A011781 -id:A011781 - 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.
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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)

A085158 Sextuple factorials, 6-factorials, n!!!!!!, n!6.

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 6, 7, 16, 27, 40, 55, 72, 91, 224, 405, 640, 935, 1296, 1729, 4480, 8505, 14080, 21505, 31104, 43225, 116480, 229635, 394240, 623645, 933120, 1339975, 3727360, 7577955, 13404160, 21827575, 33592320, 49579075, 141639680

Offset: 0

Hugo Pfoertner, Jun 21 2003

nonn

The term "Sextuple factorial numbers" is also used for the sequences A008542, A008543, A011781, A047058, A047657, A049308, which have a different definition. The definition given here is the one commonly used.

			a(14) = 224 because 14*a(14-6) = 14*a(8) = 14*16 = 224.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Multifactorial.

Cf. n!:A000142, n!!:A006882, n!!!:A007661, n!!!!:A007662, n!!!!!:A085157, 6-factorial primes: n!!!!!!+1:A085150, n!!!!!!-1:A051592.

Cf. A288093.

a:= function(n)
    if n<1 then return 1;
    else return n*a(n-6);
    fi;
  end;
List([0..40], n-> a(n) ); # G. C. Greubel, Aug 21 2019

b:=func< n | n le 6 select n else n*Self(n-6) >;
[1] cat [b(n): n in [1..40]]; // G. C. Greubel, Aug 21 2019

a:= n-> `if`(n<1, 1, n*a(n-6)); seq(a(n), n=0..40); # G. C. Greubel, Aug 21 2019

Table[Times@@Range[n,1,-6],{n,0,40}] (* Harvey P. Dale, Aug 10 2019 *)

a(n)=if(n<1, 1, n*a(n-6));
vector(40, n, n--; a(n) ) \\ G. C. Greubel, Aug 21 2019

def a(n):
    if (n<1): return 1
    else: return n*a(n-6)
[a(n) for n in (0..40)] # G. C. Greubel, Aug 21 2019

a(n)=1 for n < 1, otherwise a(n) = n*a(n-6).

Sum_{n>=0} 1/a(n) = A288093. - Amiram Eldar, Nov 10 2020

A047657 Sextuple factorial numbers: a(n) = Product_{k=0..n-1} (6*k+2).

Original entry on oeis.org

1, 2, 16, 224, 4480, 116480, 3727360, 141639680, 6232145920, 311607296000, 17450008576000, 1081900531712000, 73569236156416000, 5444123475574784000, 435529878045982720000, 37455569511954513920000, 3445912395099815280640000, 337699414719781897502720000

Offset: 0

Joe Keane (jgk(AT)jgk.org)

G. C. Greubel, Table of n, a(n) for n = 0..340

Cf. A007559, A008542, A011781, A073005.

Cf. A000165, A008544, A001813, A047055, A084947, A084948, A084949.

List([0..20], n-> Product([0..n-1], k-> 6*k+2) ); # G. C. Greubel, Aug 18 2019

[1] cat [(&*[6*k+2: k in [0..n-1]]): n in [1..20]]; // G. C. Greubel, Aug 18 2019

a:= n->product(6*j+2, j=0..n-1); seq(a(n), n=0..20); # G. C. Greubel, Aug 18 2019

b[1]=2; b[n_]:= b[n] = b[n-1] +6; a[0]=1; a[1]=2; a[n_]:= a[n] = a[n-1]*b[n]; Table[a[n], {n,0,20}] (* Roger L. Bagula, Sep 17 2008 *)
FoldList[Times,1,6*Range[0,20]+2] (* Harvey P. Dale, Aug 06 2013 *)
Table[6^n*Pochhammer[1/3, n], {n,0,20}] (* G. C. Greubel, Aug 18 2019 *)

vector(20, n, n--; prod(k=0, n-1, 6*k+2)) \\ G. C. Greubel, Aug 18 2019

[product(6*k+2 for k in (0..n-1)) for n in (0..20)] # G. C. Greubel, Aug 18 2019

E.g.f.: (1-6*x)^(-1/3).
a(n) = 2^n*A007559(n).
a(n) = A084941(n)/A000142(n)*A000079(n) = 6^n*Pochhammer(1/3, n) = 1/2*6^n*Gamma(n+1/3)*sqrt(3)*Gamma(2/3)/Pi. - Daniel Dockery (peritus(AT)gmail.com), Jun 13 2003
Let b(n) = b(n-1) + 6; then a(n) = b(n)*a(n-1). - Roger L. Bagula, Sep 17 2008
G.f.: 1/(1-2*x/(1-6*x/(1-8*x/(1-12*x/(1-14*x/(1-18*x/(1-20*x/(1-24*x/(1-26*x/(1-... (continued fraction). - Philippe Deléham, Jan 08 2012
a(n) = (-4)^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
G.f.: 1/G(0) where G(k) = 1 - x*(6*k+2)/( 1 - 6*x*(k+1)/G(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 23 2013
D-finite with recurrence: a(n) +2*(-3*n+2)*a(n-1)=0. - R. J. Mathar, Jan 17 2020
Sum_{n>=0} 1/a(n) = 1 + exp(1/6)*(Gamma(1/3) - Gamma(1/3, 1/6))/6^(2/3). - Amiram Eldar, Dec 18 2022
a(n) ~ sqrt(Pi) * 2^(n+1/2) * (3/e)^n * n^(n-1/6) / Gamma(1/3). - Amiram Eldar, Sep 01 2025

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

A049308 Sextuple factorial numbers: Product_{k=0..n-1} (6*k+4).

Original entry on oeis.org

1, 4, 40, 640, 14080, 394240, 13404160, 536166400, 24663654400, 1282510028800, 74385581670400, 4760677226905600, 333247405883392000, 25326802847137792000, 2076797833465298944000, 182758209344946307072000, 17179271678424952864768000, 1717927167842495286476800000

Offset: 0

Joe Keane (jgk(AT)jgk.org)

G. C. Greubel, Table of n, a(n) for n = 0..340

Cf. A008543, A008544, A011781, A048994, A073006.

[n le 2 select 4^(n-1) else 2*(3*n-1)*Self(n-1): n in [1..30]]; // G. C. Greubel, Mar 29 2022

Table[6^n*Pochhammer[2/3, n], {n,0,30}] (* G. C. Greubel, Mar 29 2022 *)

a(n) = prod(k=0, n-1, 6*k+4); \\ Michel Marcus, Mar 30 2022

[6^n*rising_factorial(2/3,n) for n in (0..30)] # G. C. Greubel, Mar 29 2022

E.g.f.: (1-6*x)^(-2/3).
a(n) = 2^n*A008544(n).
G.f.: 1/(1-4*x/(1-6*x/(1-10*x/(1-12*x/(1-16*x/(1-18*x/(1-22*x/(1-24*x/(1-28*x/(1-... (continued fraction). - Philippe Deléham, Jan 08 2012
a(n) = (-2)^n*Sum_{k=0..n} 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
G.f.: ( 1 - 1/Q(0) )/x/2 where Q(k) = 1 - x*(6*k-2)/(1 - x*(6*k+6)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 20 2013
D-finite with recurrence: a(n) = 2*(3*n-1)*a(n-1). - R. J. Mathar, Jan 17 2020
From G. C. Greubel, Mar 29 2022: (Start)
a(n) = 6^n * Pochhammer(n, 2/3).
G.f.: Hypergeometric2F0([1, 2/3], [], 6*x). (End)
Sum_{n>=0} 1/a(n) = 1 + exp(1/6)*(Gamma(2/3) - Gamma(2/3, 1/6))/6^(1/3). - Amiram Eldar, Dec 18 2022
a(n) ~ sqrt(Pi) * 2^(n+1/2) * (3/e)^n * n^(n+1/6) / Gamma(2/3). - Amiram Eldar, Sep 01 2025

A273464 The number of tilings of an equilateral triangle of side length n with k lozenges and n^2 - 2k unit triangles. Triangle T(n, k) with n >= 1 and 0 <= k <= n(n + 1)/2, read by rows.

Original entry on oeis.org

1, 1, 3, 1, 9, 24, 18, 1, 18, 126, 434, 762, 630, 187, 1, 30, 387, 2814, 12699, 36894, 69242, 81936, 57672, 21432, 3135, 1, 45, 915, 11127, 90270, 515970, 2139120, 6523428, 14683401, 24256853, 28975770, 24383838, 13860321, 4966929, 989970, 81462, 1, 63

Offset: 1

R. J. Mathar, May 23 2016

			Triangle T(n,k) (with rows n >= 1 and columns k >= 0) begins as follows:
  1;
  1,  3;
  1,  9,  24,   18;
  1, 18, 126,  434,   762,   630,   187;
  1, 30, 387, 2814, 12699, 36894, 69242, 81936, 57672, 21432, 3135;
  ...

R. J. Mathar, Table of n, a(n) for n = 1..575
J. A. De Loera, J. Rambau, F. Santos, Further topics, in: Triangulations, vol 25 of Algor. Computat. Math. (2010), 433-511.
R. J. Mathar, Lozenge tilings of the equilateral triangle, arXiv:1909.06336 [math.CO], 2019.
Francisco Santos, The Cayley trick and triangulations of products of simplices, arXiv:math/0312069 [math.CO], 2004.
Francisco Santos, The Cayley trick and triangulations of products of simplices, Cont. Math. 374 (2005), 151-177.
Wikipedia, Lozenge.

Cf. A045943 (column k=1), A011555, A011556, A011781, A122722, A326367 (k=2), A326368 (k=3), A326369 (k=4), A000124 (row lengths).

T(n,2) = 3*(n-1)*(n-2)*(3*n^2+3*n-4)/8 . - R. J. Mathar, May 24 2016
T(n,3) = (n-2)*(9*n^5-9*n^4-81*n^3+81*n^2+160*n-192)/16. - Greg Dresden, Jul 03 2019
Conjecture: T(n,4) = 3*(n-2)*(n-3)*(9*n^6+9*n^5-135*n^4-81*n^3+670*n^2+104*n-1216)/128. - Greg Dresden, Jul 03 2019
Conjecture: T(n,5) = 3*(n-3)*(n+3)* (27*n^8 -135*n^7 -387*n^6 +2835*n^5 -168*n^4 -18732*n^3 +19568*n^2 +36992*n -56320)/1280. - R. J. Mathar, Jul 07 2019
From Petros Hadjicostas, Sep 13 2019: (Start)
Conjecture for rightmost terms: A122722(n) =  n! * T(n, n*(n+1)/2) for n >= 1.
Conjectures for column k >= 0: Sum_{0 <= s <= 2*k + 1} (-1)^s * binomial(2*k+1, s) * T(n-s, k) = 0 for n >= 2*k+2.
Sum_{0 <= s <= 2*k} (-1)^s * binomial(2*k, s) * T(n-s, k) = A011781(k) for n >= 2*k+1. (End)

A354252 Expansion of e.g.f. 1/sqrt(7 - 6 * exp(x)).

Original entry on oeis.org

1, 3, 30, 489, 11127, 325218, 11612595, 489926559, 23846152332, 1315294430043, 81078316924035, 5523729981650004, 412148874577007037, 33425421047034028743, 2927620572178735480350, 275410244285003264624949, 27695140477706524122414867

Offset: 0

Seiichi Manyama, May 21 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..343

Cf. A305404, A354242, A354253.
Cf. A094419, A346985, A354252, A365556, A365557.
Cf. A011781.

my(N=20, x='x+O('x^N)); Vec(serlaplace(1/sqrt(7-6*exp(x))))

my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, binomial(2*k, k)*(3*(exp(x)-1)/2)^k)))

a(n) = sum(k=0, n, (3/2)^k*(2*k)!*stirling(n, k, 2)/k!);

E.g.f.: Sum_{k>=0} binomial(2*k,k) * (3 * (exp(x) - 1)/2)^k.
a(n) = Sum_{k=0..n} (3/2)^k * (2*k)! * Stirling2(n,k)/k!.
a(n) ~ sqrt(2/7) * n^n / (exp(n) * log(7/6)^(n + 1/2)). - Vaclav Kotesovec, Jun 04 2022
a(0) = 1; a(n) = Sum_{k=1..n} (6 - 3*k/n) * binomial(n,k) * a(n-k). - Seiichi Manyama, Sep 09 2023
a(0) = 1; a(n) = 3*a(n-1) - 7*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). - Seiichi Manyama, Nov 17 2023

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

A011555 Number of vertices of secondary polytope for triangle X n-simplex.

Original entry on oeis.org

6, 108, 4488, 376200

Offset: 1

N. J. A. Sloane

Gelfand, Kapranov and Zelevinsky, Discriminants, Resultants and Multidimensional Determinants, Birkhäuser, 1994, pp. 243-250.

J. A. de Loera, Nonregular triangulations of products of simplices, Discrete Comp. Geom. 15(3) (1996), 253-264.

Cf. A011556, A011781, A122722, A273464, A326367, A326368, A326369.

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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A085158 Sextuple factorials, 6-factorials, n!!!!!!, n!6.

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A047657 Sextuple factorial numbers: a(n) = Product_{k=0..n-1} (6*k+2).

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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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A049308 Sextuple factorial numbers: Product_{k=0..n-1} (6*k+4).

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A273464 The number of tilings of an equilateral triangle of side length n with k lozenges and n^2 - 2*k unit triangles. Triangle T(n, k) with n >= 1 and 0 <= k <= n*(n + 1)/2, read by rows.

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

A273464 The number of tilings of an equilateral triangle of side length n with k lozenges and n^2 - 2k unit triangles. Triangle T(n, k) with n >= 1 and 0 <= k <= n(n + 1)/2, read by rows.