A045755 -id:A045755 - cp's OEIS frontend

A034908 One half of octo-factorial numbers.

1, 10, 180, 4680, 159120, 6683040, 334152000, 19380816000, 1279133856000, 94655905344000, 7761784238208000, 698560581438720000, 68458936980994560000, 7256647319985423360000, 827257794478338263040000, 100925450926357268090880000, 13120308620426444851814400000

Offset: 1

Wolfdieter Lang

G. C. Greubel, Table of n, a(n) for n = 1..333
Index entries for sequences related to factorial numbers.

Cf. A007696, A045755, A034909, A039410, A039411, A039412, A084948.

[(&*[(8*k-6): k in [1..n]])/2: n in [1..30]]; // G. C. Greubel, Feb 26 2018

Drop[With[{nn = 40}, CoefficientList[Series[(-1 + (1 - 8*x)^(-1/4))/2, {x, 0, nn}], x]*Range[0, nn]!], 1] (* G. C. Greubel, Feb 26 2018 *)

my(x='x+O('x^30)); Vec(serlaplace((-1+(1-8*x)^(-1/4))/2)) \\ G. C. Greubel, Feb 26 2018

2*a(n) = (8*n-6)(!^8) := product(8*j-6, j=1..n) = 2^n*A007696(n); compare with A007696(n) = (4*n-3)(!^4) := product(4*j-3, j=1..n).
E.g.f.: (-1+(1-8*x)^(-1/4))/2.
G.f.: x/(1-10x/(1-8x/(1-18x/(1-16x/(1-26x/(1-24x/(1-34x/(1-32x/(1-... (continued fraction). - Philippe Deléham, Jan 07 2012
From Amiram Eldar, Dec 20 2022: (Start)
a(n) = A084948(n)/2.
Sum_{n>=1} 1/a(n) = 2*(e/8^6)^(1/8)*(Gamma(1/4) - Gamma(1/4, 1/8)). (End)

Terms a(16) onward added by G. C. Greubel, Feb 26 2018

A256268 Table of k-fold factorials, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 6, 3, 1, 1, 1, 24, 15, 4, 1, 1, 1, 120, 105, 28, 5, 1, 1, 1, 720, 945, 280, 45, 6, 1, 1, 1, 5040, 10395, 3640, 585, 66, 7, 1, 1, 1, 40320, 135135, 58240, 9945, 1056, 91, 8, 1, 1, 1, 362880, 2027025, 1106560, 208845, 22176, 1729, 120, 9, 1, 1

Offset: 0

Philippe Deléham, Jun 01 2015

A variant of A142589.

			1  1   1    1     1       1         1... A000012
1  1   2    6    24     120       720... A000142
1  1   3   15   105     945     10395... A001147
1  1   4   28   280    3640     58240... A007559
1  1   5   45   585    9945    208845... A007696
1  1   6   66  1056   22176    576576... A008548
1  1   7   91  1729   43225   1339975... A008542
1  1   8  120  2640   76560   2756160... A045754
1  1   9  153  3825  126225   5175225... A045755
1  1  10  190  5320  196840   9054640... A045756
1  1  11  231  7161  293601  14977651... A144773

G. C. Greubel, Antidiagonal rows n = 0..100, flattened

Cf. Columns : A000012, A000012, A000384, A011199, A011245.
Cf. Diagonals : A092985, A076111, A158887.
Cf. A048994, A132393.
Cf. A000142 ("1-fold"), A001147 (2-fold), A007559 (3), A007696 (4), A008548 (5), A008542 (6), A045754 (7), A045755 (8), A045756 (9), A144773 (10)

Flat(List([0..12], n-> List([0..n], k-> Product([0..n-k-1], j-> j*k+1) ))); # G. C. Greubel, Mar 04 2020

function T(n,k)
  if k eq 0 or n eq 0 then return 1;
  else return (&*[j*k+1: j in [0..n-1]]);
  end if; return T; end function;
[T(n-k,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 04 2020

seq(seq( mul(j*k+1, j=0..n-k-1), k=0..n), n=0..12); # G. C. Greubel, Mar 04 2020

T[n_, k_]= Product[j*k+1, {j,0,n-1}]; Table[T[n-k,k], {n,0,12}, {k, 0, n}]//Flatten (* G. C. Greubel, Mar 04 2020 *)

T(n,k) = prod(j=0, n-1, j*k+1);
for(n=0,12, for(k=0, n, print1(T(n-k, k), ", "))) \\ G. C. Greubel, Mar 04 2020

[[ product(j*k+1 for j in (0..n-k-1)) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Mar 04 2020

A(n, k) = (-n)^k*FallingFactorial(-1/n, k) for n >= 1. - Peter Luschny, Dec 21 2021

A034909 One third of octo-factorial numbers.

Original entry on oeis.org

1, 11, 209, 5643, 197505, 8492715, 433128465, 25554579435, 1712156822145, 128411761660875, 10658176217852625, 969894035824588875, 96019509546634298625, 10274087521489869952875, 1181520064971335044580625, 145326967991474210483416875, 19037832806883121573327610625

Offset: 1

Wolfdieter Lang

Index entries for sequences related to factorial numbers.

Cf. A045755, A034908, A034909, A034910, A034911, A034912, A144756.

3*a(n) = (8*n-5)(!^8) = product(8*j-5, j=1..n).
E.g.f.: (-1+(1-8*x)^(-3/8))/3.
G.f.: x/(1-11x/(1-8x/(1-19x/(1-16x/(1-27x/(1-24x/(1-35x/(1-32x/(1-... (continued fraction). - Philippe Deléham, Jan 07 2012
From Amiram Eldar, Dec 20 2022: (Start)
a(n) = A144756(n)/3.
Sum_{n>=1} 1/a(n) = 3*(e/8^5)^(1/8)*(Gamma(3/8) - Gamma(3/8, 1/8)). (End)

A034912 One sixth of octo-factorial numbers.

Original entry on oeis.org

1, 14, 308, 9240, 351120, 16151520, 872182080, 54075288960, 3785270227200, 295251077721600, 25391592684057600, 2386809712301414400, 243454590654744268800, 26780004972021869568000, 3160040586698580609024000, 398165113924021156737024000, 53354125265818835002761216000

Offset: 1

Wolfdieter Lang

Robert Israel, Table of n, a(n) for n = 1..333
Index entries for sequences related to factorial numbers.

Cf. A034176, A034908, A034909, A034910, A034911, A045755, A147626.

[n le 1 select 1 else (8*n-2)*Self(n-1): n in [1..40]]; // G. C. Greubel, Oct 20 2022

f:= proc(n) option remember; procname(n-1)*(8*n-2) end proc:
f(1):= 1:
map(f,[$1..20]); # Robert Israel, Mar 20 2018

Table[8^n*Pochhammer[3/4,n]/6, {n,40}] (* G. C. Greubel, Oct 20 2022 *)

[8^n*rising_factorial(3/4,n)/6 for n in range(1,40)] # G. C. Greubel, Oct 20 2022

6*a(n) = (8*n-2)(!^8) = Product_{j=1..n} (8*j - 2) = 2^n*3*A034176(n), where 3*A034176(n) = (4*n-1)(!^4) = Product_{j=1..n} (4*j - 1).
E.g.f.: (-1+(1-8*x)^(-3/4))/6.
G.f.: x/(1-14*x/(1-8*x/(1-22*x/(1-16*x/(1-30*x/(1-24*x/(1-38*x/(1-32*x/(1-...(continued fraction). - Philippe Deléham, Jan 07 2012
From G. C. Greubel, Oct 20 2022: (Start)
a(n) = (1/6) * 8^n * Pochhammer(n, 3/4).
a(n) = 2*(4*n - 1)*a(n-1). (End)
From Amiram Eldar, Dec 20 2022: (Start)
a(n) = A147626(n+1)/6.
Sum_{n>=1} 1/a(n) = 6*(e/8^2)^(1/8)*(Gamma(3/4) - Gamma(3/4, 1/8)). (End)

A113135 a(0) = a(1) = 1, a(2) = x, a(3) = 2x^2, a(n) = x(n-1)a(n-1) + Sum_{j=2..n-2} (j-1)a(j)a(n-j), n>=4 and for x = 8.

Original entry on oeis.org

1, 1, 8, 128, 3136, 103424, 4270080, 211107840, 12135936000, 794618298368, 58355305676800, 4749550536359936, 424336070117163008, 41287521140173963264, 4346005245162898325504, 492102089936714946576384

Offset: 0

Philippe Deléham and Paul D. Hanna, Oct 28 2005

nonn

			a(2) = 8.
a(3) = 2*8^2 = 128.
a(4) = 8*3*128 + 1*8*8 = 3136.
a(5) = 8*4*3136 + 1*8*128 + 2*128*8 = 103424.
a(6) = 8*5*103424 + 1*8*3136 + 2*128*128 + 3*3136*8 = 4270080
G.f.: A(x) = 1 + x + 8*x^2 + 128*x^3 + 3136*x^4 + 103424*x^5 +...
= x/series_reversion(x + x^2 + 9*x^3 + 153*x^4 + 3825*x^5 +...).

Cf. A045755, A075834(x=1), A111088(x=2), A113130(x=3), A113131(x=4), A113132(x=5), A113133(x=6), A113134(x=7).

x=8;a[0]=a[1]=1;a[2]=x;a[3]=2x^2;a[n_]:=a[n]=x*(n-1)*a[n-1]+Sum[(j-1)*a[j ]*a[n-j], {j, 2, n-2}];Table[a[n], {n, 0, 16}](Robert G. Wilson v)

a(n)=Vec(x/serreverse(x*Ser(vector(n+1,k,if(k==1,1, prod(j=0,k-2,8*j+1))))))[n+1]

a(n,x=8)=if(n<0,0,if(n==0 || n==1,1,if(n==2,x,if(n==3,2*x^2,x*(n-1)*a(n-1)+sum(j=2,n-2,(j-1)*a(j)*a(n-j))))))

a(n+1) = Sum{k, 0<=k<=n} 8^k*A113129(n, k).
G.f.: A(x) = x/series_reversion(x*G(x)) where G(x) = g.f. of 8-fold factorials.
G.f. satisfies: A(x*G(x)) = G(x) = g.f. of 8-fold factorials.

A034911 One fifth of octo-factorial numbers.

Original entry on oeis.org

1, 13, 273, 7917, 292929, 13181805, 698635665, 42616775565, 2940557513985, 226422928576845, 19245948929031825, 1789873250399959725, 180777198290395932225, 19704714613653156612525, 2305451609797419323665425, 288181451224677415458178125, 38328133012882096255937690625

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, A045755, A147625.

[n le 1 select 1 else (8*n-3)*Self(n-1): n in [1..40]]; // G. C. Greubel, Oct 20 2022

With[{nn=20},CoefficientList[Series[(-1+(1-8x)^(-5/8))/5,{x,0,nn}],x] Range[0,nn]!] (* or *) FoldList[Times,Range[5,200,8]]/5 (* Harvey P. Dale, May 25 2016 *)

[8^n*rising_factorial(5/8,n)/5 for n in range(1,40)] # G. C. Greubel, Oct 20 2022

5*a(n) = (8*n-3)(!^8) = Product_{j=1..n} 8*j-3.
E.g.f.: (-1+(1-8*x)^(-5/8))/5.
G.f.: x/(1-13*x/(1-8*x/(1-21*x/(1-16*x/(1-29*x/(1-24*x/(1-37*x/(1-32*x/(1-... (continued fraction). - Philippe Deléham, Jan 07 2012
D-finite with recurrence: a(n) = (8*n-3)*a(n-1). - R. J. Mathar, Jan 28 2020
a(n) = (1/5)* 8^n * Pochhammer(n, 5/8). - G. C. Greubel, Oct 20 2022
From Amiram Eldar, Dec 20 2022: (Start)
a(n) = A147625(n+1)/5.
Sum_{n>=1} 1/a(n) = 5*(e/8^3)^(1/8)*(Gamma(5/8) - Gamma(5/8, 1/8)). (End)

A034975 One seventh of octo-factorial numbers.

Original entry on oeis.org

1, 15, 345, 10695, 417105, 19603935, 1078216425, 67927634775, 4822862069025, 381006103452975, 33147531000408825, 3149015445038838375, 324348590839000352625, 36002693583129039141375, 4284320536392355657823625, 544108708121829168543600375, 73454675596446937753386050625

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. A007696, A034176, A034908, A034909, A034910, A034911, A034912, A034976, A045755, A049210.

[n le 1 select 1 else (8*n-1)*Self(n-1): n in [1..40]]; // G. C. Greubel, Oct 21 2022

Table[8^n*Pochhammer[7/8, n]/7, {n, 40}] (* G. C. Greubel, Oct 21 2022 *)

[8^n*rising_factorial(7/8,n)/7 for n in range(1,40)] # G. C. Greubel, Oct 21 2022

7*a(n) = (8*n-1)!^8 = Product_{j=1..n} (8*j-1) = (8*n)!/((2*n)!*2^(6*n)*3^2*5 * A045755(n)*A007696(n)*A034909(n)*A034911(n)*A034176(n)).
E.g.f.: (-1+(1-8*x)^(-7/8))/7.
G.f.: x/(1-15*x/(1-8*x/(1-23*x/(1-16*x/(1-31*x/(1-24*x/(1-39*x/(1-32*x/(1-... (continued fraction). - Philippe Deléham, Jan 07 2012
a(n) = (1/7) * 8^n * Pochhammer(n, 7/8). - G. C. Greubel, Oct 21 2022
From Amiram Eldar, Dec 20 2022: (Start)
a(n) = A049210(n)/7.
Sum_{n>=1} 1/a(n) = 7*(e/8)^(1/8)*(Gamma(7/8) - Gamma(7/8, 1/8)). (End)

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)

A230114 Expansion of e.g.f. 1/(1 - sin(8*x))^(1/8).

Original entry on oeis.org

1, 1, 9, 89, 1521, 32401, 869049, 27608489, 1019581281, 42824944801, 2017329504489, 105299243488889, 6032850630082641, 376363074361201201, 25396689469918450329, 1843101478742259481289, 143145930384321475601601, 11846611289341729822881601, 1040750126963789832859930569

Offset: 0

Paul D. Hanna, Dec 20 2013

nonn

Generally, for e.g.f. 1/(1-sin(p*x))^(1/p) is a(n) ~ n! * 2^(n+3/p) * p^n / (Gamma(2/p) * n^(1-2/p) * Pi^(n+2/p)). - Vaclav Kotesovec, Jan 03 2014

			E.g.f.: A(x) = 1 + x + 9*x^2/2! + 89*x^3/3! + 1521*x^4/4! + 32401*x^5/5! + ...
where A(x)^4 = 1 + 4*x + 48*x^2/2! + 704*x^3/3! + 14592*x^4/4! + 369664*x^5/5! + ...
and 1/A(x)^4 = 1 - 4*x - 16*x^2/2! + 64*x^3/3! + 256*x^4/4! - 1024*x^5/5! + ...
which illustrates 1/A(x)^4 = cos(4*x) - sin(4*x).
O.g.f.: 1/(1-x - 8*1*1*x^2/(1-9*x - 8*2*5*x^2/(1-17*x - 8*3*9*x^2/(1-25*x - 8*4*13*x^2/(1-33*x - 8*5*17*x^2/(1-...)))))), a continued fraction.

Cf. A001586 (p=2), A007788 (p=3), A144015 (p=4), A230134 (p=5), A227544 (p=6), A235128 (p=7).

Cf. A045755, A136630.

CoefficientList[Series[1/(1-Sin[8*x])^(1/8), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jan 03 2014 *)

{a(n)=local(X=x+x*O(x^n)); n!*polcoeff((cos(4*X)-sin(4*X))^(-1/4), n)}
for(n=0, 20, print1(a(n), ", "))

{a(n)=local(A=1+x+x*O(x^n)); for(i=0, n, A=exp(intformal(A^4/subst(A^4, x, -x)))); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))

a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a045755(n) = prod(k=0, n-1, 8*k+1);
a(n) = sum(k=0, n, a045755(k)*(8*I)^(n-k)*a136630(n, k)); \\ Seiichi Manyama, Jun 24 2025

E.g.f. A(x) satisfies:
(1) A(x) = (cos(4*x) - sin(4*x))^(-1/4).
(2) A(x)^4/A(-x)^4 = 1/cos(8*x) + tan(8*x).
(3) A(x) = exp( Integral A(x)^4/A(-x)^4 dx ).
O.g.f.: 1/G(0) where G(k) = 1 - (8*k+1)*x - 8*(k+1)*(4*k+1)*x^2/G(k+1) [continued fraction formula from A144015 due to Sergei N. Gladkovskii].
a(n) ~ n! * 2^(4*n+3/8) / (Gamma(1/4) * n^(3/4) * Pi^(n+1/4)). - Vaclav Kotesovec, Jan 03 2014
a(n) = Sum_{k=0..n} A045755(k) * (8*i)^(n-k) * A136630(n,k), where i is the imaginary unit. - Seiichi Manyama, Jun 24 2025

A051187 Generalized Stirling number triangle of the first kind.

Original entry on oeis.org

1, -8, 1, 128, -24, 1, -3072, 704, -48, 1, 98304, -25600, 2240, -80, 1, -3932160, 1122304, -115200, 5440, -120, 1, 188743680, -57802752, 6651904, -376320, 11200, -168, 1, -10569646080, 3425697792, -430309376, 27725824, -1003520, 20608, -224, 1

Offset: 1

Wolfdieter Lang

T(n,m)= R_n^m(a=0, b=8) in the notation of the given 1962 reference.
T(n,m) is a Jabotinsky matrix, i.e. the monic row polynomials E(n,x) := Sum_{m=1..n} T(n,m)*x^m = Product_{j=0..n-1} (x - 8*j), n >= 1, and E(0,x) := 1 are exponential convolution polynomials (see A039692 for the definition and a Knuth reference).
From Petros Hadjicostas, Jun 07 2020: (Start)
For integers n, m >= 0 and complex numbers a, b (with b <> 0), the numbers R_n^m(a,b) were introduced by Mitrinovic (1961) and further examined by Mitrinovic and Mitrinovic (1962). Such numbers are related to the work of Nörlund (1924).
They are defined via Product_{r=0..n-1} (x - (a + b*r)) = Sum_{m=0..n} R_n^m(a,b)*x^m for n >= 0. As a result, R_n^m(a,b) = R_{n-1}^{m-1}(a,b) - (a + b*(n-1))*R_{n-1}^m(a,b) for n >= m >= 1 with R_1^0(a,b) = a, R_1^1(a,b) = 1, R_n^m(a,b) = 0 for n < m, and R_0^0(a,b) = 1.
With a = 0 and b = 1, we get the Stirling numbers of the first kind S1(n,m) = R_n^m(a=0, b=1) = A048994(n,m).
We have R_n^m(a,b) = Sum_{k=0}^{n-m} (-1)^k * a^k * b^(n-m-k) * binomial(m+k, k) * S1(n, m+k) for n >= m >= 0.
For the current array, T(n,m) = R_n^m(a=0, b=8) but with no zero row or column. (End)

			Triangle T(n,m) (with rows n >= 1 and columns m = 1..n) begins:
          1;
         -8,         1;
        128,       -24,       1;
      -3072,       704,     -48,       1;
      98304,    -25600,    2240,     -80,     1;
   -3932160,   1122304, -115200,    5440,  -120,    1;
  188743680, -57802752, 6651904, -376320, 11200, -168, 1;
  ...
3rd row o.g.f.: E(3,x) = Product_{j=0..2} (x - 8*j) = 128*x - 24*x^2 + x^3.

D. S. Mitrinovic, Sur une classe de nombres reliés aux nombres de Stirling, Comptes rendus de l'Académie des sciences de Paris, t. 252 (1961), 2354-2356.
D. S. Mitrinovic and R. S. Mitrinovic, Sur les polynômes de Stirling, Bulletin de la Société des mathématiciens et physiciens de la R. P. de Serbie, t. 10 (1958), 43-49.
D. S. Mitrinovic and R. S. Mitrinovic, Sur les nombres de Stirling et les nombres de Bernoulli d'ordre supérieur, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., No. 43 (1960), 1-63.
D. S. Mitrinovic and R. S. Mitrinovic, Sur une classe de nombres se rattachant aux nombres de Stirling--Appendice: Table des nombres de Stirling, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., No. 60 (1961), 1-15 and 17-62.
D. S. Mitrinovic and R. S. Mitrinovic, Tableaux d'une classe de nombres reliés aux nombres de Stirling, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., No. 77 (1962), 1-77.
D. S. Mitrinovic and R. S. Mitrinovic, Tableaux d'une classe de nombres reliés aux nombres de Stirling, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., No. 77 (1962), 1-77 [jstor stable version].
Niels Nörlund, Vorlesungen über Differenzenrechnung, Springer, Berlin, 1924.

First (m=1) column sequence is: A051189(n-1).
Row sums (signed triangle): A049210(n-1)*(-1)^(n-1).
Row sums (unsigned triangle): A045755(n).
The b=1..7 triangles are: A008275 (Stirling1 triangle), A039683, A051141, A051142, A051150, A051151, A051186.

T(n, m) = T(n-1, m-1) - 8*(n-1)*T(n-1, m) for n >= m >= 1; T(n, m) := 0 for n < m; T(n, 0) := 0 for n >= 1; T(0, 0) = 1.
E.g.f. for the m-th column of the signed triangle: (log(1 + 8*x)/8)^m/m!.
From Petros Hadjicostas, Jun 07 2020: (Start)
T(n,m) = 8^(n-m)*Stirling1(n,m) = 8^(n-m)*A048994(n,m) = 8^(n-m)*A008275(n,m) for n >= m >= 1.
Bivariate e.g.f.-o.g.f.: Sum_{n,m >= 1} T(n,m)*x^n*y^m/n! = exp((y/8)*log(1 + 8*x)) - 1 = (1 + 8*x)^(y/8) - 1. (End)

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A034908 One half of octo-factorial numbers.

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A256268 Table of k-fold factorials, read by antidiagonals.

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A034909 One third of octo-factorial numbers.

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A034912 One sixth of octo-factorial numbers.

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A113135 a(0) = a(1) = 1, a(2) = x, a(3) = 2x^2, a(n) = x*(n-1)*a(n-1) + Sum_{j=2..n-2} (j-1)*a(j)*a(n-j), n>=4 and for x = 8.

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A034911 One fifth of octo-factorial numbers.

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A034975 One seventh of octo-factorial numbers.

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A113135 a(0) = a(1) = 1, a(2) = x, a(3) = 2x^2, a(n) = x(n-1)a(n-1) + Sum_{j=2..n-2} (j-1)a(j)a(n-j), n>=4 and for x = 8.