A049209 -id:A049209 - cp's OEIS frontend

A045754 7-fold factorials: a(n) = Product_{k=0..n-1} (7*k+1).

1, 1, 8, 120, 2640, 76560, 2756160, 118514880, 5925744000, 337767408000, 21617114112000, 1534815101952000, 119715577952256000, 10175824125941760000, 936175819586641920000, 92681406139077550080000, 9824229050742220308480000, 1110137882733870894858240000

Offset: 0

Wolfdieter Lang

nonn

G. C. Greubel, Table of n, a(n) for n = 0..338
Index entries for sequences related to factorial numbers.

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), A008542 (and A085158), A045755.
See also A113134.
Unsigned row sums of triangle A051186 (scaled Stirling1).
First column of triangle A132056 (S2(8)).
Cf. A114799, A084947, A144739, A144827, A147585, A049209, A051188.

List([0..20], n-> Product([0..n-1], k-> 7*k+1) ); # G. C. Greubel, Aug 21 2019

[1] cat [&*[7*j+1: j in [0..n-1]]: n in [1..20]]; // G. C. Greubel, Aug 21 2019

f := n->product( (7*k+1), k=0..(n-1));
G(x):=(1-7*x)^(-1/7): f[0]:=G(x): for n from 1 to 29 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=0..14); # Zerinvary Lajos, Apr 03 2009

FoldList[Times, 1, 7Range[0, 20] + 1] (* Harvey P. Dale, Jan 21 2013 *)

PARI
```
a(n)=prod(k=0,n-1,7*k+1)
```

[7^n*rising_factorial(1/7, n) for n in (0..20)] # G. C. Greubel, Aug 21 2019

a(n) = Sum_{k=0..n} (-7)^(n-k)*A048994(n, k), where A048994 = Stirling-1 numbers.
E.g.f.: (1-7*x)^(-1/7).
G.f.: 1/(1-x/(1-7*x/(1-8*x/(1-14*x/(1-15*x/(1-21*x/(1-22*x/(1-... (continued fraction). - Philippe Deléham, Jan 08 2012
a(n) = (-6)^n*Sum_{k=0..n} (7/6)^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/G(0), where G(k)= 1 - x*(7*k+1)/(1 - x*(7*k+7)/G(k+1)); (continued fraction). - Sergei N. Gladkovskii, Jun 05 2013
G.f.: G(0)/2, where G(k)= 1 + 1/(1 - x*(7*k+1)/(x*(7*k+1) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 05 2013
a(n) = 7^n * Gamma(n + 1/7) / Gamma(1/7). - Artur Jasinski, Aug 23 2016
a(n) = A114799(7n-6). - M. F. Hasler, Feb 23 2018
D-finite with recurrence: a(n) +(-7*n+6)*a(n-1)=0. - R. J. Mathar, Jan 17 2020
Sum_{n>=0} 1/a(n) = 1 + (e/7^6)^(1/7)*(Gamma(1/7) - Gamma(1/7, 1/7)). - Amiram Eldar, Dec 19 2022

Additional comments from Philippe Deléham and Paul D. Hanna, Oct 29 2005
Edited by N. J. A. Sloane, Oct 16 2008 at the suggestion of M. F. Hasler, Oct 14 2008
Corrected by Zerinvary Lajos, Apr 03 2009

A256890 Triangle T(n,k) = t(n-k, k); t(n,m) = f(m)t(n-1,m) + f(n)t(n,m-1), where f(x) = x + 2.

Original entry on oeis.org

1, 2, 2, 4, 12, 4, 8, 52, 52, 8, 16, 196, 416, 196, 16, 32, 684, 2644, 2644, 684, 32, 64, 2276, 14680, 26440, 14680, 2276, 64, 128, 7340, 74652, 220280, 220280, 74652, 7340, 128, 256, 23172, 357328, 1623964, 2643360, 1623964, 357328, 23172, 256, 512, 72076, 1637860, 10978444, 27227908, 27227908, 10978444, 1637860, 72076, 512

Offset: 0

Dale Gerdemann, Apr 12 2015

Related triangles may be found by varying the function f(x). If f(x) is a linear function, it can be parameterized as f(x) = a*x + b. With different values for a and b, the following triangles are obtained:
  a\b 1.......2.......3.......4.......5.......6
  -1  A144431
  0   A007318 A038208 A038221
  1   A008292 A256890 A257180 A257606 A257607
  2   A060187 A257609 A257611 A257613 A257615
  3   A142458 A257610 A257620 A257622 A257624 A257626
  4   A142459 A257612 A257621
  5   A142460 A257614 A257623
  6   A142461 A257616 A257625
  7   A142462 A257617 A257627
  8   A167884 A257618
  9   A257608 A257619
The row sums of these, and similarly constructed number triangles, are shown in the following table:
  a\b 1.......2.......3.......4.......5.......6.......7.......8.......9
  0   A000079 A000302 A000400
  1   A000142 A001715 A001725 A049388 A049198
  2   A000165 A002866 A002866 A051580 A051582
  3   A008544 A051578 A037559 A051605 A051607 A051609
  4   A001813 A047053 A000407 A034177 A051618 A051620 A051622
  5   A047055 A008546 A008548 A034300 A034325 A051688 A051690
  6   A047657 A049308 A047058 A034689 A034724 A034788 A053101 A053103
  7   A084947 A144827 A049209 A045754 A034830 A034832 A034834 A053105
  8   A084948 A144828 A147626 A051189 A034908 A034910 A034912 A034976 A053115
  9   A084949 A144829 A147630 A049211 A045756 A035013 A035018 A035021 A035023
  10                                  A051262 A035265 A035273         A035277
  11                                  A254322
  12                                          A145448
The formula can be further generalized to: t(n,m) = f(m+s)*t(n-1,m) + f(n-s)*t(n,m-1), where f(x) = a*x + b. The following table specifies triangles with nonzero values for s (given after the slash).
  a\b  0           1           2          3
  -2   A130595/1
  -1
  0
  1    A110555/-1  A120434/-1  A144697/1  A144699/2
With the absolute value, f(x) = |x|, one obtains A038221/3, A038234/4, A038247/5, A038260/6, A038273/7, A038286/8, A038299/9 (with value for s after the slash).
If f(x) = A000045(x) (Fibonacci) and s = 1, the result is A010048 (Fibonomial).
In the notation of Carlitz and Scoville, this is the triangle of generalized Eulerian numbers A(r, s | alpha, beta) with alpha = beta = 2. Also the array A(2,1,4) in the notation of Hwang et al. (see page 31). - Peter Bala, Dec 27 2019

			Array, t(n, k), begins as:
   1,    2,      4,        8,        16,         32,          64, ...;
   2,   12,     52,      196,       684,       2276,        7340, ...;
   4,   52,    416,     2644,     14680,      74652,      357328, ...;
   8,  196,   2644,    26440,    220280,    1623964,    10978444, ...;
  16,  684,  14680,   220280,   2643360,   27227908,   251195000, ...;
  32, 2276,  74652,  1623964,  27227908,  381190712,  4677894984, ...;
  64, 7340, 357328, 10978444, 251195000, 4677894984, 74846319744, ...;
Triangle, T(n, k), begins as:
    1;
    2,     2;
    4,    12,      4;
    8,    52,     52,       8;
   16,   196,    416,     196,      16;
   32,   684,   2644,    2644,     684,      32;
   64,  2276,  14680,   26440,   14680,    2276,     64;
  128,  7340,  74652,  220280,  220280,   74652,   7340,   128;
  256, 23172, 357328, 1623964, 2643360, 1623964, 357328, 23172,   256;

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened.)
L. Carlitz and R. Scoville, Generalized Eulerian numbers: combinatorial applications, J. für die reine und angewandte Mathematik, 265 (1974): 110-37. See Section 3.
Dale Gerdemann, A256890, Plot of t(m,n) mod k , YouTube, 2015.
Hsien-Kuei Hwang, Hua-Huai Chern, and Guan-Huei Duh, An asymptotic distribution theory for Eulerian recurrences with applications, arXiv:1807.01412 [math.CO], 2018-2019.

Cf. A000079, A001715, A008292, A038208, A257180, A257606, A257607, A257609.

Cf. A257610, A257612, A257614, A257616, A257617, A257618, A257619.

A256890:= func< n,k | (&+[(-1)^(k-j)*Binomial(j+3,j)*Binomial(n+4,k-j)*(j+2)^n: j in [0..k]]) >;
[A256890(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Oct 18 2022

Table[Sum[(-1)^(k-j)*Binomial[j+3, j] Binomial[n+4, k-j] (j+2)^n, {j,0,k}], {n,0, 9}, {k,0,n}]//Flatten (* Michael De Vlieger, Dec 27 2019 *)

t(n,m) = if ((n<0) || (m<0), 0, if ((n==0) && (m==0), 1, (m+2)*t(n-1, m) + (n+2)*t(n, m-1)));
tabl(nn) = {for (n=0, nn, for (k=0, n, print1(t(n-k, k), ", ");); print(););} \\ Michel Marcus, Apr 14 2015

def A256890(n,k): return sum((-1)^(k-j)*Binomial(j+3,j)*Binomial(n+4,k-j)*(j+2)^n for j in range(k+1))
flatten([[A256890(n,k) for k in range(n+1)] for n in range(11)]) # G. C. Greubel, Oct 18 2022

T(n,k) = t(n-k, k); t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0 else t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = x + 2.
Sum_{k=0..n} T(n, k) = A001715(n).
T(n,k) = Sum_{j = 0..k} (-1)^(k-j)*binomial(j+3,j)*binomial(n+4,k-j)*(j+2)^n. - Peter Bala, Dec 27 2019
Modified rule of Pascal: T(0,0) = 1, T(n,k) = 0 if k < 0 or k > n else T(n,k) = f(n-k) * T(n-1,k-1) + f(k) * T(n-1,k), where f(x) = x + 2. - Georg Fischer, Nov 11 2021
From G. C. Greubel, Oct 18 2022: (Start)
T(n, n-k) = T(n, k).
T(n, 0) = A000079(n). (End)

A084947 a(n) = Product_{i=0..n-1} (7*i+2).

Original entry on oeis.org

1, 2, 18, 288, 6624, 198720, 7352640, 323516160, 16499324160, 956960801280, 62202452083200, 4478576549990400, 353807547449241600, 30427449080634777600, 2829752764499034316800, 282975276449903431680000, 30278354580139667189760000, 3451732422135922059632640000

Offset: 0

Daniel Dockery (peritus(AT)gmail.com), Jun 13 2003

Harvey P. Dale, Table of n, a(n) for n = 0..340

Cf. A000079, A000142, A000165, A008544, A001813, A045754, A047055, A047657, A084942, A084947, A084948, A084949, A144739, A144827, A049209, A051188.

List([0..20], n-> Product([0..n-1], k-> 7*k+2) ); # G. C. Greubel, Aug 18 2019

[ 1 ] cat [ &*[ (7*k+2): k in [0..n-1] ]: n in [1..15] ]; // Klaus Brockhaus, Nov 10 2008

a := n->product(7*i+2,i=0..n-1); [seq(a(j),j=0..30)];

Join[{1},FoldList[Times,7*Range[0,15]+2]] (* Harvey P. Dale, Nov 27 2015 *)
Table[7^n*Pochhammer[2/7, n], {n,0,15}] (* G. C. Greubel, Aug 18 2019 *)

vector(20, n, n--; prod(k=0, n-1, 7*k+2)) \\ G. C. Greubel, Aug 18 2019

[product(7*k+2 for k in (0..n-1)) for n in (0..20)] # G. C. Greubel, Aug 18 2019

a(n) = A084942(n)/A000142(n)*A000079(n) = 7^n*Pochhammer(2/7, n) = 7^n*Gamma(n+2/7)/Gamma(2/7).
D-finite with recurrence a(0) = 1; a(n) = (7*n - 5)*a(n-1) for n > 0. - Klaus Brockhaus, Nov 10 2008
G.f.: 1/(1-2*x/(1-7*x/(1-9*x/(1-14*x/(1-16*x/(1-21*x/(1-23*x/(1-28*x/(1-... (continued fraction). - Philippe Deléham, Jan 08 2012
a(n) = (-5)^n*Sum_{k=0..n} (7/5)^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
From Ilya Gutkovskiy, Mar 23 2017: (Start)
E.g.f.: 1/(1 - 7*x)^(2/7).
a(n) ~ sqrt(2*Pi)*7^n*n^n/(exp(n)*n^(3/14)*Gamma(2/7)). (End)
Sum_{n>=0} 1/a(n) = 1 + (e/7^5)^(1/7)*(Gamma(2/7) - Gamma(2/7, 1/7)). - Amiram Eldar, Dec 19 2022

a(15) from Klaus Brockhaus, Nov 10 2008

A051188 Sept-factorial numbers.

Original entry on oeis.org

1, 7, 98, 2058, 57624, 2016840, 84707280, 4150656720, 232436776320, 14643516908160, 1025046183571200, 78928556134982400, 6629998715338521600, 603329883095805465600, 59126328543388935628800

Offset: 0

Wolfdieter Lang

For n >= 1, a(n) is the order of the wreath product of the symmetric group S_n and the Abelian group (C_7)^n. - Ahmed Fares (ahmedfares(AT)my-deja.com), May 07 2001

Vincenzo Librandi, Table of n, a(n) for n = 0..300
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 614.
Index entries for sequences related to factorial numbers

Cf. A000142, A000165, A032031, A045754, A047053, A047058, A049209, A051186, A052562, A053106, A084947, A092516, A092750, A144827, A144739, A147585.

[7^n*Factorial(n): n in [0..20]]; // Vincenzo Librandi, Oct 05 2011

a(n) = n!*7^n; \\ Michel Marcus, Jun 08 2020

a(n) = n!*7^n =: (7*n)(!^7).
a(n) = 7*A034834(n) = Product_{k=1..n} 7*k, n >= 1.
E.g.f.: 1/(1 - 7*x).
G.f.: 1/(1 - 7*x/(1 - 7*x/(1 - 14*x/(1 - 14*x/(1 - 21*x/(1 - 21*x/(1 - 28*x/(1 - 28*x/(1 - ... (continued fraction). - Philippe Deléham, Jan 08 2012
From Amiram Eldar, Jun 25 2020: (Start)
Sum_{n>=0} 1/a(n) = e^(1/7) (A092516).
Sum_{n>=0} (-1)^n/a(n) = e^(-1/7) (A092750). (End)

A049211 a(n) = Product_{k=1..n} (9*k - 1); 9-factorial numbers.

Original entry on oeis.org

1, 8, 136, 3536, 123760, 5445440, 288608320, 17893715840, 1270453824640, 101636305971200, 9045631231436800, 886471860680806400, 94852489092846284800, 11002888734770169036800, 1375361091846271129600000, 184298386307400331366400000, 26354669241958247385395200000

Offset: 0

Wolfdieter Lang

G. C. Greubel, Table of n, a(n) for n = 0..325

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), A049210 (m=8), this sequence (m=9), A049212 (m=10), A254322 (m=11), A346896 (m=12).

Cf. A035022, A048994.

m:=9; [Round(m^n*Gamma(n +(m-1)/m)/Gamma((m-1)/m)): n in [0..20]]; // G. C. Greubel, Feb 08 2022

CoefficientList[Series[(1-9*x)^(-8/9),{x,0,20}],x] * Range[0,20]! (* Vaclav Kotesovec, Jan 28 2015 *)

a(n) = prod(k=1, n, 9*k-1); \\ Michel Marcus, Jan 08 2015

m=9; [m^n*rising_factorial((m-1)/m, n) for n in (0..20)] # G. C. Greubel, Feb 08 2022

a(n) = 8*A035022(n) = (9*n-1)(!^9), n >= 1, a(0) = 1.
a(n) = (-1)^n*Sum_{k=0..n} 9^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
a(n) = 9^n * Gamma(n+8/9) / Gamma(8/9). - Vaclav Kotesovec, Jan 28 2015
E.g.f: (1-9*x)^(-8/9). - Vaclav Kotesovec, Jan 28 2015
From Nikolaos Pantelidis, Dec 09 2020: (Start)
G.f.: 1/(1-8*x-72*x^2/(1-26*x-306*x^2/(1-44*x-702*x^2/(1-62*x-1260*x^2/(1-80*x-1980*x^2/(1-...)))))) (Jacobi continued fraction).
G.f.: 1/(1-8*x/(1-9*x/(1-17*x/(1-18*x/(1-26*x/(1-27*x/(1-35*x/(1-36*x/(1-44*x/(1-45*x/(1-...))))))))))) (Stieltjes continued fraction). (End)
From Nikolaos Pantelidis, Dec 19 2020: (Start)
G.f.: 1/G(0) where G(k) = 1 - (18*k+8)*x - 9*(k+1)*(9*k+8)*x^2/G(k+1) (continued fraction).
G.f.: 1/Q(0) where Q(k) = 1 - x*(9*k+8)/(1 - x*(9*k+9)/Q(k+1) ) (continued fraction). (End)
G.f.: hypergeometric2F0([1, 8/9], [--], 9*x). - G. C. Greubel, Feb 08 2022
Sum_{n>=0} 1/a(n) = 1 + (e/9)^(1/9)*(Gamma(8/9) - Gamma(8/9, 1/9)). - Amiram Eldar, Dec 21 2022

a(9) (originally given incorrectly as 1011636305971200) corrected by Peter Bala, Feb 20 2015
a(15)-a(16) from Vincenzo Librandi, Feb 20 2015
a(16) corrected and incorrect MAGMA program removed by Georg Fischer, May 10 2021

A144739 7-factorial numbers A114799(7n+3): Partial products of A017017(k) = 7k+3, a(0) = 1.

Original entry on oeis.org

1, 3, 30, 510, 12240, 379440, 14418720, 648842400, 33739804800, 1990648483200, 131382799891200, 9590944392057600, 767275551364608000, 66752972968720896000, 6274779459059764224000, 633752725365036186624000, 68445294339423908155392000, 7871208849033749437870080000

Offset: 0

Philippe Deléham, Sep 20 2008

nonn

			a(0)=1, a(1)=3, a(2)=3*10=30, a(3)=3*10*17=510, a(4)=3*10*17*24=12240, ...

G. C. Greubel, Table of n, a(n) for n = 0..335
Index entries for sequences related to factorial numbers.

Cf. A114799, A001710, A001147, A032031, A008545, A047056, A011781, A045754, A084947, A144827, A147585, A049209, A051188.

List([0..20], n-> Product([0..n-1], k-> 7*k+3) ); # G. C. Greubel, Aug 19 2019

[ 1 ] cat [ &*[ (7*k+3): k in [0..n] ]: n in [0..20] ]; // Klaus Brockhaus, Nov 10 2008

a:= n-> product(7*j+3, j=0..n-1); seq(a(n), n=0..20); # G. C. Greubel, Aug 19 2019

Table[7^n*Pochhammer[3/7, n], {n,0,20}] (* G. C. Greubel, Aug 19 2019 *)

a(n)=prod(i=1,n,7*i-4) \\ Charles R Greathouse IV, Jul 02 2013

[product(7*k+3 for k in (0..n-1)) for n in (0..20)] # G. C. Greubel, Aug 19 2019

a(n) = Sum_{k=0..n} A132393(n,k)*3^k*7^(n-k).
G.f.: 1/(1-3*x/(1-7*x/(1-10*x/(1-14*x/(1-17*x/(1-21*x/(1-24*x/(1-... (continued fraction). - Philippe Deléham, Jan 08 2012
a(n) = (-4)^n*Sum_{k=0..n} (7/4)^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)^(3/7).
a(n) ~ sqrt(2*Pi)*7^n*n^n/(exp(n)*n^(1/14)*Gamma(3/7)). (End)
a(n) = A114799(7*n-4). - M. F. Hasler, Feb 23 2018
D-finite with recurrence: a(n) +(-7*n+4)*a(n-1)=0. - R. J. Mathar, Feb 21 2020
Sum_{n>=0} 1/a(n) = 1 + (e/7^4)^(1/7)*(Gamma(3/7) - Gamma(3/7, 1/7)). - Amiram Eldar, Dec 19 2022

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

Original entry on oeis.org

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

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

A034833 a(n) = n-th sept-factorial number divided by 6.

Original entry on oeis.org

1, 13, 260, 7020, 238680, 9785880, 469722240, 25834723200, 1601752838400, 110520945849600, 8399591884569600, 697166126419276800, 62744951377734912000, 6086260283640286464000, 632971069498589792256000, 70259788714343466940416000, 8290655068292529098969088000

Offset: 1

Wolfdieter Lang

Harvey P. Dale, Table of n, a(n) for n = 1..339
Index entries for sequences related to factorial numbers.

Cf. A049209, A045754, A034829, A034830, A034831, A034832, A034834.

[(&*[(7*k-1): k in [1..n]])/6: n in [1..30]]; // G. C. Greubel, Feb 24 2018

FoldList[Times,1,Rest[7*Range[20]-1]] (* Harvey P. Dale, Dec 15 2014 *)

my(x='x+('x^30)); Vec(serlaplace((-1 + (1-7*x)^(-6/7))/6)) \\ G. C. Greubel, Feb 22 2018

a(n) = A049209(n)/6;
a(n) = (7*n-1)(!^7)/6;
a(n) = (1/6)*Product_{j=1..n} (7*j-1);
a(n) = (1/6)*(7*n)! / (7^n * n! * A045754(n) * 2*A034829(n) * 3*A034830(n) * 4*A034831(n) * 5*A034832(n)).
E.g.f.: (-1 + (1-7*x)^(-6/7))/6.
Sum_{n>=1} 1/a(n) = 6*(e/7)^(1/7)*(Gamma(6/7) - Gamma(6/7, 1/7)). - Amiram Eldar, Dec 20 2022

A049212 a(n) = -Product_{k=0..n} (10*k - 1); deca-factorial numbers.

Original entry on oeis.org

1, 9, 171, 4959, 193401, 9476649, 559122291, 38579438079, 3047775608241, 271252029133449, 26853950884211451, 2927080646379048159, 348322596919106730921, 44933615002564768288809, 6245772485356502792144451, 930620100318118916029523199, 147968595950580907648694188641

Offset: 0

Wolfdieter Lang

Andrew Howroyd, Table of n, a(n) for n = 0..100
Index entries for sequences related to factorial numbers.

Cf. A008543, A035278, A048994, A049209, A049210, A049211, A048176.

[Round(10^n*Gamma(n+9/10)/Gamma(9/10)): n in [0..25]]; // G. C. Greubel, Feb 03 2022

CoefficientList[Series[(1-10*x)^(-9/10),{x,0,20}],x] * Range[0,20]! (* Vaclav Kotesovec, Jan 28 2015 *)

a(n) = {-prod(k=0, n, 10*k-1)} \\ Andrew Howroyd, Jan 02 2020

[10^n*rising_factorial(9/10, n) for n in (0..25)] # G. C. Greubel, Feb 03 2022

a(n) = 9*A035278(n) = (10*n-1)(!^10), n >= 1, a(0) = 1.
a(n) = (-1)^n*Sum_{k=0..n} 10^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
a(n) = 10^n * Gamma(n+9/10) / Gamma(9/10). - Vaclav Kotesovec, Jan 28 2015
E.g.f.: (1-10*x)^(-9/10). - Vaclav Kotesovec, Jan 28 2015
From Nikolaos Pantelidis, Jan 17 2021: (Start)
G.f.: 1/G(0) where G(k) = 1 - (20*k+9)*x - 10*(k+1)*(10*k+9)*x^2/G(k+1) (continued fraction).
G.f.: 1/(1-9*x-90*x^2/(1-29*x-380*x^2/(1-49*x-870*x^2/(1-69*x-1560*x^2/(1-89*x-2450*x^2/(1-...)))))) (Jacobi continued fraction).
G.f.: 1/Q(0) where Q(k) = 1 - x*(10*k+9)/(1 - x*(10*k+10)/Q(k+1)) (continued fraction).
G.f.: 1/(1-9*x/(1-10*x/(1-19*x/(1-20*x/(1-29*x/(1-30*x/(1-39*x/(1-40*x/(1-49*x/(1-50*x/(1-...))))))))))) (Stieltjes continued fraction).
(End)
G.f.: Hypergeometric2F0([1, 9/10], --; 10*x). - G. C. Greubel, Feb 03 2022
Sum_{n>=0} 1/a(n) = 1 + (e/10)^(1/10)*(Gamma(9/10) - Gamma(9/10, 1/10)). - Amiram Eldar, Dec 22 2022

Terms a(14) and beyond from Andrew Howroyd, Jan 02 2020

cp's OEIS Frontend

A045754 7-fold factorials: a(n) = Product_{k=0..n-1} (7*k+1).

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A256890 Triangle T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = x + 2.

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A084947 a(n) = Product_{i=0..n-1} (7*i+2).

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A051188 Sept-factorial numbers.

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A049211 a(n) = Product_{k=1..n} (9*k - 1); 9-factorial numbers.

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A144739 7-factorial numbers A114799(7*n+3): Partial products of A017017(k) = 7*k+3, a(0) = 1.

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A256890 Triangle T(n,k) = t(n-k, k); t(n,m) = f(m)t(n-1,m) + f(n)t(n,m-1), where f(x) = x + 2.

A144739 7-factorial numbers A114799(7n+3): Partial products of A017017(k) = 7k+3, a(0) = 1.