A007559 -id:A007559 - cp's OEIS frontend

A035017 One quarter of 9-factorial numbers.

1, 13, 286, 8866, 354640, 17377360, 1007886880, 67528420960, 5132159992960, 436233599401600, 41005958343750400, 4223613709406291200, 473044735453504614400, 57238412989874058342400, 7440993688683627584512000, 1034298122727024234247168000, 153076122163599586668580864000

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. A007559, A034171, A035012, A035013, A035017, A035018.

Cf. A035020, A035021, A035022, A035023, A045756, A144829.

[n le 1 select 1 else (9*n-5)*Self(n-1): n in [1..40]]; // G. C. Greubel, Oct 18 2022

s=1;lst={s};Do[s+=n*s;AppendTo[lst, s], {n, 12, 2*5!, 9}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)
Table[9^n*Pochhammer[4/9, n]/4, {n,40}] (* G. C. Greubel, Oct 18 2022 *)

[9^n*rising_factorial(4/9,n)/4 for n in range(1,40)] # G. C. Greubel, Oct 18 2022

4*a(n) = (9*n-5)(!^9) := Product_{j=1..n} (9*j-5).
E.g.f.: (-1+(1-9*x)^(-4/9))/4.
From G. C. Greubel, Oct 18 2022: (Start)
a(n) = (1/4) * 9^n * Pochhammer(n, 4/9).
a(n) = (9*n-5)*a(n-1). (End)
From Amiram Eldar, Dec 21 2022: (Start)
a(n) = A144829(n)/4.
Sum_{n>=1} 1/a(n) = 4*(e/9^5)^(1/9)*(Gamma(4/9) - Gamma(4/9, 1/9)). (End)

A035018 One fifth of 9-factorial numbers.

Original entry on oeis.org

1, 14, 322, 10304, 422464, 21123200, 1246268800, 84746278400, 6525463436800, 561189855564800, 53313036278656000, 5544555772980224000, 626534802346765312000, 76437245886305368064000, 10013279211106003216384000, 1401859089554840450293760000, 208877004343671227093770240000

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. A007559, A034171, A035012, A035013, A035017, A035018.

Cf. A035020, A035021, A035022, A035023, A045756, A147629.

[n le 1 select 1 else (9*n-4)*Self(n-1): n in [1..40]]; // G. C. Greubel, Oct 18 2022

s=1;lst={s};Do[s+=n*s;AppendTo[lst, s], {n, 13, 2*5!, 9}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)
Rest[FoldList[Times,1,9*Range[20]-4]/5] (* Harvey P. Dale, May 22 2013 *)

[9^n*rising_factorial(5/9,n)/5 for n in range(1,40)] # G. C. Greubel, Oct 18 2022

5*a(n) = (9*n-4)(!^9) := Product_{j=1..n} (9*j-4).
E.g.f.: (-1+(1-9*x)^(-5/9))/5.
From G. C. Greubel, Oct 18 2022: (Start)
a(n) = (1/5) * 9^n * Pochhammer(n, 5/9).
a(n) = (9*n-4)*a(n-1). (End)
From Amiram Eldar, Dec 21 2022: (Start)
a(n) = A147629(n+1)/5.
Sum_{n>=1} 1/a(n) = 5*(e/9^4)^(1/9)*(Gamma(5/9) - Gamma(5/9, 1/9)). (End)

A035020 One sixth of 9-factorial numbers.

Original entry on oeis.org

1, 15, 360, 11880, 498960, 25446960, 1526817600, 105350414400, 8217332323200, 714907912118400, 68631159563366400, 7206271754153472000, 821514979973495808000, 101046342536739984384000, 13338117214849677938688000, 1880674527293804589355008000, 282101179094070688403251200000

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. A007559, A034000, A034171, A035012, A035013, A035017, A147630.

Cf. A035018, A035020, A035021, A035022, A035023, A045756.

[n le 1 select 1 else (9*n-3)*Self(n-1): n in [1..40]]; // G. C. Greubel, Oct 18 2022

s=1;lst={s};Do[s+=n*s;AppendTo[lst, s], {n, 14, 2*5!, 9}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)
Table[9^n*Pochhammer[2/3, n]/6, {n, 40}] (* G. C. Greubel, Oct 18 2022 *)

[9^n*rising_factorial(2/3,n)/6 for n in range(1,40)] # G. C. Greubel, Oct 18 2022

6*a(n) = (9*n-3)(!^9) := Product_{j=1..n} (9*j-3) = 3^n*2*A034000(n), where 2*A034000(n) = (3*n-1)(!^3) := Product_{j=1..n} (3*j-1).
E.g.f.: (-1+(1-9*x)^(-2/3))/6.
From G. C. Greubel, Oct 18 2022: (Start)
a(n) = (1/6) * 9^n * Pochhammer(n, 2/3).
a(n) = (9*n - 3)*a(n-1). (End)
From Amiram Eldar, Dec 21 2022: (Start)
a(n) = A147630(n+1)/6.
Sum_{n>=1} 1/a(n) = 6*(e/9^3)^(1/9)*(Gamma(2/3) - Gamma(2/3, 1/9)). (End)

A144827 Partial products of successive terms of A017029; a(0)=1.

Original entry on oeis.org

1, 4, 44, 792, 19800, 633600, 24710400, 1136678400, 60243955200, 3614637312000, 242180699904000, 17921371792896000, 1451631115224576000, 127743538139762688000, 12135636123277455360000, 1237834884574300446720000, 134924002418598748692480000, 15651184280557454848327680000

Offset: 0

Philippe Deléham, Sep 21 2008

nonn

			a(0)=1, a(1)=4, a(2)=4*11=44, a(3)=4*11*18=792, a(4)=4*11*18*25=19800, ...

T. D. Noe, Table of n, a(n) for n = 0..100

Cf. A001715, A002866, A007559, A008546, A045754, A047053, A132393.

Cf. A049209, A049308, A051188, A084947, A144739, A147585.

[ 1 ] cat [ &*[ (7*k+4): k in [0..n] ]: n in [0..14] ]; // Klaus Brockhaus, Nov 10 2008

FoldList[Times,1,Range[4,150,7]] (* Harvey P. Dale, Apr 25 2014 *)

[1]+[4*7^(n-1)*rising_factorial(11/7, n-1) for n in (1..30)] # G. C. Greubel, Feb 22 2022

a(n) = Sum_{k=0..n} A132393(n,k)*4^k*7^(n-k).
G.f.: 1/(1-4*x/(1-7*x/(1-11*x/(1-14*x/(1-18*x/(1-21*x/(1-25*x/(1-... (continued fraction). - Philippe Deléham, Jan 08 2012
a(n) = (-3)^n*Sum_{k=0..n} (7/3)^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
From Ilya Gutkovskiy, Mar 23 2017: (Start)
E.g.f.: 1/(1 - 7*x)^(4/7).
a(n) ~ sqrt(2*Pi)*7^n*n^(n+1/14)/(exp(n)*Gamma(4/7)). (End)
a(n) = 4*7^(n-1)*Pochhammer(n-1, 11/7) with a(0) = 1. - G. C. Greubel, Feb 22 2022
Sum_{n>=0} 1/a(n) = 1 + (e/7^3)^(1/7)*(Gamma(4/7) - Gamma(4/7, 1/7)). - Amiram Eldar, Dec 19 2022

Corrected a(9) by Vincenzo Librandi, Jul 14 2011

A317996 Expansion of e.g.f. exp((1 - exp(-3*x))/3).

Original entry on oeis.org

1, 1, -2, 1, 19, -128, 379, 1549, -32600, 261631, -845909, -10713602, 237695149, -2513395259, 11792378662, 151915180429, -4826456213273, 70741388773960, -558513179369297, -2833805536521839, 200720356696607416, -4256279445015662093, 54120395442382043743, -173423789950999240226

Offset: 0

Ilya Gutkovskiy, Aug 20 2018

sign

Seiichi Manyama, Table of n, a(n) for n = 0..495
Eric Weisstein's World of Mathematics, Bell Polynomial

Column k=3 of A309386.

Cf. A004212, A007559, A009235, A014182, A318179, A318180, A318181.

a:=series(exp((1 - exp(-3*x))/3), x=0, 24): seq(n!*coeff(a, x, n), n=0..23); # Paolo P. Lava, Mar 26 2019

nmax = 23; CoefficientList[Series[Exp[(1 - Exp[-3 x])/3], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[(-3)^(n - k) StirlingS2[n, k], {k, 0, n}], {n, 0, 23}]
a[n_] := a[n] = Sum[(-3)^(k - 1) Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 23}]
Table[(-3)^n BellB[n, -1/3], {n, 0, 23}] (* Peter Luschny, Aug 20 2018 *)

a(n) = Sum_{k=0..n} (-3)^(n-k)*Stirling2(n,k).
a(0) = 1; a(n) = Sum_{k=1..n} (-3)^(k-1)*binomial(n-1,k-1)*a(n-k).
a(n) = (-3)^n BellPolynomial_n(-1/3). - Peter Luschny, Aug 20 2018

A225470 Triangle read by rows, s_3(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.

Original entry on oeis.org

1, 2, 1, 10, 7, 1, 80, 66, 15, 1, 880, 806, 231, 26, 1, 12320, 12164, 4040, 595, 40, 1, 209440, 219108, 80844, 14155, 1275, 57, 1, 4188800, 4591600, 1835988, 363944, 39655, 2415, 77, 1, 96342400, 109795600, 46819324, 10206700, 1276009, 95200, 4186, 100, 1

Offset: 0

Peter Luschny, May 16 2013

The Stirling-Frobenius subset numbers S_{m}(n,k), for m >= 1 fixed, regarded as an infinite lower triangular matrix, can be inverted by Sum_{k} S_{m}(n,k)*s_{m}(k,j)*(-1)^(n-k) = [j = n]. The inverse numbers s_{m}(k,j), which are unsigned, are the Stirling-Frobenius cycle numbers. For m = 1 this gives the classical Stirling cycle numbers A132393. The Stirling-Frobenius subset numbers are defined in A225468.

Triangle T(n,k), read by rows, given by (2, 3, 5, 6, 8, 9, 11, 12, 14, 15, ... (A007494)) DELTA (1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, May 14 2015

			Triangle starts:
  [n\k][    0,      1,     2,     3,    4,  5,  6]
  [0]       1,
  [1]       2,      1,
  [2]      10,      7,     1,
  [3]      80,     66,    15,     1,
  [4]     880,    806,   231,    26,    1,
  [5]   12320,  12164,  4040,   595,   40,  1,
  [6]  209440, 219108, 80844, 14155, 1275, 57,  1.
  ...
From _Wolfdieter Lang_, Aug 11 2017: (Start)
Recurrence (see Maple program): T(4, 2) = T(3, 1) + (3*4 - 1)*T(3, 2) = 66 + 11*15 = 231.
Boas-Buck type recurrence for column k = 2 and n = 4: T(4, 2) = (4!/2)*(3*(2 + 6*(5/12))*T(2, 2)/2! + 1*(2 + 6*(1/2))*T(3,2)/3!) = (4!/2)*(3*9/4 + 5*15/3!) = 231. (End)

Peter Bala, A 3 parameter family of generalized Stirling numbers
Peter Luschny, Generalized Eulerian polynomials
Peter Luschny, The Stirling-Frobenius numbers

Cf. A225468; A132393 (m=1), A028338 (m=2), A225471(m=4).
Column k=0..4 give A008544, A024395(n-1), A286722(n-2), A383221, A383222.
T(n, n) ~ A000012; T(n, n-1) ~ A005449; T(n, n-2) ~ A024391; T(n, n-3) ~ A024392.
row sums ~ A032031; alternating row sums ~ A007559.
Cf. A132393.

SF_C := proc(n, k, m) option remember;
if n = 0 and k = 0 then return(1) fi;
if k > n or  k < 0 then return(0) fi;
SF_C(n-1, k-1, m) + (m*n-1)*SF_C(n-1, k, m) end:
seq(print(seq(SF_C(n, k, 3), k = 0..n)), n = 0..8);

SFC[0, 0, ] = 1; SFC[n, k_, ] /; (k > n || k < 0) = 0; SFC[n, k_, m_] := SFC[n, k, m] = SFC[n-1, k-1, m] + (m*n-1)*SFC[n-1, k, m]; Table[SFC[n, k, 3], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 26 2013, after Maple *)

For a recurrence see the Maple program.
From Wolfdieter Lang, May 18 2017: (Start)
This is the Sheffer triangle (1/(1 - 3*x)^{-2/3}, -(1/3)*log(1-3*x)). See the P. Bala link where this is called exponential Riordan array, and the signed version is  denoted by s_{(3,0,2)}.
E.g.f. of row polynomials in the variable x (i.e., of the triangle): (1 - 3*z)^{-(2+x)/3}.
E.g.f. of column k: (1-3*x)^(-2/3)*(-(1/3)*log(1-3*x))^k/k!, k >= 0.
Recurrence for row polynomials R(n, x) = Sum_{k=0..n} T(n, k)*x^k: R(n, x) = (x+2)*R(n-1,x+3), with R(0, x) = 1.
R(n, x) = risefac(3,2;x,n) := Product_{j=0..(n-1)} (x + (2 + 3*j)). (See the P. Bala link, eq. (16) for the signed s_{3,0,2} row polynomials.)
T(n, k) = Sum_{j=0..(n-m)} binomial(n-j, k)* S1p(n, n-j)*2^(n-k-j)*3^j, with S1p(n, m) = A132393(n, m). (End)
Boas-Buck type recurrence for column sequence k: T(n, k) = (n!/(n - k)) * Sum_{p=k..n-1} 3^(n-1-p)*(2 + 3*k*beta(n-1-p))*T(p, k)/p!, for n > k >= 0, with input T(k, k) = 1, and beta(k) = A002208(k+1)/A002209(k+1), beginning {1/2, 5/12, 3/8, 251/720, ...}. See a comment and references in A286718. - Wolfdieter Lang, Aug 11 2017

A007788 Number of augmented Andre 3-signed permutations: E.g.f. (1-sin(3*x))^(-1/3).

Original entry on oeis.org

1, 1, 4, 19, 136, 1201, 13024, 165619, 2425216, 40132801, 740882944, 15091932019, 336257744896, 8134269015601, 212309523595264, 5946914908771219, 177934946000306176, 5663754614516217601, 191097349696090537984, 6812679868133940475219, 255885704427935576621056

Offset: 0

R. Ehrenborg (ehrenbor(AT)lacim.uqam.ca) and M. A. Readdy (readdy(AT)lacim.uqam.ca)

nonn

It appears that all members are of the form 3k+1. - Ralf Stephan, Nov 12 2007

Vincenzo Librandi, Table of n, a(n) for n = 0..200
R. Ehrenborg and M. A. Readdy, Sheffer posets and r-signed permutations, Preprint submitted to Ann. Sci. Math. Quebec, 1994. (Annotated scanned copy)
R. Ehrenborg and M. A. Readdy, Sheffer posets and r-signed permutations, Ann. Sci. Math. Québec, 19 (1995), no. 2, 173-196.
R. Ehrenborg and M. A. Readdy, The r-cubical lattice and a generalization of the cd-index, European J. Combin. 17 (1996), no. 8, 709-725.

Cf. A001586, A144015, A230134, A227544, A235128, A230114.
Cf. A007863, A235135, A235132.
Cf. A007559, A136630.

R:=PowerSeriesRing(Rationals(), 20); Coefficients(R!(Laplace( (1-Sin(3*x))^(-1/3) ))); // G. C. Greubel, Mar 05 2020

m:=20; S:=series( (1-sin(3*x))^(-1/3), x, m+1): seq(j!*coeff(S, x, j), j=0..m); # G. C. Greubel, Mar 05 2020

With[{nn=20},CoefficientList[Series[(1-Sin[3x])^(-1/3),{x,0,nn}], x] Range[0,nn]!] (* Harvey P. Dale, Nov 23 2011 *)

Vec(serlaplace( (1-sin(3*x))^(-1/3) +O('x^20) )) \\ G. C. Greubel, Mar 05 2020

a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a007559(n) = prod(k=0, n-1, 3*k+1);
a(n) = sum(k=0, n, a007559(k)*(3*I)^(n-k)*a136630(n, k)); \\ Seiichi Manyama, Jun 24 2025

m=20;
def A007788_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( (1-sin(3*x))^(-1/3) ).list()
a=A007788_list(m+1); [factorial(n)*a[n] for n in (0..m)] # G. C. Greubel, Mar 05 2020

E.g.f.: (1-sin(3*x))^(-1/3).
a(n) ~ n! * 2*6^n/(Pi^(n+2/3)*n^(1/3)*Gamma(2/3)). - Vaclav Kotesovec, Jun 25 2013
a(n) = Sum_{k=0..n} A007559(k) * (3*i)^(n-k) * A136630(n,k), where i is the imaginary unit. - Seiichi Manyama, Jun 24 2025

A034787 a(n) = n-th sextic factorial number divided by 5.

Original entry on oeis.org

1, 11, 187, 4301, 124729, 4365515, 178986115, 8412347405, 445854412465, 26305410335435, 1709851671803275, 121399468698032525, 9347759089748504425, 775864004449125867275, 69051896395972202187475, 6559930157617359207810125, 662552945919353279988822625

Offset: 1

Wolfdieter Lang

G. C. Greubel, Table of n, a(n) for n = 1..345

Cf. A008542, A034689, A034723, A034724, A034788.

List([1..20], n-> Product([1..n], j-> 6*j-1)/5 ); # G. C. Greubel, Nov 11 2019

[(&*[6*j-1: j in [1..n]])/5: n in [1..20]]; // G. C. Greubel, Nov 11 2019

seq( mul(6*j-1, j=1..n)/5, n=1..20); # G. C. Greubel, Nov 11 2019

Table[6^n*Pochhammer[5/6, n]/5, {n,20}] (* G. C. Greubel, Nov 11 2019 *)
With[{nn=20},CoefficientList[Series[(-1+(1-6x)^(-5/6))/5,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Dec 21 2024 *)

vector(20, n, prod(j=1,n, 6*j-1)/5 ) \\ G. C. Greubel, Nov 11 2019

[product( (6*j-1) for j in (1..n))/5 for n in (1..20)] # G. C. Greubel, Nov 11 2019

5*a(n) = (6*n-1)(!^6) = Product_{j=1..n} (6*j-1) = (6*n)!/(3^(2*n)*2^(2*n+1)*(2*n)!*A008542(n)*A007559(n)*A034000(n)).
E.g.f.: (-1 + (1-6*x)^(-5/6))/5.
a(n+1) ~ sqrt(2*Pi) * 6/(5*Gamma(5/6)) * n^(4/3) * (6*n/e)^n * (1 + (61/72)/n + ...). - Joe Keane (jgk(AT)jgk.org), Nov 24 2001
D-finite with recurrence: a(n) +(-6*n+1)*a(n-1)=0. - R. J. Mathar, Feb 24 2020
Sum_{n>=1} 1/a(n) = 5*(e/6)^(1/6)*(Gamma(5/6) - Gamma(5/6, 1/6)). - Amiram Eldar, Dec 18 2022

A035021 One seventh of 9-factorial numbers.

Original entry on oeis.org

1, 16, 400, 13600, 584800, 30409600, 1854985600, 129848992000, 10258070368000, 902710192384000, 87562888661248000, 9281666198092288000, 1067391612780613120000, 132356559984796026880000, 17603422477977871575040000, 2499685991872857763655680000, 377452584772801522312007680000

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. A007559, A034171, A035012, A035013, A035017, A035018.

Cf. A035020, A035021, A035022, A035023, A045756, A147631.

[n le 1 select 1 else (9*n-2)*Self(n-1): n in [1..40]]; // G. C. Greubel, Oct 19 2022

f := gfun:-rectoproc({(9*n - 2)*a(n - 1) - a(n) = 0, a(1) = 1}, a(n), remember);
map(f, [$ (1 .. 20)]); # Georg Fischer, Feb 15 2020

Table[9^n*Pochhammer[7/9, n]/7, {n, 40}] (* G. C. Greubel, Oct 19 2022 *)

[9^n*rising_factorial(7/9,n)/7 for n in range(1,40)] # G. C. Greubel, Oct 19 2022

7*a(n) = (9*n-2)(!^9) := Product_{j=1..n} (9*j-2).
E.g.f.: (-1+(1-9*x)^(-7/9))/7.
D-finite with recurrence: a(1) = 1, a(n) = (9*n - 2)*a(n-1) for n > 1. - Georg Fischer, Feb 15 2020
a(n) = (1/7) * 9^n * Pochhammer(n, 7/9). - G. C. Greubel, Oct 19 2022
From Amiram Eldar, Dec 21 2022: (Start)
a(n) = A147631(n+1)/7.
Sum_{n>=1} 1/a(n) = 7*(e/9^2)^(1/9)*(Gamma(7/9) - Gamma(7/9, 1/9)). (End)

Terms a(15) onward added by G. C. Greubel, Oct 19 2022

A051604 a(n) = (3*n+4)!!!/4!!!.

Original entry on oeis.org

1, 7, 70, 910, 14560, 276640, 6086080, 152152000, 4260256000, 132067936000, 4490309824000, 166141463488000, 6645658539520000, 285763317199360000, 13145112591170560000, 644110516967357440000, 33493746882302586880000, 1842156078526642278400000

Offset: 0

Wolfdieter Lang

Related to A007559(n+1) ((3*n+1)!!! triple factorials).

Row m=4 of the array A(4; m,n) := ((3*n+m)(!^3))/m(!^3), m >= 0, n >= 0.

G. C. Greubel, Table of n, a(n) for n = 0..378

Cf. A032031, A007559(n+1), A034000(n+1), A034001(n+1). (rows m=0..3).

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1/(1-3*x)^(7/3))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Aug 15 2018

With[{nn = 30}, CoefficientList[Series[1/(1-3*x)^(7/3), {x, 0, nn}], x]* Range[0,nn]! ] (* G. C. Greubel, Aug 15 2018 *)
With[{c=Times@@Range[4,1,-3]},Table[(Times@@Range[3n+4,1,-3])/c,{n,0,20}]] (* Harvey P. Dale, Feb 06 2023 *)

x='x+O('x^30); Vec(serlaplace(1/(1-3*x)^(7/3))) \\ G. C. Greubel, Aug 15 2018

a(n) = ((3*n+4)(!^3))/4(!^3).
E.g.f.: 1/(1-3*x)^(7/3).
Sum_{n>=0} 1/a(n) = 1 + 3*(3*e)^(1/3)*(Gamma(7/3) - Gamma(7/3, 1/3)). - Amiram Eldar, Dec 23 2022

cp's OEIS Frontend

A035017 One quarter of 9-factorial numbers.

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A035018 One fifth of 9-factorial numbers.

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A035020 One sixth of 9-factorial numbers.

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A144827 Partial products of successive terms of A017029; a(0)=1.

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A317996 Expansion of e.g.f. exp((1 - exp(-3*x))/3).

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A225470 Triangle read by rows, s_3(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.

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A007788 Number of augmented Andre 3-signed permutations: E.g.f. (1-sin(3*x))^(-1/3).

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