A034975 -id:A034975 - cp's OEIS frontend

A049210 a(n) = -Product_{k=0..n} (8*k-1); octo-factorial numbers.

1, 7, 105, 2415, 74865, 2919735, 137227545, 7547514975, 475493443425, 33760034483175, 2667042724170825, 232032717002861775, 22043108115271868625, 2270440135873002468375, 252018855081903273989625, 29990243754746489604765375, 3808760956852804179805202625

Offset: 0

Wolfdieter Lang

Seiichi Manyama, Table of n, a(n) for n = 0..333
Index entries for sequences related to factorial numbers.

Cf. A034975, A048994, A051187.

Sequences of the form m^n*Pochhammer((m-1)/m, n): A000007 (m=1), A001147 (m=2), A008544 (m=3), A008545 (m=4), A008546 (m=5), A008543 (m=6), A049209 (m=7), this sequence (m=8), A049211 (m=9), A049212 (m=10), A254322 (m=11), A346896 (m=12).

m:=8; [Round(m^n*Gamma(n +(m-1)/m)/Gamma((m-1)/m)): n in [0..30]]; // G. C. Greubel, Feb 16 2022

FoldList[Times,1,8*Range[20]-1] (* Harvey P. Dale, Aug 03 2014 *)
CoefficientList[Series[(1-8*x)^(-7/8),{x,0,20}],x] * Range[0,20]! (* Vaclav Kotesovec, Jan 28 2015 *)

a(n) = -prod(k=0, n, 8*k-1); \\ Michel Marcus, Jan 08 2015

m=8; [m^n*rising_factorial((m-1)/m, n) for n in (0..30)] # G. C. Greubel, Feb 16 2022

a(n) = 7*A034975(n) = (8*n-1)(!^8), n >= 1, a(0) = 1.
G.f.: 1/(1-7*x/(1-8*x/(1-15*x/(1-16*x/(1-23*x/(1-24*x/(1-31*x/(1-32*x/(1-... (continued fraction). - Philippe Deléham, Jan 07 2012
a(n) = (-1)^n*Sum_{k=0..n} 8^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*(8*k-1)/(1 - x*(8*k+8)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 20 2013
a(n) = 8^n*Gamma(n+7/8)/Gamma(7/8). - R. J. Mathar, Mar 20 2013
E.g.f: (1-8*x)^(-7/8). - Vaclav Kotesovec, Jan 28 2015
G.f.: 1/(1-7*x-56*x^2/(1-23*x-240*x^2/(1-39*x-552*x^2/(1-55*x-992*x^2/(1-71*x-1560*x^2/(1-... )))))) (Jacobi continued fraction). - Nikolaos Pantelidis, Dec 09 2020
G.f.: 1/G(0) where G(k) = 1 - (16*k+7)*x - 8*(k+1)*(8*k+7)*x^2/G(k+1); (continued fraction). - Nikolaos Pantelidis, Dec 19 2020
Sum_{n>=0} 1/a(n) = 1 + (e/8)^(1/8)*(Gamma(7/8) - Gamma(7/8, 1/8)). - Amiram Eldar, Dec 20 2022

A034976 One eighth of octo-factorial numbers.

Original entry on oeis.org

1, 16, 384, 12288, 491520, 23592960, 1321205760, 84557168640, 6088116142080, 487049291366400, 42860337640243200, 4114592413463347200, 427917611000188108800, 47926772432021068185600, 5751212691842528182272000, 736155224555843607330816000

Offset: 1

Wolfdieter Lang

G. C. Greubel, Table of n, a(n) for n = 1..325
Index entries for sequences related to factorial numbers.

Cf. A034908, A034909, A034910, A034911, A034912, A034975, A045755, A051189.

[8^(n-1)*Factorial(n): n in [1..40]]; // G. C. Greubel, Oct 20 2022

Table[8^(n-1)*n!, {n,40}] (* G. C. Greubel, Oct 20 2022 *)

[8^(n-1)*factorial(n) for n in range(1,40)] # G. C. Greubel, Oct 20 2022

8*a(n) = (8*n)!^8 = Product_{j=1..n} 8*j = 8^n*n!.
E.g.f.: (-1+(1-8*x)^(-1))/8.
G.f.: x/(1-16*x/(1-8*x/(1-24*x/(1-16*x/(1-32*x/(1-24*x/(1-40*x/(1-32*x/(1-... (continued fraction). - Philippe Deléham, Jan 07 2012
From Amiram Eldar, Jan 08 2022: (Start)
Sum_{n>=1} 1/a(n) = 8*(exp(1/8)-1).
Sum_{n>=1} (-1)^(n+1)/a(n) = 8*(1-exp(-1/8)). (End)

A025753 8th-order Patalan numbers (generalization of Catalan numbers).

Original entry on oeis.org

1, 1, 28, 1120, 51520, 2555392, 132880384, 7137574912, 392566620160, 21983730728960, 1248675905404928, 71742106565083136, 4161042180774821888, 243260927491451125760, 14317643160925409116160, 847604475126784219676672, 50432466270043661070761984, 3014081513550844685170245632

Offset: 0

Olivier Gérard

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seq., Vol. 3 (2000), Article 00.2.4.
Elżbieta Liszewska and Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.
Thomas M. Richardson, The Super Patalan Numbers, J. Int. Seq. 18 (2015), Article 15.3.3; arXiv preprint, arXiv:1410.5880 [math.CO], 2014.

Cf. A034975, A203146.

CoefficientList[Series[(9 - (1 - 64*x)^(1/8))/8, {x, 0, 20}], x] (* Vincenzo Librandi, Dec 29 2012 *)
a[n_] := 64^(n-1) * Pochhammer[7/8, n-1]/n!; a[0] = 1; Array[a, 20, 0] (* Amiram Eldar, Aug 20 2025 *)

G.f.: (9-(1-64*x)^(1/9))/8.
a(n) = 8^(n-1)*7*A034975(n-1)/n!, n >= 2, where 7*A034975(n-1)= (8*n-9)!^8 = Product_{j=2..n} (8*j - 9). - Wolfdieter Lang
a(n) ~ 64^(n-1) / (Gamma(7/8) * n^(9/8)). - Amiram Eldar, Aug 20 2025

A053115 a(n) = ((8n+10)(!^8))/20, related to A034908 ((8n+2)(!^8) octo- or 8-factorials).

Original entry on oeis.org

1, 18, 468, 15912, 668304, 33415200, 1938081600, 127913385600, 9465590534400, 776178423820800, 69856058143872000, 6845893698099456000, 725664731998542336000, 82725779447833826304000

Offset: 0

Wolfdieter Lang

Row m=10 of the array A(9; m,n) := ((8*n+m)(!^8))/m(!^8), m >= 0, n >= 0.

G. C. Greubel, Table of n, a(n) for n = 0..332

Cf. A051189, A045755, A034908-12, A034975-6, A053114 (rows m=0..9).

m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1/(1-8*x)^(9/4))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Aug 26 2018

s=1;lst={s};Do[s+=n*s;AppendTo[lst, s], {n, 17, 5!, 8}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)
With[{nmax = 50}, CoefficientList[Series[1/(1 - 8*x)^(9/4), {x, 0, nmax}], x]*Range[0, nmax]!] (* G. C. Greubel, Aug 26 2018 *)

x='x+O('x^25); Vec(serlaplace(1/(1-8*x)^(9/4))) \\ G. C. Greubel, Aug 26 2018

a(n) = ((8*n+10)(!^8))/10(!^8) = A034908(n+2)/10.
E.g.f.: 1/(1-8*x)^(9/4).
G.f.: 1/(1-18x/(1-8x/(1-26x/(1-16x/(1-34x/(1-24x/(1-42x/(1-32x/(1-... (continued fraction). - Philippe Deléham, Jan 07 2012

A053114 a(n) = ((8n+9)(!^8))/9, related to A045755 ((8n+1)(!^8) octo- or 8-factorials).

Original entry on oeis.org

1, 17, 425, 14025, 575025, 28176225, 1606044825, 104392913625, 7620682694625, 617275298264625, 54937501545551625, 5328937649918507625, 559538453241443300625, 63227845216283092970625

Offset: 0

Wolfdieter Lang

Row m=9 of the array A(9; m,n) := ((8*n+m)(!^8))/m(!^8), m >= 0, n >= 0.

G. C. Greubel, Table of n, a(n) for n = 0..332

Cf. A051189, A045755, A034908-12, A034975-6 (rows m=0..8).

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1/(1-8*x)^(17/8))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Aug 16 2018

s=1;lst={s};Do[s+=n*s;AppendTo[lst, s], {n, 16, 5!, 8}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)
With[{nn = 30}, CoefficientList[Series[1/(1 - 8*x)^(17/8), {x, 0, nn}], x]*Range[0, nn]!] (* G. C. Greubel, Aug 16 2018 *)

x='x+O('x^30); Vec(serlaplace(1/(1-8*x)^(17/8))) \\ G. C. Greubel, Aug 16 2018

a(n) = ((8*n+9)(!^8))/9(!^8) = A045755(n+2)/9.
E.g.f.: 1/(1-8*x)^(17/8).
G.f.: 1/(1-17x/(1-8x/(1-25x/(1-16x/(1-33x/(1-24x/(1-41x/(1-32x/(1-... (continued fraction). - Philippe Deléham, Jan 07 2012

cp's OEIS Frontend

A049210 a(n) = -Product_{k=0..n} (8*k-1); octo-factorial numbers.

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A034976 One eighth of octo-factorial numbers.

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A025753 8th-order Patalan numbers (generalization of Catalan numbers).

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A053115 a(n) = ((8*n+10)(!^8))/20, related to A034908 ((8*n+2)(!^8) octo- or 8-factorials).

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Magma

Mathematica

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Formula

A053114 a(n) = ((8*n+9)(!^8))/9, related to A045755 ((8*n+1)(!^8) octo- or 8-factorials).

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Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A053115 a(n) = ((8n+10)(!^8))/20, related to A034908 ((8n+2)(!^8) octo- or 8-factorials).

A053114 a(n) = ((8n+9)(!^8))/9, related to A045755 ((8n+1)(!^8) octo- or 8-factorials).