A052562 -id:A052562 - cp's OEIS frontend

A114799 Septuple factorial, 7-factorial, n!7, n!!!!!!!, a(n) = n*a(n-7) if n > 1, else 1.

1, 1, 2, 3, 4, 5, 6, 7, 8, 18, 30, 44, 60, 78, 98, 120, 288, 510, 792, 1140, 1560, 2058, 2640, 6624, 12240, 19800, 29640, 42120, 57624, 76560, 198720, 379440, 633600, 978120, 1432080, 2016840, 2756160, 7352640, 14418720, 24710400, 39124800

Offset: 0

Jonathan Vos Post, Feb 18 2006

Many of the terms yield multifactorial primes a(n) + 1, e.g.: a(2) + 1 = 3, a(4) + 1 = 5, a(6) + 1 = 7, a(9) + 1 = 19, a(10) + 1 = 31, a(12) + 1 = 61, a(13) + 1 = 79, a(24) + 1 = 12241, a(25) + 1 = 19801, a(26) + 1 = 29641, a(29) + 1 = 76561, a(31) + 1 = 379441, a(35) + 1 = 2016841, a(36) + 1 = 2756161, ...

Equivalently, product of all positive integers <= n congruent to n (mod 7). - M. F. Hasler, Feb 23 2018

			a(40) = 40 * a(40-7) = 40 * a(33) = 40 * (33*a(26)) = 40 * 33 * (26*a(19)) = 40 * 33 * 26 * (19*a(12)) = 40 * 33 * 26 * 19 * (12*a(5)) = 40 * 33 * 26 * 19 * 12 5 = 39124800.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Multifactorial.
Index entries for sequences related to factorial numbers

Cf. A045754, A084947, A288094.

Cf. k-fold factorials: A000142, A001147 (and A000165, A006882), A007559 (and A032031, A008544, A007661), A007696 (and A001813, A008545, A047053, A007662), A008548 (and A052562, A047055, A085157), A085158 (and A008542, A047058, A047657), A045755.

a:= function(n)
    if n<1 then return 1;
    else return n*a(n-7);
    fi;
  end;
List([0..40], n-> a(n) ); # G. C. Greubel, Aug 20 2019

b:= func< n | (n lt 8) select n else n*Self(n-7) >;
[1] cat [b(n): n in [1..40]]; // G. C. Greubel, Aug 20 2019

A114799 := proc(n)
    option remember;
    if n < 1 then
        1;
    else
        n*procname(n-7) ;
    end if;
end proc:
seq(A114799(n),n=0..40) ; # R. J. Mathar, Jun 23 2014
A114799 := n -> product(n-7*k,k=0..(n-1)/7); # M. F. Hasler, Feb 23 2018

a[n_]:= If[n<1, 1, n*a[n-7]]; Table[a[n], {n,0,40}] (* G. C. Greubel, Aug 20 2019 *)

A114799(n,k=7)=prod(j=0,(n-1)\k,n-j*k) \\ M. F. Hasler, Feb 23 2018

def a(n):
    if (n<1): return 1
    else: return n*a(n-7)
[a(n) for n in (0..40)] # G. C. Greubel, Aug 20 2019

a(n) = 1 for n <= 1, else a(n) = n*a(n-7).

Sum_{n>=0} 1/a(n) = A288094. - Amiram Eldar, Nov 10 2020

Edited by M. F. Hasler, Feb 23 2018

A034325 a(n) is the n-th quintic factorial number divided by 5.

Original entry on oeis.org

1, 10, 150, 3000, 75000, 2250000, 78750000, 3150000000, 141750000000, 7087500000000, 389812500000000, 23388750000000000, 1520268750000000000, 106418812500000000000, 7981410937500000000000, 638512875000000000000000

Offset: 1

Wolfdieter Lang

Michael De Vlieger, Table of n, a(n) for n = 1..354
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 594.
Norihiro Nakashima and Shuhei Tsujie, Enumeration of Flats of the Extended Catalan and Shi Arrangements with Species, arXiv:1904.09748 [math.CO], 2019.

Cf. A008548, A034300, A034301, A034323, A052562.

List([1..20], n-> 5^(n-1)*Factorial(n) ); # G. C. Greubel, Aug 23 2019

[5^(n-1)*Factorial(n): n in [1..20]]; // G. C. Greubel, Aug 23 2019

seq(5^(n-1)*n!, n=1..20); # G. C. Greubel, Aug 23 2019

Array[5^(# - 1) #! &, 16] (* Michael De Vlieger, May 30 2019 *)

vector(20, n, 5^(n-1)*n!) \\ G. C. Greubel, Aug 23 2019

[5^(n-1)*factorial(n) for n in (1..20)] # G. C. Greubel, Aug 23 2019

5*a(n) = (5*n)(!^5) = Product_{j=1..n} 5*j = 5^(n-1)*n!.
E.g.f.: (-1 + (1-5*x)^(-1))/5, a(0) = 0.
D-finite with recurrence: a(n) - 5*n*a(n-1) = 0. - R. J. Mathar, Feb 24 2020
From Amiram Eldar, Jan 08 2022: (Start)
Sum_{n>=1} 1/a(n) = 5*(exp(1/5)-1).
Sum_{n>=1} (-1)^(n+1)/a(n) = 5*(1-exp(-1/5)). (End)

A051150 Generalized Stirling number triangle of first kind.

Original entry on oeis.org

1, -5, 1, 50, -15, 1, -750, 275, -30, 1, 15000, -6250, 875, -50, 1, -375000, 171250, -28125, 2125, -75, 1, 11250000, -5512500, 1015000, -91875, 4375, -105, 1, -393750000, 204187500, -41037500, 4230625

Offset: 1

Wolfdieter Lang

a(n,m) = R_n^m(a=0, b=5) in the notation of the given 1961 and 1962 references.
a(n,m) is a Jabotinsky matrix, i.e., the monic row polynomials E(n,x) := Sum_{m=1..n} a(n,m)*x^m = Product_{j=0..n-1} (x - 5*j), n >= 1, and E(0,x) := 1 are exponential convolution polynomials (see A039692 for the definition and a Knuth reference).
This is the signed Stirling1 triangle A008275 with diagonal d >= 0 (main diagonal d = 0) scaled with 5^d.

			Triangle a(n,m) (with rows n >= 1 and columns m = 1..n) begins:
        1;
       -5,      1;
       50,    -15,      1;
     -750,    275,    -30,   1;
    15000,  -6250,    875, -50,    1;
  -375000, 171250, -28125, 2125, -75, 1;
  ...
3rd row o.g.f.: E(3,x) = 50*x - 15*x^2 + x^3.

Wolfdieter Lang, First 10 rows.
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. [The numbers R_n^m(a,b) are first introduced.]
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. [Special cases of the numbers R_n^m(a,b) are tabulated.]

First (m=1) column sequence: A052562(n-1).
Row sums (signed triangle): A008546(n-1)*(-1)^(n-1).
Row sums (unsigned triangle): A008548(n).
Cf. A008275 (Stirling1 triangle, b=1), A039683 (b=2), A051141 (b=3), A051142 (b=4).

a(n, m) = a(n-1, m-1) - 5*(n-1)*a(n-1, m) for n >= m >= 1; a(n, m) := 0 for n < m; a(n, 0) := 0 for n >= 1; a(0, 0) = 1.
E.g.f. for the m-th column of the signed triangle: (log(1 + 5*x)/5)^m/m!.
a(n, m) = S1(n, m)*5^(n-m), with S1(n, m) := A008275(n, m) (signed Stirling1 triangle).

A051687 a(n) = (5n+6)(!^5)/6, related to A008548 ((5n+1)(!^5) quintic, or 5-factorials).

Original entry on oeis.org

1, 11, 176, 3696, 96096, 2978976, 107243136, 4396968576, 202260554496, 10315288279296, 577656143640576, 35237024762075136, 2325643634296958976, 165120698035084087296, 12549173050666390634496

Offset: 0

Wolfdieter Lang

Row m=6 of the array A(6; m,n) := ((5*n+m)(!^5))/m(!^5), m >= 0, n >= 0.

G. C. Greubel, Table of n, a(n) for n = 0..353

Cf. A052562, A008548(n+1), A034323(n+1), A034300(n+1), A034301(n+1), A034325(n+1), A051687-A051691 (rows m=0..10).

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1/(1-5*x)^(11/5))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Aug 15 2018

s=1;lst={s};Do[s+=n*s;AppendTo[lst, s], {n, 10, 5!, 5}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)
With[{nn = 30}, CoefficientList[Series[1/(1 - 5*x)^(11/5), {x, 0, nn}], x]*Range[0, nn]!] (* G. C. Greubel, Aug 15 2018 *)

x='x+O('x^30); Vec(serlaplace(1/(1-5*x)^(11/5))) \\ G. C. Greubel, Aug 15 2018

a(n) = ((5*n+6)(!^5))/6(!^5).

E.g.f.: 1/(1-5*x)^(11/5).

A196347 Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).

Original entry on oeis.org

1, 1, 1, 2, 4, 2, 6, 18, 18, 6, 24, 96, 144, 96, 24, 120, 600, 1200, 1200, 600, 120, 720, 4320, 10800, 14400, 10800, 4320, 720, 5040, 35280, 105840, 176400, 176400, 105840, 35280, 5040, 40320, 322560, 1128960, 2257920, 2822400, 2257920, 1128960, 322560, 40320

Offset: 0

Philippe Deléham, Oct 28 2011

Unsigned version of A021012.

Equal to A136572*A007318.

			Triangle begins:
    1;
    1,   1;
    2,   4,    2;
    6,  18,   18,    6;
   24,  96,  144,   96,  24;
  120, 600, 1200, 1200, 600, 120;
  ...

G. C. Greubel, Table of n, a(n) for n = 0..495
P. Bala, Deformations of the Hadamard product of power series
Paul Barry, On the inversion of Riordan arrays, arXiv:2101.06713 [math.CO], 2021.
M. Dukes, C. D. White, Web Matrices: Structural Properties and Generating Combinatorial Identities, arXiv:1603.01589 [math.CO], 2016.

Cf. A007318, A008619, A021012, A084938, A136572.

Columns include: A000142, A001563, A001804, A001805, A001806, A001807.

/* As triangle */ [[Factorial(n)*Binomial(n, k): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Sep 28 2015

Table[n!*Binomial[n, j], {n, 0, 30}, {j, 0, n}] (* G. C. Greubel, Sep 27 2015 *)

factorial(n)*binomial(n,k) # Danny Rorabaugh, Sep 27 2015

T(n,k) is given by (1,1,2,2,3,3,4,4,5,5,6,6,...) DELTA (1,1,2,2,3,3,4,4,5,5,6,6, ...) where DELTA is the operator defined in A084938.
Sum_{k>=0} T(m,k)*T(n,k) = (m+n)!.
T(2n,n) = A122747(n).
Sum_{k>=0} T(n,k)^2 = A010050(n) = (2n)!.
Sum_{k>=0} T(n,k)*x^k = A000007(n), A000142(n), A000165(n), A032031(n), A047053(n), A052562(n), A047058(n), A051188(n), A051189(n), A051232(n), A051262(n), A196258(n), A145448(n) for x = -1,0,1,2,3,4,5,6,7,8,9,10,11 respectively.
The row polynomials have the form (x + 1) o (x + 2) o ... o (x + n), where o denotes the black diamond multiplication operator of Dukes and White. See example E10 in the Bala link. - Peter Bala, Jan 18 2018

Name exchanged with a formula by Peter Luschny, Feb 01 2015

A051688 a(n) = (5n+7)(!^5)/7(!^5), related to A034323 ((5n+2)(!^5) quintic, or 5-factorials).

Original entry on oeis.org

1, 12, 204, 4488, 121176, 3877632, 143472384, 6025840128, 283214486016, 14727153272832, 839447736551424, 52045759666188288, 3487065897634615296, 251068744629692301312, 19332293336486307201024

Offset: 0

Wolfdieter Lang

Row m=7 of the array A(6; m,n) := ((5*n+m)(!^5))/m(!^5), m >= 0, n >= 0.

G. C. Greubel, Table of n, a(n) for n = 0..353

Cf. A052562, A008548(n+1), A034323(n+1), A034300(n+1), A034301(n+1), A034325(n+1), A051687-A051691 (rows m=0..10).

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1/(1-5*x)^(12/5))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Aug 15 2018

s=1;lst={s};Do[s+=n*s;AppendTo[lst, s], {n, 11, 5!, 5}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)
With[{nn = 30}, CoefficientList[Series[1/(1 - 5*x)^(12/5), {x, 0, nn}], x]*Range[0, nn]!] (* G. C. Greubel, Aug 15 2018 *)

x='x+O('x^30); Vec(serlaplace(1/(1-5*x)^(12/5))) \\ G. C. Greubel, Aug 15 2018

a(n) = ((5*n+7)(!^5))/7(!^5) = A034323(n+2)/7.

E.g.f.: 1/(1-5*x)^(12/5).

A051690 a(n) = (5n+9)(!^5)/9(!^5), related to A034301 ((5n+2)(!^5) quintic, or 5-factorials).

Original entry on oeis.org

1, 14, 266, 6384, 185136, 6294624, 245490336, 10801574784, 529277164416, 28580966878464, 1686277045829376, 107921730933080064, 7446599434382524416, 551048358144306806784, 43532820293400237735936, 3656756904645619969818624, 325451364513460177313857536

Offset: 0

Wolfdieter Lang

Row m=9 of the array A(6; m,n) := ((5*n+m)(!^5))/m(!^5), m >= 0, n >= 0.

G. C. Greubel, Table of n, a(n) for n = 0..352

Cf. A052562, A008548(n+1), A034323(n+1), A034300(n+1), A034301(n+1), A034325(n+1), A051687-A051691 (rows m=0..10).

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1/(1-5*x)^(14/5))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Aug 15 2018

s=1;lst={s};Do[s+=n*s;AppendTo[lst, s], {n, 13, 5!, 5}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)
With[{nn = 30}, CoefficientList[Series[1/(1 - 5*x)^(14/5), {x, 0, nn}], x]*Range[0, nn]!] (* G. C. Greubel, Aug 15 2018 *)

my(x='x+O('x^30)); Vec(serlaplace(1/(1-5*x)^(14/5))) \\ G. C. Greubel, Aug 15 2018

a(n) = ((5*n+9)(!^5))/9(!^5) = A034301(n+2)/9.

E.g.f.: 1/(1-5*x)^(14/5).

A129116 Multifactorial array: A(k,n) = k-tuple factorial of n, for positive n, read by ascending antidiagonals.

Original entry on oeis.org

1, 1, 2, 1, 2, 6, 1, 2, 3, 24, 1, 2, 3, 8, 120, 1, 2, 3, 4, 15, 720, 1, 2, 3, 4, 10, 48, 5040, 1, 2, 3, 4, 5, 18, 105, 40320, 1, 2, 3, 4, 5, 12, 28, 384, 362880, 1, 2, 3, 4, 5, 6, 21, 80, 945, 3628800, 1, 2, 3, 4, 5, 6, 14, 32, 162, 3840, 39916800, 1, 2, 3, 4, 5, 6, 7, 24, 45, 280, 10395, 479001600

Offset: 1

Jonathan Vos Post, May 24 2007

The term "Quintuple factorial numbers" is also used for the sequences A008546, A008548, A052562, A047055, A047056 which have a different definition. The definition given here is the one commonly used. This problem exists for the other rows as well. "n!2" = n!!, "n!3" = n!!!, "n!4" = n!!!!, etcetera. Main diagonal is A[n,n] = n!n = n.

Similar to A114423 (with rows and columns exchanged). - Georg Fischer, Nov 02 2021

			Table begins:
  k / A(k,n)
  1 | 1 2 6 24 120 720 5040 40320 362880 3628800 ... = A000142.
  2 | 1 2 3  8  15  48  105   384    945    3840 ... = A006882.
  3 | 1 2 3  4  10  18   28    80    162     280 ... = A007661.
  4 | 1 2 3  4   5  12   21    32     45     120 ... = A007662.
  5 | 1 2 3  4   5   6   14    24     36      50 ... = A085157.
  6 | 1 2 3  4   5   6    7    16     27      40 ... = A085158.

Alois P. Heinz, Antidiagonals n = 1..141, flattened
Eric Weisstein's World of Mathematics, Multifactorial.

Cf. A000142 (n!), A006882 (n!!), A007661 (n!!!), A007662(n!4), A085157 (n!5), A085158 (n!6), A114799 (n!7), A114800 (n!8), A114806 (n!9), A288327 (n!10).

Cf. A114423 (transposed).

A:= proc(k,n) option remember; if n >= 1 then n* A(k, n-k) elif n >= 1-k then 1 else 0 fi end: seq(seq(A(1+d-n, n), n=1..d), d=1..16); # Alois P. Heinz, Feb 02 2009

A[k_, n_] := A[k, n] = If[n >= 1, n*A[k, n-k], If[n >= 1-k, 1, 0]]; Table[ A[1+d-n, n], {d, 1, 16}, {n, 1, d}] // Flatten (* Jean-François Alcover, May 27 2016, after Alois P. Heinz *)

A(k,n) = n!k.

A(k,n) = M(n,k) in A114423. - Georg Fischer, Nov 02 2021

Corrected and extended by Alois P. Heinz, Feb 02 2009

A303488 a(n) = n! * [x^n] 1/(1 - 5*x)^(n/5).

Original entry on oeis.org

1, 1, 14, 312, 9576, 375000, 17873856, 1004306688, 65006637696, 4763494479744, 389812500000000, 35237024762075136, 3487065897634615296, 374960171943074285568, 43532820293400237735936, 5427359437500000000000000, 723181462895975365595529216, 102563963819340862347122245632

Offset: 0

Ilya Gutkovskiy, Apr 24 2018

nonn

			a(1) = 1;
a(2) = 2*7 = 14;
a(3) = 3*8*13 = 312;
a(4) = 4*9*14*19 = 9576;
a(5) = 5*10*15*20*25 = 375000, etc.

Index entries for sequences related to factorial numbers

Column k=5 of A303489.

Cf. A008546, A008548, A034300, A034301, A034323, A034325, A047055, A047056, A051687, A051688, A051689, A051690, A051691, A052562, A113551, A303486, A303487.

Table[n! SeriesCoefficient[1/(1 - 5 x)^(n/5), {x, 0, n}], {n, 0, 17}]
Table[Product[5 k + n, {k, 0, n - 1}], {n, 0, 17}]
Table[5^n Pochhammer[n/5, n], {n, 0, 17}]

a(n) = Product_{k=0..n-1} (5*k + n).
a(n) = 5^n*Gamma(6*n/5)/Gamma(n/5).
a(n) ~ 6^(6*n/5-1/2)*n^n/exp(n).

A051689 a(n) = (5n+8)(!^5)/8(!^5), related to A034300 ((5n+3)(!^5) quintic, or 5-factorials).

Original entry on oeis.org

1, 13, 234, 5382, 150696, 4972968, 188972784, 8125829712, 390039826176, 20672110787328, 1198982425665024, 75535892816896512, 5136440711548962816, 374960171943074285568, 29246893411559794274304

Offset: 0

Wolfdieter Lang

Row m=8 of the array A(6; m,n) := ((5*n+m)(!^5))/m(!^5), m >= 0, n >= 0.

G. C. Greubel, Table of n, a(n) for n = 0..352

Cf. A052562, A008548(n+1), A034323(n+1), A034300(n+1), A034301(n+1), A034325(n+1), A051687-A051691 (rows m=0..10).

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1/(1-5*x)^(13/5))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Aug 15 2018

s=1;lst={s};Do[s+=n*s;AppendTo[lst, s], {n, 12, 5!, 5}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)
With[{nn = 30}, CoefficientList[Series[1/(1 - 5*x)^(13/5), {x, 0, nn}], x]*Range[0, nn]!] (* G. C. Greubel, Aug 15 2018 *)

x='x+O('x^30); Vec(serlaplace(1/(1-5*x)^(13/5))) \\ G. C. Greubel, Aug 15 2018

a(n) = ((5*n+8)(!^5))/8(!^5) = A034300(n+2)/8.

E.g.f.: 1/(1-5*x)^(13/5).

cp's OEIS Frontend

A114799 Septuple factorial, 7-factorial, n!7, n!!!!!!!, a(n) = n*a(n-7) if n > 1, else 1.

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A034325 a(n) is the n-th quintic factorial number divided by 5.

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A051150 Generalized Stirling number triangle of first kind.

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A051687 a(n) = (5*n+6)(!^5)/6, related to A008548 ((5*n+1)(!^5) quintic, or 5-factorials).

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A196347 Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).

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A051688 a(n) = (5*n+7)(!^5)/7(!^5), related to A034323 ((5*n+2)(!^5) quintic, or 5-factorials).

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A051690 a(n) = (5*n+9)(!^5)/9(!^5), related to A034301 ((5*n+2)(!^5) quintic, or 5-factorials).

A051687 a(n) = (5n+6)(!^5)/6, related to A008548 ((5n+1)(!^5) quintic, or 5-factorials).

A051688 a(n) = (5n+7)(!^5)/7(!^5), related to A034323 ((5n+2)(!^5) quintic, or 5-factorials).

A051690 a(n) = (5n+9)(!^5)/9(!^5), related to A034301 ((5n+2)(!^5) quintic, or 5-factorials).

A051689 a(n) = (5n+8)(!^5)/8(!^5), related to A034300 ((5n+3)(!^5) quintic, or 5-factorials).