A144739 -id:A144739 - cp's OEIS frontend

A132393 Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= k <= n.

1, 0, 1, 0, 1, 1, 0, 2, 3, 1, 0, 6, 11, 6, 1, 0, 24, 50, 35, 10, 1, 0, 120, 274, 225, 85, 15, 1, 0, 720, 1764, 1624, 735, 175, 21, 1, 0, 5040, 13068, 13132, 6769, 1960, 322, 28, 1, 0, 40320, 109584, 118124, 67284, 22449, 4536, 546, 36, 1, 0, 362880, 1026576, 1172700, 723680, 269325, 63273, 9450, 870, 45, 1

Offset: 0

Philippe Deléham, Nov 10 2007, Oct 15 2008, Oct 17 2008

Another name: Triangle of signless Stirling numbers of the first kind.
Triangle T(n,k), 0<=k<=n, read by rows given by [0,1,1,2,2,3,3,4,4,5,5,...] DELTA [1,0,1,0,1,0,1,0,1,...] where DELTA is the operator defined in A084938.
A094645*A007318 as infinite lower triangular matrices.
Row sums are the factorial numbers. - Roger L. Bagula, Apr 18 2008
Exponential Riordan array [1/(1-x), log(1/(1-x))]. - Ralf Stephan, Feb 07 2014
Also the Bell transform of the factorial numbers (A000142). For the definition of the Bell transform see A264428 and for cross-references A265606. - Peter Luschny, Dec 31 2015
This is the lower triagonal Sheffer matrix of the associated or Jabotinsky type |S1| = (1, -log(1-x)) (see the W. Lang link under A006232 for the notation and references). This implies the e.g.f.s given below. |S1| is the transition matrix from the monomial basis {x^n} to the rising factorial basis {risefac(x,n)}, n >= 0. - Wolfdieter Lang, Feb 21 2017
T(n, k), for n >= k >= 1, is also the total volume of the n-k dimensional cell (polytope) built from the n-k orthogonal vectors of pairwise different lengths chosen from the set {1, 2, ..., n-1}. See the elementary symmetric function formula for T(n, k) and an example below. - Wolfdieter Lang, May 28 2017
From Wolfdieter Lang, Jul 20 2017: (Start)
The compositional inverse w.r.t. x of y = y(t;x) = x*(1 - t(-log(1-x)/x)) = x + t*log(1-x) is x = x(t;y) = ED(y,t) := Sum_{d>=0} D(d,t)*y^(d+1)/(d+1)!, the e.g.f. of the o.g.f.s D(d,t) = Sum_{m>=0} T(d+m, m)*t^m of the diagonal sequences of the present triangle. See the P. Bala link for a proof (there d = n-1, n >= 1, is the label for the diagonals).
This inversion gives D(d,t) =  P(d, t)/(1-t)^(2*d+1), with the numerator polynomials P(d, t) =  Sum_{m=0..d} A288874(d, m)*t^m. See an example below. See also the P. Bala formula in A112007. (End)
For n > 0, T(n,k) is the number of permutations of the integers from 1 to n which have k visible digits when viewed from a specific end, in the sense that a higher value hides a lower one in a subsequent position. - Ian Duff, Jul 12 2019

			Triangle T(n,k) begins:
  1;
  0,    1;
  0,    1,     1;
  0,    2,     3,     1;
  0,    6,    11,     6,    1;
  0,   24,    50,    35,   10,    1;
  0,  120,   274,   225,   85,   15,   1;
  0,  720,  1764,  1624,  735,  175,  21,  1;
  0, 5040, 13068, 13132, 6769, 1960, 322, 28, 1;
  ...
---------------------------------------------------
Production matrix is
  0, 1
  0, 1, 1
  0, 1, 2,  1
  0, 1, 3,  3,  1
  0, 1, 4,  6,  4,  1
  0, 1, 5, 10, 10,  5,  1
  0, 1, 6, 15, 20, 15,  6, 1
  0, 1, 7, 21, 35, 35, 21, 7, 1
  ...
From _Wolfdieter Lang_, May 09 2017: (Start)
Three term recurrence: 50 = T(5, 2) = 1*6 + (5-1)*11 = 50.
Recurrence from the Sheffer a-sequence [1, 1/2, 1/6, 0, ...]: 50 = T(5, 2) = (5/2)*(binomial(1, 1)*1*6 + binomial(2, 1)*(1/2)*11 + binomial(3, 1)*(1/6)*6 + 0) = 50. The vanishing z-sequence produces the k=0 column from T(0, 0) = 1. (End)
Elementary symmetric function T(4, 2) = sigma^{(3)}_2 = 1*2 + 1*3 + 2*3 = 11. Here the cells (polytopes) are 3 rectangles with total area 11. - _Wolfdieter Lang_, May 28 2017
O.g.f.s of diagonals: d=2 (third diagonal) [0, 6, 50, ...] has D(2,t) = P(2, t)/(1-t)^5, with P(2, t) = 2 + t, the n = 2 row of A288874. - _Wolfdieter Lang_, Jul 20 2017
Boas-Buck recurrence for column k = 2 and n = 5: T(5, 2) = (5!*2/3)*((3/8)*T(2,2)/2! + (5/12)*T(3,2)/3! + (1/2)*T(4,2)/4!) = (5!*2/3)*(3/16 + (5/12)*3/3! + (1/2)*11/4!) = 50. The beta sequence begins: {1/2, 5/12, 3/8, ...}. - _Wolfdieter Lang_, Aug 11 2017

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, pages 31, 187, 441, 996.
R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd. ed., Table 259, p. 259.
Steve Roman, The Umbral Calculus, Dover Publications, New York (1984), pp. 149-150

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
Roland Bacher and P. De La Harpe, Conjugacy growth series of some infinitely generated groups, hal-01285685v2, 2016.
Eli Bagno and David Garber, Combinatorics of q,r-analogues of Stirling numbers of type B, arXiv:2401.08365 [math.CO], 2024. See page 5.
Peter Bala, Diagonals of triangles with generating function exp(t*F(x)).
J. Fernando Barbero G., Jesús Salas, and Eduardo J. S. Villaseñor, Bivariate Generating Functions for a Class of Linear Recurrences. I. General Structure, arXiv:1307.2010 [math.CO], 2013.
Jean-Luc Baril and Sergey Kirgizov, The pure descent statistic on permutations, Preprint, 2016.
Jean-Luc Baril and Sergey Kirgizov, Transformation à la Foata for special kinds of descents and excedances, arXiv:2101.01928 [math.CO], 2021.
Jean-Luc Baril and José L. Ramírez, Some distributions in increasing and flattened permutations, arXiv:2410.15434 [math.CO], 2024. See p. 9.
Ricky X. F. Chen, A Note on the Generating Function for the Stirling Numbers of the First Kind, Journal of Integer Sequences, 18 (2015), #15.3.8.
Bishal Deb and Alan D. Sokal, Higher-order Stirling cycle and subset triangles: Total positivity, continued fractions and real-rootedness, arXiv:2507.18959 [math.CO], 2025. See p. 5.
FindStat - Combinatorial Statistic Finder, The number of saliances of a permutation, The number of cycles in the cycle decomposition of a permutation.
Bill Gosper, Colored illustrations of triangle of Stirling numbers of first kind read mod 2, 3, 4, 5, 6, 7
W. Steven Gray and Makhin Thitsa, System Interconnections and Combinatorial Integer Sequences, in: System Theory (SSST), 2013 45th Southeastern Symposium on, Date of Conference: 11-11 March 2013, Digital Object Identifier: 10.1109/SSST.2013.6524939.
John M. Holte, Carries, Combinatorics and an Amazing Matrix, The American Mathematical Monthly, Vol. 104, No. 2 (Feb., 1997), pp. 138-149.
Tanya Khovanova and J. B. Lewis, Skyscraper Numbers, J. Int. Seq. 16 (2013) #13.7.2.
Sergey Kitaev and Philip B. Zhang, Distributions of mesh patterns of short lengths, arXiv:1811.07679 [math.CO], 2018.
Wolfdieter Lang, On Sums of Powers of Arithmetic Progressions, and Generalized Stirling, Eulerian and Bernoulli numbers, arXiv:1707.04451 [math.NT], 2017.
Shuzhen Lv and Philip B. Zhang, Joint equidistributions of mesh patterns 123 and 321 with symmetric and antipodal shadings, arXiv:2501.00357 [math.CO], 2024. See p. 11.
Shuzhen Lv and Philip B. Zhang, Joint equidistributions of mesh patterns 123 and 132 with antipodal shadings, arXiv:2506.23148 [math.CO], 2025. See p. 6.
Shi-Mei Ma, Some combinatorial sequences associated with context-free grammars, arXiv:1208.3104v2 [math.CO], 2012. - From _N. J. A. Sloane_, Aug 21 2012
Mathematics Stack Exchange, Symmetric (under the swapping) recursions for Stirling numbers of both kinds, Aug 10 2025.
Emanuele Munarini, Shifting Property for Riordan, Sheffer and Connection Constants Matrices, Journal of Integer Sequences, Vol. 20 (2017), Article 17.8.2.
Emanuele Munarini, Combinatorial identities involving the central coefficients of a Sheffer matrix, Applicable Analysis and Discrete Mathematics (2019) Vol. 13, 495-517.
Fedor Petrov, Recursive algorithm for columns of Stirling numbers of the first kind, answer to question on MathOverflow (2025).
E. G. Santos, Counting non-attacking chess pieces placements: Bishops and Anassas, arXiv:2411.16492 [math.CO], 2024. See p. 2.
Umesh Shankar, Log-concavity of rows of triangular arrays satisfying a certain super-recurrence, arXiv:2508.12467 [math.CO], 2025. See p. 4.
Xun-Tuan Su, Deng-Yun Yang, and Wei-Wei Zhang, A note on the generalized factorial, Australasian Journal of Combinatorics, Volume 56 (2013), Pages 133-137.
Benjamin Testart, Completing the enumeration of inversion sequences avoiding one or two patterns of length 3, arXiv:2407.07701 [math.CO], 2024. See p. 37.

Essentially a duplicate of A048994. Cf. A008275, A008277, A112007, A130534, A288874, A354795.

a132393 n k = a132393_tabl !! n !! k
a132393_row n = a132393_tabl !! n
a132393_tabl = map (map abs) a048994_tabl
-- Reinhard Zumkeller, Nov 06 2013

a132393_row := proc(n) local k; seq(coeff(expand(pochhammer (x,n)),x,k),k=0..n) end: # Peter Luschny, Nov 28 2010

p[t_] = 1/(1 - t)^x; Table[ ExpandAll[(n!)SeriesCoefficient[ Series[p[t], {t, 0, 30}], n]], {n, 0, 10}]; a = Table[(n!)* CoefficientList[SeriesCoefficient[ Series[p[t], {t, 0, 30}], n], x], {n, 0, 10}]; Flatten[a] (* Roger L. Bagula, Apr 18 2008 *)
Flatten[Table[Abs[StirlingS1[n,i]],{n,0,10},{i,0,n}]] (* Harvey P. Dale, Feb 04 2014 *)

create_list(abs(stirling1(n,k)),n,0,12,k,0,n); /* Emanuele Munarini, Mar 11 2011 */

column(n,k) = my(v1, v2); v1 = vector(n-1, i, 0); v2 = vector(n, i, 0); v2[1] = 1; for(i=1, n-1, v1[i] = (i+k)*(i+k-1)/2*v2[i]; for(j=1, i-1, v1[j] *= (i-j)*(i+k)/(i-j+2)); v2[i+1] = vecsum(v1)/i); v2 \\ generates n first elements of the k-th column starting from the first nonzero element. - Mikhail Kurkov, Mar 05 2025

T(n,k) = T(n-1,k-1)+(n-1)*T(n-1,k), n,k>=1; T(n,0)=T(0,k); T(0,0)=1.
Sum_{k=0..n} T(n,k)*x^(n-k) = A000012(n), A000142(n), A001147(n), A007559(n), A007696(n), A008548(n), A008542(n), A045754(n), A045755(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8 respectively. - Philippe Deléham, Nov 13 2007
Expand 1/(1-t)^x = Sum_{n>=0}p(x,n)*t^n/n!; then the coefficients of the p(x,n) produce the triangle. - Roger L. Bagula, Apr 18 2008
Sum_{k=0..n} T(n,k)*2^k*x^(n-k) = A000142(n+1), A000165(n), A008544(n), A001813(n), A047055(n), A047657(n), A084947(n), A084948(n), A084949(n) for x = 1, 2, 3, 4, 5, 6, 7, 8, 9 respectively. - Philippe Deléham, Sep 18 2008
a(n) = Sum_{k=0..n} T(n,k)*3^k*x^(n-k) = A001710(n+2), A001147(n+1), A032031(n), A008545(n), A047056(n), A011781(n), A144739(n), A144756(n), A144758(n) for x=1,2,3,4,5,6,7,8,9,respectively. - Philippe Deléham, Sep 20 2008
Sum_{k=0..n} T(n,k)*4^k*x^(n-k) = A001715(n+3), A002866(n+1), A007559(n+1), A047053(n), A008546(n), A049308(n), A144827(n), A144828(n), A144829(n) for x=1,2,3,4,5,6,7,8,9 respectively. - Philippe Deléham, Sep 21 2008
Sum_{k=0..n} x^k*T(n,k) = x*(1+x)*(2+x)*...*(n-1+x), n>=1. - Philippe Deléham, Oct 17 2008
From Wolfdieter Lang, Feb 21 2017: (Start)
E.g.f. k-th column: (-log(1 - x))^k, k >= 0.
E.g.f. triangle (see the Apr 18 2008 Baluga comment): exp(-x*log(1-z)).
E.g.f. a-sequence: x/(1 - exp(-x)). See A164555/A027642. The e.g.f. for the z-sequence is 0. (End)
From Wolfdieter Lang, May 28 2017: (Start)
The row polynomials R(n, x) = Sum_{k=0..n} T(n, k)*x^k, for n >= 0, are R(n, x) = risefac(x,n-1) := Product_{j=0..n-1} x+j, with the empty product for n=0 put to 1. See the Feb 21 2017 comment above. This implies:
T(n, k) = sigma^{(n-1)}_(n-k), for n >= k >= 1, with the elementary symmetric functions sigma^{(n-1)}_m of degree m in the n-1 symbols 1, 2, ..., n-1, with binomial(n-1, m) terms. See an example below.(End)
Boas-Buck type recurrence for column sequence k: T(n, k) = (n!*k/(n - k)) * Sum_{p=k..n-1} 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). See a comment and references in A286718. - Wolfdieter Lang, Aug 11 2017
T(n,k) = Sum_{j=k..n} j^(j-k)*binomial(j-1, k-1)*A354795(n,j) for n > 0. - Mélika Tebni, Mar 02 2023
n-th row polynomial: n!*Sum_{k = 0..2*n} (-1)^k*binomial(-x, k)*binomial(-x, 2*n-k) = n!*Sum_{k = 0..2*n} (-1)^k*binomial(1-x, k)*binomial(-x, 2*n-k). - Peter Bala, Mar 31 2024
From Mikhail Kurkov, Mar 05 2025: (Start)
For a general proof of the formulas below via generating functions, see Mathematics Stack Exchange link.
Recursion for the n-th row (independently of other rows): T(n,k) = 1/(n-k)*Sum_{j=2..n-k+1} binomial(-k,j)*T(n,k+j-1)*(-1)^j for 1 <= k < n with T(n,n) = 1.
Recursion for the k-th column (independently of other columns): T(n,k) = 1/(n-k)*Sum_{j=2..n-k+1} (j-2)!*binomial(n,j)*T(n-j+1,k) for 1 <= k < n with T(n,n) = 1 (see Fedor Petrov link). (End)

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

Original entry on oeis.org

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

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

A049209 a(n) = -Product_{k=0..n} (7*k-1); sept-factorial numbers.

Original entry on oeis.org

1, 6, 78, 1560, 42120, 1432080, 58715280, 2818333440, 155008339200, 9610517030400, 663125675097600, 50397551307417600, 4182996758515660800, 376469708266409472000, 36517561701841718784000, 3797826416991538753536000, 421558732286060801642496000

Offset: 0

Wolfdieter Lang

G. C. Greubel, Table of n, a(n) for n = 0..330
Index entries for sequences related to factorial numbers.

Cf. A034833, A045754, A048994, A051188, A084947, A144739, A144827, A147585.
Row sums of triangle A051186 (scaled Stirling1 triangle).
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), this sequence (m=7), A049210 (m=8), A049211 (m=9), A049212 (m=10), A254322 (m=11), A346896 (m=12).

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

CoefficientList[Series[(1-7*x)^(-6/7),{x,0,20}],x] * Range[0,20]! (* Vaclav Kotesovec, Jan 28 2015 *)
With[{m=7}, Table[m^n*Pochhammer[(m-1)/m, n], {n, 0, 30}]] (* G. C. Greubel, Feb 16 2022 *)

m=7; [m^n*rising_factorial((m-1)/m, n) for n in (0..30)] # G. C. Greubel, Feb 16 2022

a(n) = 6*A034833(n) = (7*n-1)*(!^7), n >= 1, a(0) := 1.
a(n) = Product_{k=1..n} (7*k - 1). a(0) = 1; a(n) = (7*n - 1)*a(n-1) for n > 0. - Klaus Brockhaus, Nov 10 2008
G.f.: 1/(1-6*x/(1-7*x/(1-13*x/(1-14*x/(1-20*x/(1-21*x/(1-27*x/(1-28*x/(1-...(continued fraction). - Philippe Deléham, Jan 08 2012
a(n) = (-1)^n*Sum_{k=0..n} 7^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) = 7^n * Gamma(n+6/7) / Gamma(6/7). - Vaclav Kotesovec, Jan 28 2015
E.g.f.: (1-7*x)^(-6/7). - Vaclav Kotesovec, Jan 28 2015
From Nikolaos Pantelidis, Dec 19 2020: (Start)
G.f.: 1/G(0) where G(k) = 1 - (14*k+6)*x - 7*(k+1)*(7*k+6)*x^2/G(k+1); (continued fraction).
which starts as 1/(1-6*x-42*x^2/(1-20*x-182*x^2/(1-34*x-420*x^2/(1-48*x-756*x^2/(1-62*x-1190*x^2/(1-... )))))) (Jacobi continued fraction).
G.f.: 1/Q(0) where Q(k) = 1 - (7*k+6)*x/(1 - (7*k+7)*x/Q(k+1) ); (continued fraction). (End)
Sum_{n>=0} 1/a(n) = 1 + (e/7)^(1/7)*(Gamma(6/7) - Gamma(6/7, 1/7)). - Amiram Eldar, Dec 19 2022

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)

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

A034830 a(n) = n-th sept-factorial number divided by 3.

Original entry on oeis.org

1, 10, 170, 4080, 126480, 4806240, 216280800, 11246601600, 663549494400, 43794266630400, 3196981464019200, 255758517121536000, 22250990989573632000, 2091593153019921408000, 211250908455012062208000, 22815098113141302718464000, 2623736283011249812623360000

Offset: 1

Wolfdieter Lang

G. C. Greubel, Table of n, a(n) for n = 1..339
Index entries for sequences related to factorial numbers.

Cf. A045754, A034829, A034831, A034832, A034833, A034834, A144739.

Drop[With[{nn = 40}, CoefficientList[Series[(-1 + (1 - 7*x)^(-3/7))/3, {x, 0, nn}], x]*Range[0, nn]!], 1] (* G. C. Greubel, Feb 23 2018 *)

my(x='x+O('x^30)); Vec(serlaplace((-1 + (1-7*x)^(-3/7))/3)) \\ G. C. Greubel, Feb 23 2018

3*a(n) = (7*n-4)(!^7) = Product_{j=1..n} (7*j-4).
E.g.f.: (-1 + (1-7*x)^(-3/7))/3.
From Amiram Eldar, Dec 20 2022: (Start)
a(n) = A144739(n)/3.
Sum_{n>=1} 1/a(n) = 3*(e/7^4)^(1/7)*(Gamma(3/7) - Gamma(3/7, 1/7)). (End)

More terms added by G. C. Greubel, Feb 23 2018

A147585 a(1) = 1; a(n) = (7n-9)a(n-1) for n > 1.

Original entry on oeis.org

1, 5, 60, 1140, 29640, 978120, 39124800, 1838865600, 99298742400, 6057223286400, 411891183475200, 30891838760640000, 2533130778372480000, 225448639275150720000, 21643069370414469120000, 2229236145152690319360000, 245215975966795935129600000, 28690269188115124410163200000

Offset: 1

Vladimir Joseph Stephan Orlovsky, Nov 08 2008

nonn

G. C. Greubel, Table of n, a(n) for n = 1..340

Cf. A048994, A132393.

Cf. A045754, A049209, A084947, A144739, A144827, A051188.

[ n eq 1 select 1 else Self(n-1)*(7*n-9): n in [1..15] ]; // Klaus Brockhaus, Nov 10 2008

[ 1 ] cat [ &*[ (5+7*k): k in [0..n-1] ]: n in [1..14] ]; // Klaus Brockhaus, Nov 10 2008

seq( -7^n*pochhammer(-2/7, n)/2, n = 1..15); # G. C. Greubel, Dec 03 2019

Table[-7^n*Pochhammer[-2/7, n]/2, {n, 15}] (* G. C. Greubel, Dec 03 2019 *)

{for(n=1, 15, print1(prod(k=1, n-1, 7*k-2,), ","))} \\ Klaus Brockhaus, Nov 10 2008

[-7^n*rising_factorial(-2/7, n)/2 for n in (1..15)] # G. C. Greubel, Dec 03 2019

a(n) = Product_{k=1..n-1} (7*k - 2). - Klaus Brockhaus, Nov 10 2008
a(n) = (5*7^(n-1)*Gamma(5/7+n))/Gamma(12/7). - Klaus Brockhaus, Nov 10 2008
a(n+1) = Sum_{k=0..n} A132393(n,k)*5^k*7^(n-k). - Philippe Deléham, Nov 09 2008
G.f.: x/(1-5x/(1-7x/(1-12x/(1-14x/(1-19x/(1-21x/(1-26x/(1-... (continued fraction). - Philippe Deléham, Jan 08 2012
a(n) = (-2)^n*Sum_{k=0..n} (7/2)^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
Sum_{n>=1} 1/a(n) = 1 + (e/7^2)^(1/7)*(Gamma(5/7) - Gamma(5/7, 1/7)). - Amiram Eldar, Dec 19 2022

Edited by Klaus Brockhaus, Nov 10 2008

A144758 Partial products of successive terms of A017197.

Original entry on oeis.org

1, 3, 36, 756, 22680, 884520, 42456960, 2420046720, 159723083520, 11979231264000, 1006255426176000, 93581754634368000, 9545338972705536000, 1059532625970314496000, 127143915116437739520000, 16401565050020468398080000, 2263415976902824638935040000

Offset: 0

Philippe Deléham, Sep 20 2008

			a(0)=1, a(1)=3, a(2)=3*12=36, a(3)=3*12*21=756, a(4)=3*12*21*30=22680, ...

G. C. Greubel, Table of n, a(n) for n = 0..250

Cf. A048994, A132393.

Cf. A001710, A001147, A008545, A011781, A032031, A047056, A144739, A144756.

[Round(9^n*Gamma(n+1/3)/Gamma(1/3)): n in [0..20]]; // G. C. Greubel, Dec 03 2019

seq(9^n*pochhammer(1/3, n), n = 0..20); # G. C. Greubel, Dec 03 2019

Table[9^n*Pochhammer[1/3, n], {n, 0, 20}] (* G. C. Greubel, Dec 03 2019 *)
Join[{1},FoldList[Times,NestList[#+9&,3,20]]] (* Harvey P. Dale, Mar 09 2025 *)

a(n)=3^n*prod(i=1,n,3*i-2) \\ Charles R Greathouse IV, Jan 17 2012

[9^n*rising_factorial(1/3, n) for n in (0..20)] # G. C. Greubel, Dec 03 2019

a(n) = Sum_{k=0..n} A132393(n,k)*3^k*9^(n-k).
a(n) = (-6)^n*Sum_{k=0..n} (3/2)^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
Sum_{n>=0} 1/a(n) = 1 + (e/9^6)^(1/9)*(Gamma(1/3) - Gamma(1/3, 1/9)). - Amiram Eldar, Dec 21 2022

A144756 Partial products of successive terms of A017101; a(0)=1 .

Original entry on oeis.org

1, 3, 33, 627, 16929, 592515, 25478145, 1299385395, 76663738305, 5136470466435, 385235284982625, 31974528653557875, 2909682107473766625, 288058528639902895875, 30822262564469609858625, 3544560194914005133741875, 435980903974422631450250625, 57113498420649364719982831875

Offset: 0

Philippe Deléham, Sep 20 2008

			a(0)=1, a(1)=3, a(2)=3*11=33, a(3)=3*11*19=627, a(4)=3*11*19*27=16929, ...

Cf. A001710, A001147, A008545, A011781, A017101, A032031, A047056, A048994, A144739.

Join[{1},FoldList[Times,8Range[0,20]+3]] (* Harvey P. Dale, Aug 11 2019 *)

a(n) = Sum_{k=0..n} A132393(n,k)*3^k*8^(n-k).
a(n) = (-5)^n*sum_{k=0..n} (8/5)^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
G.f.: 2/G(0), where G(k)= 1 + 1/(1 - 2*x*(8*k+3)/(2*x*(8*k+3) - 1 + 16*x*(k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 30 2013
a(n) +(-8*n+5)*a(n-1)=0. - R. J. Mathar, Sep 04 2016
From Ilya Gutkovskiy, Mar 23 2017: (Start)
E.g.f.: 1/(1 - 8*x)^(3/8).
a(n) ~ sqrt(2*Pi)*8^n*n^n/(exp(n)*n^(1/8)*Gamma(3/8)). (End)
Sum_{n>=0} 1/a(n) = 1 + (e/8^5)^(1/8)*(Gamma(3/8) - Gamma(3/8, 1/8)). - Amiram Eldar, Dec 20 2022

a(11) corrected by Ilya Gutkovskiy, Mar 23 2017

cp's OEIS Frontend

A132393 Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= k <= n.

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

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

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A049209 a(n) = -Product_{k=0..n} (7*k-1); sept-factorial numbers.

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A051188 Sept-factorial numbers.

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

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A147585 a(1) = 1; a(n) = (7n-9)a(n-1) for n > 1.