9,x,x*A) ); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))

Let G(x) = 1 + x*G(x)^10 be the g.f. of A059968, then the e.g.f. A(x) of this sequence satisfies:
(1) A(x) = exp( 10*x*A(x) * G(x*A(x))^9 ) / G(x*A(x))^9.
(2) A(x) = F(x*A(x)) where F(x) = exp(10*x*G(x)^9)/G(x)^9 is the e.g.f. of A251580.
(3) a(n) = [x^n/n!] F(x)^(n+1)/(n+1) where F(x) is the e.g.f. of A251580.
E.g.f.: -LambertW(-10*x) * (10 + LambertW(-10*x))^9 / (x*10^10). - Vaclav Kotesovec, Dec 07 2014

A251700 a(n) = (8n+1) (9n+1)^(n-2) 10^n.

Original entry on oeis.org

1, 9, 1700, 700000, 451770000, 399077600000, 448380625000000, 612032839680000000, 983672470878500000000, 1819783935144064000000000, 3809045473682732010000000000, 8900000000000000000000000000000, 22963427643543537169897000000000000, 64847222511564256853913600000000000000

Offset: 0

Paul D. Hanna, Dec 07 2014

nonn

In general, sequence a(n) = (b*n+1)*(c*n+1)^(n-2)*d^n, b > 0, c > 0, d > 0, has e.g.f. (1 + (1-b/c)*LambertW(-c*d*x)/(c+1)) * (-LambertW(-c*d*x)/(c*d*x))^(1/c). - Vaclav Kotesovec, Dec 07 2014

			E.g.f.: A(x) = 1 + 9*x + 1700*x^2/2! + 700000*x^3/3! + 451770000*x^4/4! + 399077600000*x^5/5! +...
such that A(x) = exp( 10*x*A(x)^9 * G(x*A(x)^9)^9 ) / G(x*A(x)^9),
where G(x) = 1 + x*G(x)^10 is the g.f. A059968:
G(x) = 1 + x + 10*x^2 + 145*x^3 + 2470*x^4 + 46060*x^5 + 910252*x^6 +...
Also, e.g.f. A(x) satisfies A(x) = F(x*A(x)^9) where
F(x) = 1 + 9*x + 242*x^2/2! + 11824*x^3/3! + 856824*x^4/4! + 82986080*x^5/5! +...
F(x) = exp( 10*x*G(x)^9 ) / G(x) is the e.g.f. of A251670.

G. C. Greubel, Table of n, a(n) for n = 0..230

Cf. A251670, A059968.

Cf. Variants: A127670, A251693, A251694, A251695, A251696, A251697, A251698, A251699.

[(8*n + 1)*(9*n + 1)^(n - 2)*10^n: n in [0..50]]; // G. C. Greubel, Nov 14 2017

Table[(8*n + 1)*(9*n + 1)^(n - 2)*10^n, {n, 0, 50}] (* G. C. Greubel, Nov 14 2017 *)

{a(n) = (8*n+1) * (9*n+1)^(n-2) * 10^n}
for(n=0,20,print1(a(n),", "))

{a(n)=local(G=1,A=1); for(i=0, n, G = 1 + x*G^10 +x*O(x^n));
A = ( serreverse( x*G^9 / exp(90*x*G^9) )/x )^(1/9); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))

Let G(x) = 1 + x*G(x)^10 be the g.f. of A059968, then the e.g.f. A(x) of this sequence satisfies:
(1) A(x) = exp( 10*x*A(x)^9 * G(x*A(x)^9)^9 ) / G(x*A(x)^9).
(2) A(x) = F(x*A(x)^9) where F(x) = exp(10*x*G(x)^9)/G(x) is the e.g.f. of A251670.
(3) A(x) = ( Series_Reversion( x*G(x)^9 / exp(90*x*G(x)^9) )/x )^(1/9).
E.g.f.: (-LambertW(-90*x)/(90*x))^(1/9) * (1 + LambertW(-90*x)/90). - Vaclav Kotesovec, Dec 07 2014

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)

A235535 a(n) = binomial(9n, 3n) / (6*n + 1).

Original entry on oeis.org

1, 12, 1428, 246675, 50067108, 11124755664, 2619631042665, 642312451217745, 162250238001816900, 41932353590942745504, 11034966795189838872624, 2946924270225408943665279, 796607831560617902288322405, 217550867863011281855594752680

Offset: 0

Bruno Berselli, Jan 12 2014

This is the case l=6, k=3 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.

Also, the sequence follows A002296 and A235536, namely binomial(7*n,n)/(6*n+1) and binomial(8*n,2*n)/(6*n+1); naturally, even binomial(10*n,4*n)/(6*n+1) is always integer.

Robert Israel, Table of n, a(n) for n = 0..403
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), A235534 (l=4, k=2), A235536 (l=6, k=2), A187357 (l=3, k=3), this sequence (l=6, k=3).

l:=6; k:=3; [Binomial((l+k)*n,k*n)/(l*n+1): n in [0..20]]; /* here l is divisible by all the prime factors of k */

seq(binomial(9*n,3*n)/(6*n+1), n=0..30); # Robert Israel, Feb 15 2021

Table[Binomial[9 n, 3 n]/(6 n + 1), {n, 0, 20}]

a(n) = A001764(3*n) = A047749(6*n).
From Ilya Gutkovskiy, Jun 21 2018: (Start)
G.f.: 6F5(1/9,2/9,4/9,5/9,7/9,8/9; 1/3,1/2,2/3,5/6,7/6; 19683*x/64).
a(n) ~ 3^(9*n-1)/(sqrt(Pi)*4^(3*n+1)*n^(3/2)). (End)
D-finite with recurrence 8*(6*n + 5)*(2*n + 1)*(n + 1)*(3*n + 2)*(3*n + 1)*(6*n + 7)*a(n + 1) = 3*(9*n + 8)*(9*n + 7)*(9*n + 5)*(9*n + 4)*(9*n + 2)*(9*n + 1)*a(n). - Robert Israel, Feb 15 2021

cp's OEIS Frontend

A234525 Binomial(10*n+2,n)/(5*n+1).

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A234527 2*binomial(10*n+4,n)/(5*n+2).

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A234526 3*binomial(10*n+3,n)/(10*n+3).

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A234528 Binomial(10*n+5,n)/(2*n+1).

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A234529 3*binomial(10*n+6,n)/(5*n+3).

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A234570 7*binomial(10*n+7,n)/(10*n+7).

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A251590 a(n) = 10^(n-8) * (n+1)^(n-10) * (4782969*n^8 + 50309748*n^7 + 237013938*n^6 + 655232760*n^5 + 1166624361*n^4 + 1374998212*n^3 + 1051760172*n^2 + 479277840*n + 100000000).

A234525 Binomial(10n+2,n)/(5n+1).

A234527 2binomial(10n+4,n)/(5*n+2).

A234526 3binomial(10n+3,n)/(10*n+3).

A234528 Binomial(10n+5,n)/(2n+1).

A234529 3binomial(10n+6,n)/(5*n+3).

A234570 7binomial(10n+7,n)/(10*n+7).

A251590 a(n) = 10^(n-8) * (n+1)^(n-10) * (4782969n^8 + 50309748n^7 + 237013938n^6 + 655232760n^5 + 1166624361n^4 + 1374998212n^3 + 1051760172n^2 + 479277840n + 100000000).

A251700 a(n) = (8n+1) (9n+1)^(n-2) 10^n.

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

A235535 a(n) = binomial(9n, 3n) / (6*n + 1).