A254287 -id:A254287 - cp's OEIS frontend

A008546 Quintuple factorial numbers: Product_{k = 0..n-1} (5*k + 4).

1, 4, 36, 504, 9576, 229824, 6664896, 226606464, 8837652096, 388856692224, 19053977918976, 1028914807624704, 60705973649857536, 3885182313590882304, 268077579637770878976, 19837740893195045044224, 1567181530562408558493696, 131643248567242318913470464

Offset: 0

Joe Keane (jgk(AT)jgk.org)

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), Article 00.2.4.
Keiichi Shigechi, On the lattice of weighted partitions, arXiv:2212.14666 [math.CO], 2022. See p. 27.

Cf. A011801, A034301, A048994.

Cf. A008548, A047055, A047056, A051150, A052562, A254287.

List([0..20], n-> Product([0..n-1], k-> 5*k+4 )); # G. C. Greubel, Aug 20 2019

[1] cat [(&*[5*k+4: k in [0..n-1]]): n in [1..20]]; // G. C. Greubel, Aug 20 2019

Maple
```
f:= n-> product(5*k+4, k=0..n-1);
```

FoldList[Times, 1, 5Range[0, 20] + 4] (* Vincenzo Librandi, Jun 10 2013 *)
CoefficientList[Series[(1 - 5x)^(-4/5), {x, 0, 20}], x] Range[0, 20]! (* Vaclav Kotesovec, Jan 28 2015 *)
Table[5^n Pochhammer[4/5, n], {n, 0, 20}] (* G. C. Greubel, Aug 20 2019 *)

vector(20, n, n--; prod(j=0,n-1, 5*j+4) ) \\ G. C. Greubel, Aug 20 2019

[5^n*rising_factorial(4/5, n) for n in (0..20)] # G. C. Greubel, Aug 20 2019

a(n) = 4*A034301(n) = (5*n - 1)(!^5), n >= 1, with a(0) = 1.
a(n) = A011801(n + 1, 1) (first column of triangle).
a(n) ~ (sqrt(2*Pi)/Gamma(4/5))*n^(n + 3/10)*(5/e)^n*(1 + 1/(300*n) + ...). - Joe Keane (jgk(AT)jgk.org), Nov 24 2001
G.f.: 1/(1 - 4*x/(1 - 5*x/(1 - 9*x/(1 - 10*x/(1 - 14*x/(1 - 15*x/(1 - 19*x/(1 - 20*x/(1 - 24*x/(1 - ... (continued fraction). - Philippe Deléham, Jan 08 2012
a(n) = (-1)^n*Sum_{k = 0..n} 5^k*s(n + 1, n + 1 - k), where s(n, k) are the Stirling numbers of the first kind, A048994. - Mircea Merca, May 03 2012
G.f.: ( 1 - 1/Q(0) )/x where Q(k) = 1 - x*(5*k - 1)/(1 - x*(5*k + 5)/Q(k + 1) ); (continued fraction); e.g.f. (1 - 5*x)^(-4/5). - Sergei N. Gladkovskii, Mar 20 2013
G.f.: 1/x - G(0)/(2*x), where G(k) = 1 + 1/(1 - x*(5*k - 1)/(x*(5*k - 1) + 1/G(k + 1))); (continued fraction). - Sergei N. Gladkovskii, May 27 2013
a(n) = 5^n * Gamma(n + 4/5) / Gamma(4/5). - Vaclav Kotesovec, Jan 28 2015
a(n) + (-5*n + 1)*a(n - 1) = 0. - R. J. Mathar, Sep 04 2016
G.f.: 1/(1 - 4*x - 20*x^2/(1 - 14*x - 90*x^2/(1 - 24*x - 210*x^2/(1 - 34*x - 380*x^2/(1 - 44*x - 600*x^2/(1 - 54*x - 870*x^2/(1 - ...))))))) (Jacobi continued fraction). - Nikolaos Pantelidis, Feb 29 2020
Sum_{n>=0} 1/a(n) = 1 + (e/5)^(1/5)*(Gamma(4/5) - Gamma(4/5, 1/5)). - Amiram Eldar, Dec 19 2022

A254322 Expansion of e.g.f.: (1-11*x)^(-10/11).

Original entry on oeis.org

1, 10, 210, 6720, 288960, 15603840, 1014249600, 77082969600, 6706218355200, 657209398809600, 71635824470246400, 8596298936429568000, 1126115160672273408000, 159908352815462823936000, 24465977980765812062208000, 4012420388845593178202112000

Offset: 0

Vaclav Kotesovec, Jan 28 2015

Generally, for k > 1, if e.g.f. = (1-k*x)^(-(k-1)/k) then a(n) ~ n! * k^n / (n^(1/k) * Gamma((k-1)/k)).

G. C. Greubel, Table of n, a(n) for n = 0..300

Sequences of the form k^n*Pochhammer((k-1)/k, n): A000007 (k=1), A001147 (k=2), A008544 (k=3), A008545 (k=4), A008546 (k=5), A008543 (k=6), A049209 (k=7), A049210 (k=8), A049211 (k=9), A049212 (k=10), this sequence (k=11), A346896 (k=12).

Cf. A000108, A254282, A254286, A254287.

m=11; [Round(m^n*Gamma(n +(m-1)/m)/Gamma((m-1)/m)): n in [0..20]]; // G. C. Greubel, Feb 08 2022

CoefficientList[Series[(1-11*x)^(-10/11), {x, 0, 20}], x] * Range[0, 20]!
FullSimplify[Table[11^n * Gamma[n+10/11] / Gamma[10/11], {n, 0, 18}]]

m=11; [m^n*rising_factorial((m-1)/m, n) for n in (0..20)] # G. C. Greubel, Feb 08 2022

D-finite with recurrence: a(0) = 1; a(n) = (11*n-1) * a(n-1) for n > 0. [corrected by Georg Fischer, Dec 23 2019]
a(n) = 11^n * Gamma(n+10/11) / Gamma(10/11).
a(n) ~ n! * 11^n / (n^(1/11) * Gamma(10/11)).
From Nikolaos Pantelidis, Jan 17 2021: (Start)
G.f.: 1/G(0) where G(k) = 1 - (22*k+10)*x - 11*(k+1)*(11*k+10)*x^2/G(k+1) (continued fraction).
G.f.: 1/(1-10*x-110*x^2/(1-32*x-462*x^2/(1-54*x-1056*x^2/(1-76*x-1892*x^2/(1-98*x-2970*x^2/(1-...)))))) (Jacobi continued fraction).
G.f.: 1/Q(0) where Q(k) = 1 - x*(11*k+10)/(1 - x*(11*k+11)/Q(k+1)) (continued fraction).
G.f.: 1/(1-10*x/(1-11*x/(1-21*x/(1-22*x/(1-32*x/(1-33*x/(1-43*x/(1-44*x/(1-54*x/(1-55*x/(1-...))))))))))) (Stieltjes continued fraction).
(End)
G.f.: hypergeometric2F0([1, 10/11], [--], 11*x). - G. C. Greubel, Feb 08 2022
Sum_{n>=0} 1/a(n) = 1 + (e/11)^(1/11)*(Gamma(10/11) - Gamma(10/11, 1/11)). - Amiram Eldar, Dec 22 2022

A254286 Expansion of (1 - (1-256x)^(1/4)) / (64x).

Original entry on oeis.org

1, 96, 14336, 2523136, 484442112, 98180268032, 20645907791872, 4459516083044352, 983075545417777152, 220208922173582082048, 49967406340478261526528, 11459191854083014643417088, 2651480699775516003646046208, 618173786004806016850049630208

Offset: 0

Vaclav Kotesovec, Jan 27 2015

G. C. Greubel, Table of n, a(n) for n = 0..250

Cf. A000108, A008545, A025749, A048779, A254282, A254287.

[Round(2^(8*n)*Gamma(n+3/4)/(Gamma(3/4)*Gamma(n+2))): n in [0..30]]; // G. C. Greubel, Aug 10 2022

CoefficientList[Series[(1-(1-256*x)^(1/4)) / (64*x),{x,0,20}],x]
CoefficientList[Series[Hypergeometric1F1[3/4,2,256*x],{x,0,20}],x] * Range[0,20]! (* Vaclav Kotesovec, Jan 28 2015 *)

[2^(8*n)*rising_factorial(3/4,n)/factorial(n+1) for n in (0..30)] # G. C. Greubel, Aug 10 2022

G.f.: (1 - (1-256*x)^(1/4)) / (64*x).
a(n) ~ 256^n / (Gamma(3/4) * n^(5/4)).
Recurrence: (n+1)*a(n) = 64*(4*n-1)*a(n-1).
a(n) = 256^n * Gamma(n+3/4) / (Gamma(3/4) * Gamma(n+2)).
E.g.f.: hypergeom([3/4], [2], 256*x). - Vaclav Kotesovec, Jan 28 2015
From Peter Bala, Sep 01 2017: (Start)
a(n) = (-1)^n*binomial(1/4, n+1)*4^(4*n+1). Cf. A000108(n) = (-1)^n*binomial(1/2, n+1)*2^(2*n+1).
a(n) = 16^n*A025749(n+1); a(n) = 32^n*A048779(n+1).
(End)

A254282 Expansion of (1 - (1 - 27x)^(1/3)) / (9x).

Original entry on oeis.org

1, 9, 135, 2430, 48114, 1010394, 22084326, 496897335, 11428638705, 267430145697, 6345388002447, 152289312058728, 3690087176807640, 90143558176300920, 2217531531137002632, 54883905395640815142, 1365640704844474400298, 34141017621111860007450

Offset: 0

Vaclav Kotesovec, Jan 27 2015

Harvey P. Dale, Table of n, a(n) for n = 0..700

Cf. A000108, A008544, A254286, A254287, A025748.

[Round(3^(3*n)*Gamma(n+2/3)/(Gamma(2/3)*Gamma(n+2))): n in [0..30]]; // G. C. Greubel, Aug 10 2022

CoefficientList[Series[(1-(1-27*x)^(1/3))/(9*x),{x,0,20}],x]
CoefficientList[Series[Hypergeometric1F1[2/3,2,27*x],{x,0,20}],x] * Range[0,20]! (* Vaclav Kotesovec, Jan 28 2015 *)
nxt[{n_,a_}]:={n+1,((27n+18)*a)/(n+2)}; NestList[nxt,{0,1},20][[All,2]] (* Harvey P. Dale, Jun 03 2019 *)

[3^(3*n)*rising_factorial(2/3,n)/factorial(n+1) for n in (0..30)] # G. C. Greubel, Aug 10 2022

G.f.: (1 - (1-27*x)^(1/3)) / (9*x).
a(n) ~ 3^(3*n) / (Gamma(2/3) * n^(4/3)).
Recurrence: (n+1)*a(n) = 9*(3*n-1)*a(n-1).
a(n) = 27^n * Gamma(n+2/3) / (Gamma(2/3) * Gamma(n+2)).
E.g.f.: hypergeom([2/3], [2], 27*x). - Vaclav Kotesovec, Jan 28 2015
From Peter Bala, Sep 01 2017: (Start)
a(n) = (-1)^n*binomial(1/3, n+1)*3^(3*n+1). Cf. A000108(n) = (-1)^n*binomial(1/2, n+1)*2^(2*n+1).
a(n) = 3^n*A025748(n+1). (End)

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A008546 Quintuple factorial numbers: Product_{k = 0..n-1} (5*k + 4).

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A254322 Expansion of e.g.f.: (1-11*x)^(-10/11).

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A254286 Expansion of (1 - (1-256*x)^(1/4)) / (64*x).

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A254282 Expansion of (1 - (1 - 27*x)^(1/3)) / (9*x).

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Magma

Mathematica

SageMath

Formula

A254286 Expansion of (1 - (1-256x)^(1/4)) / (64x).

A254282 Expansion of (1 - (1 - 27x)^(1/3)) / (9x).