A032031 -id:A032031 - cp's OEIS frontend

A352069 Expansion of e.g.f. 1 / (1 + log(1 - 3*x) / 3).

1, 1, 5, 42, 492, 7374, 134478, 2887128, 71281656, 1988802720, 61860849552, 2121993490176, 79566300371952, 3237181141173264, 142019158472311248, 6682603650677875584, 335698708873243355136, 17930674324049810882688, 1014685181110897126616448, 60641642160287342580586752

Offset: 0

Ilya Gutkovskiy, Mar 02 2022

nonn

Cf. A007840, A032031, A087674, A227917, A255927, A352071.

nmax = 19; CoefficientList[Series[1/(1 + Log[1 - 3 x]/3), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS1[n, k] k! (-3)^(n - k), {k, 0, n}], {n, 0, 19}]

my(x='x+O('x^25)); Vec(serlaplace(1/(1+log(1-3*x)/3))) \\ Michel Marcus, Mar 02 2022

a(n) = Sum_{k=0..n} Stirling1(n,k) * k! * (-3)^(n-k).
a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * (k-1)! * 3^(k-1) * a(n-k).
a(n) ~ n! * 3^(n+1) * exp(3*n) / (exp(3) - 1)^(n+1). - Vaclav Kotesovec, Mar 03 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

A145448 a(n) = 12^n*n!.

Original entry on oeis.org

1, 12, 288, 10368, 497664, 29859840, 2149908480, 180592312320, 17336861982720, 1872381094133760, 224685731296051200, 29658516531078758400, 4270826380475341209600, 666248915354153228697600

Offset: 0

Richard V. Scholtz, III, Oct 10 2008

12-factorial numbers.

Let G(z) = Gamma(z)/(sqrt(2*Pi)*z^(z-1/2)*exp(-z)). For any z > 0 the bounds 1 < G(z) < exp(1/(12*z)) = 1 + 1/(12*z) + 1/(288*z^2) + 1/(10368*z^3) + ... hold. G. Nemes improved the upper bound to 1 + 1/(12*z) + 1/(288*z^2) which gives a simple estimate for the Gamma function on the positive real line. - Peter Luschny, Sep 24 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Gergő Nemes, Error bounds and exponential improvements for the asymptotic expansions of the gamma function and its reciprocal, Proceedings of the Royal Society of Edinburgh, 145A, pp. 571-596, 2015.

Cf. A000142, A000165, A032031.

[(Factorial(n)*12^n): n in [0..20]]; // Vincenzo Librandi, Oct 28 2011

Table[12^n*n!, {n,0,30}] (* G. C. Greubel, Mar 24 2022 *)

[12^n*factorial(n) for n in (0..30)] # G. C. Greubel, Mar 24 2022

E.g.f.: 1/(1-12*x). - Philippe Deléham, Oct 28 2011
G.f.: 1/(1 - 12*x/(1 - 12*x/(1 - 24*x/(1 - 24*x/(1 - 36*x/(1 - 36*x/(1 - ...))))))), a continued fraction. - Ilya Gutkovskiy, Aug 09 2017
From Amiram Eldar, Jun 25 2020: (Start)
Sum_{n>=0} 1/a(n) = e^(1/12).
Sum_{n>=0} (-1)^n/a(n) = e^(-1/12). (End)

a(0)=1 prepended by Richard V. Scholtz, III, Mar 11 2009

a(10)-a(13) corrected by Vincenzo Librandi, Oct 28 2011

A203412 Triangle read by rows, a(n,k), n>=k>=1, which represent the s=3, h=1 case of a two-parameter generalization of Stirling numbers arising in conjunction with normal ordering.

Original entry on oeis.org

1, 1, 1, 4, 3, 1, 28, 19, 6, 1, 280, 180, 55, 10, 1, 3640, 2260, 675, 125, 15, 1, 58240, 35280, 10360, 1925, 245, 21, 1, 1106560, 658000, 190680, 35385, 4620, 434, 28, 1, 24344320, 14266560, 4090240, 756840, 100065, 9828, 714, 36, 1

Offset: 1

Mark Shattuck, Jan 01 2012

Also the Bell transform of the triple factorial numbers A007559 which adds a first column (1,0,0 ...) on the left side of the triangle. For the definition of the Bell transform see A264428. See A051141 for the triple factorial numbers A032031 and A004747 for the triple factorial numbers A008544 as well as A039683 and A132062 for the case of double factorial numbers. - Peter Luschny, Dec 23 2015

			Triangle starts:
[    1]
[    1,     1]
[    4,     3,     1]
[   28,    19,     6,    1]
[  280,   180,    55,   10,   1]
[ 3640,  2260,   675,  125,  15,  1]
[58240, 35280, 10360, 1925, 245, 21, 1]

Richell O. Celeste, Roberto B. Corcino, and Ken Joffaniel M. Gonzales. Two Approaches to Normal Order Coefficients, Journal of Integer Sequences, Vol. 20 (2017), Article 17.3.5.
T. Mansour, M. Schork, and M. Shattuck, On a new family of generalized Stirling and Bell numbers, Electron. J. Combin. 18 (2011) #P77 (33 pp.).
Toufik Mansour, Matthias Schork and Mark Shattuck, On the Stirling numbers associated with the meromorphic Weyl algebra, Applied Mathematics Letters, Volume 25, Issue 11, November 2012, Pages 1767-1771.

Cf. A007559, A032031, A039683, A051141, A132062, A264428.

A203412 := (n,k) -> (n!*3^n)/(k!*2^k)*add((-1)^j*binomial(k,j)*binomial(n-2*j/3-1, n), j=0..k): seq(seq(A203412(n,k),k=1..n),n=1..9); # Peter Luschny, Dec 21 2015

Table[(n! 3^n)/(k! 2^k) Sum[ (-1)^j Binomial[k, j] Binomial[n - 2 j/3 - 1, n], {j, 0, k}], {n, 9}, {k, n}] // Flatten (* Michael De Vlieger, Dec 23 2015 *)

# uses[bell_transform from A264428]
triplefactorial = lambda n: prod(3*k + 1 for k in (0..n-1))
def A203412_row(n):
    trifact = [triplefactorial(k) for k in (0..n)]
    return bell_transform(n, trifact)
[A203412_row(n) for n in (0..8)] # Peter Luschny, Dec 21 2015

(1) Is given by the recurrence relation
  a(n+1,k) = a(n,k-1)+(3*n-2*k)*a(n,k) if n>=0 and k>=1, along with the initial values a(n,0) = delta_{n,0} and a(0,k) = delta_{0,k} for all n,k>=0.
(2) Is given explicitly by
  a(n,k) = (n!*3^n)/(k!*2^k)*Sum{j=0..k} (-1)^j*C(k,j)*C(n-2*j/3-1,n) for all n>=k>=1.
a(n,1) = A007559(n-1). - Peter Luschny, Dec 21 2015

A304117 If n = Product (p_j^k_j) then a(n) = Product (pi(p_j)*k_j), where pi() = A000720.

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 4, 3, 4, 3, 5, 4, 6, 4, 6, 4, 7, 4, 8, 6, 8, 5, 9, 6, 6, 6, 6, 8, 10, 6, 11, 5, 10, 7, 12, 8, 12, 8, 12, 9, 13, 8, 14, 10, 12, 9, 15, 8, 8, 6, 14, 12, 16, 6, 15, 12, 16, 10, 17, 12, 18, 11, 16, 6, 18, 10, 19, 14, 18, 12, 20, 12, 21, 12, 12, 16, 20, 12, 22, 12

Offset: 1

Ilya Gutkovskiy, May 06 2018

			a(36) = 8 because 36 = 2^2*3^2 = prime(1)^2*prime(2)^2 and 1*2*2*2 = 8.

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Ilya Gutkovskiy, Extended graphical example
Index entries for sequences computed from indices in prime factorization
Index entries for sequences computed from exponents in factorization of n

Cf. A000026, A000040, A000720, A001414, A002110, A003963, A005361, A056239, A156061, A304037.

a[n_] := Times @@ (PrimePi[#[[1]]] #[[2]] & /@ FactorInteger[n]); a[1] = 1; Table[a[n], {n, 1, 80}]

a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1] = primepi(f[k,1])*f[k,2]; f[k, 2] = 1); factorback(f); \\ Michel Marcus, May 06 2018

a(n) = A005361(n)*A156061(n).
a(p^k) = A000720(p)*k where p is a prime.
a(A002110(m)^k) = k^m*m!.
As an example:
a(A000040(k)) = k.
a(A006450(k)) = A000040(k).
a(A001248(k)) = a(A031215(k)) = A005843(k).
a(A030078(k)) = a(A031336(k)) = A008585(k)
a(A061742(k)) = A000165(k).
a(A115964(k)) = A032031(k).
a(A002110(k)) = A000142(k).
a(A080696(k)) = A002110(k).

A091535 First column (k=2) of array A091534 ((5,2)-Stirling2).

Original entry on oeis.org

1, 20, 1120, 123200, 22422400, 6098892800, 2317579264000, 1172695107584000, 762251819929600000, 618948477782835200000, 613996889960572518400000, 730656299053081296896000000, 1027302756468632303435776000000, 1684776520608556977634672640000000

Offset: 1

Wolfdieter Lang, Jan 23 2004

The scaled sequence (2/(3n-1)!!!)*a(n) = (3*n-2)!!! = A007559(n), n>=1.

Third column of array A091752 ((-1, 2)-Stirling2).

Cf. A007559, A032031, A091534.

a := n -> 9^n*GAMMA(n+1/3)*GAMMA(n+2/3)*sqrt(3)/(4*Pi);
seq(a(n), n=1..16); # Peter Luschny, Sep 17 2014

a[n_] := (3*n-1)!/(2!*3^(n-1)*(n-1)!); Array[a, 15] (* Amiram Eldar, Sep 01 2025 *)

a(n) = (3*n-1)!/(2!*3^(n-(2-1))*(n-1)!) = ((3*n-1)!/2)/A032031(n-1).
a(n) = A091534(n, 2), n>=1.
E.g.f.: (hypergeom([1/3, 2/3, 1], [], 9*x)-1)/2.
a(n) = 9^n*Gamma(n+1/3)*Gamma(n+2/3)*sqrt(3)/(4*Pi). - Peter Luschny, Sep 17 2014
a(n) ~ (sqrt(3)/2) * (3*n/e)^(2*n). - Amiram Eldar, Sep 01 2025

A091539 Second column (k=3) of array A091534 ((5,2)-Stirling2) divided by 10.

Original entry on oeis.org

1, 104, 16192, 3745280, 1222291840, 537758144000, 307503360102400, 221965373351321600, 197530935371241472000, 212553938009841139712000, 272115940122123843665920000, 408828811133790954169303040000, 712427095375430807967713198080000, 1425431682224708301179257251430400000

Offset: 2

Wolfdieter Lang, Feb 13 2004

Pawel Blasiak, Karol A. Penson, and Allan I. Solomon, The general boson normal ordering problem, Physics Letters A, Vol. 309, No. 3-4 (2003), pp. 198-205; arXiv preprint, arXiv:quant-ph/0402027, 2004.

Cf. A091534, A091540.

Cf. A032031, A008544, A007559, A007661.

a[n_] := 3^(2*n) * Pochhammer[2/3, n] * (n! - 3 * Pochhammer[1/3, n])/(3!*10); Array[a, 20, 2] (* Amiram Eldar, Aug 30 2025 *)

a(n) = A091534(n, 3)/10, n >= 2.
a(n) = Product_{j=0..n-1} (3*j + 2)*(Product_{j=0..n-1} (3*(j+1)) - 3*Product_{j=0..n-1} (3*j + 1))/(3!*10). From eq. (12) of the Blasiak et al. reference for r=5, s=2 and k=3.
a(n) = (3^(2*n))*risefac(2/3, n)*(n!-3*risefac(1/3, n))/(3!*10), with risefac(x, n) = Pochhammer(x, n).
a(n) = (fac3(3*n-1)/10)*(fac3(3*n) - 3*fac3(3*n-2))/3!, with fac3(3*n) = A032031(n) = n!*3^n, fac3(3*n-1) = A008544(n) and fac3(3*n-2) = A007559(n) (triple factorials: fac3(n) = A007661(n)).
E.g.f.: (hypergeom([2/3, 1], [], 9*x)-3*hypergeom([1/3, 2/3], [], 9*x)+2)/(3!*10).
a(n) ~ Pi * 3^(2*n) *  n^(2*n + 2/3) / (30 * Gamma(2/3) * exp(2*n)). - Amiram Eldar, Aug 30 2025

A091540 Rescaled second column A091539 of array A091534 ((5,2)-Stirling2).

Original entry on oeis.org

1, 13, 184, 3040, 58360, 1283800, 31917760, 886123840, 27192323200, 914387689600, 33446228569600, 1322364153510400, 56203860301388800, 2555756347720576000, 123819357959385088000, 6367367706293321728000

Offset: 2

Wolfdieter Lang, Feb 13 2004

A certain difference of two triple factorial sequences.

If offset 0: exponential (also called binomial) convolution of A091541 and A051606.

G. C. Greubel, Table of n, a(n) for n = 2..380

Cf. A091541.

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( (3-2*(1-3*x)^(2/3))/(1-3*x)^3 )); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Aug 15 2018

Drop[With[{nmax = 50}, CoefficientList[Series[(1 - 2*x - (1 - 3*x)^(2/3))/(2*(1 - 3*x)), {x, 0, nmax}], x]*Range[0, nmax]!],2] (* G. C. Greubel, Aug 15 2018 *)

x='x+O('x^30); Vec(serlaplace((1 - 2*x - (1 - 3*x)^(2/3))/(2*(1 - 3*x)))) \\ G. C. Greubel, Aug 15 2018

a(n)= (5*2/fac3(3*n-1))*A091539(n), n>=2, with fac3(3*n-1) := A008544(n) (triple factorials).
E.g.f.: (1-2*x-(1-3*x)^(2/3))/(2*(1-3*x))= (1/2-x+int((1-3*x)^(-1/3), x))/(1-3*x).
E.g.f. with offset 0: (3-2*(1-3*x)^(2/3))/(1-3*x)^3.
a(n)=(fac3(3*n) - 3*fac3(3*n-2))/3! with fac3(3*n) := A032031(n)= n!*3^n and fac3(3*n-2) := A007559(n).
a(n) ~ 3^(n-1) * n! / 2. - Vaclav Kotesovec, Aug 16 2018

A131182 Table T(n,k) = n!*k^n, read by upwards antidiagonals.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 6, 8, 3, 1, 0, 24, 48, 18, 4, 1, 0, 120, 384, 162, 32, 5, 1, 0, 720, 3840, 1944, 384, 50, 6, 1, 0, 5040, 46080, 29160, 6144, 750, 72, 7, 1, 0, 40320, 645120, 524880, 122880, 15000, 1296, 98, 8, 1, 0, 362880, 10321920, 11022480, 2949120, 375000, 31104, 2058, 128, 9, 1

Offset: 0

Philippe Deléham, Sep 25 2007

For k>0, T(n,k) is the n-th moment of the exponential distribution with mean = k. - Geoffrey Critzer, Jan 06 2019

T(n,k) is the minimum value of Product_{i=1..n} Sum_{j=1..k} r_j[i] where each r_j is a permutation of {1..n}. For the maximum value, see A331988. - Chai Wah Wu, Sep 01 2022

			The (inverted) table begins:
k=0: 1, 0,   0,    0,      0,       0, ... (A000007)
k=1: 1, 1,   2,    6,     24,     120, ... (A000142)
k=2: 1, 2,   8,   48,    384,    3840, ... (A000165)
k=3: 1, 3,  18,  162,   1944,   29160, ... (A032031)
k=4: 1, 4,  32,  384,   6144,  122880, ... (A047053)
k=5: 1, 5,  50,  750,  15000,  375000, ... (A052562)
k=6: 1, 6,  72, 1296,  31104,  933120, ... (A047058)
k=7: 1, 7,  98, 2058,  57624, 2016840, ... (A051188)
k=8: 1, 8, 128, 3072,  98304, 3932160, ... (A051189)
k=9: 1, 9, 162, 4374, 157464, 7085880, ... (A051232)
Main diagonal is 1, 1, 8, 162, 6144, 375000, ... (A061711).

Chai Wah Wu, Permutations r_j such that ∑i∏j r_j(i) is maximized or minimized, arXiv:1508.02934 [math.CO], 2015-2020.
Chai Wah Wu, On rearrangement inequalities for multiple sequences, arXiv:2002.10514 [math.CO], 2020.

Columns k=0-9 give: A000007, A000142, A000165, A032031, A047053, A052562, A047058, A051188, A051189, A051232.
Main diagonal gives A061711.
Cf. A292783, A303489, A331988.

T:= (n,k)-> n!*k^n:
seq(seq(T(d-k, k), k=0..d), d=0..12);  # Alois P. Heinz, Jan 06 2019

from math import factorial
def A131182_T(n, k): # compute T(n, k)
    return factorial(n)*k**n # Chai Wah Wu, Sep 01 2022

From Ilya Gutkovskiy, Aug 11 2017: (Start)
G.f. of column k: 1/(1 - k*x/(1 - k*x/(1 - 2*k*x/(1 - 2*k*x/(1 - 3*k*x/(1 - 3*k*x/(1 - ...))))))), a continued fraction.
E.g.f. of column k: 1/(1 - k*x). (End)

A153271 Triangle T(n, k) = Product_{j=0..k} (j*n + prime(m)), with T(n, 0) = prime(m) and m = 3, read by rows.

Original entry on oeis.org

5, 5, 30, 5, 35, 315, 5, 40, 440, 6160, 5, 45, 585, 9945, 208845, 5, 50, 750, 15000, 375000, 11250000, 5, 55, 935, 21505, 623645, 21827575, 894930575, 5, 60, 1140, 29640, 978120, 39124800, 1838865600, 99298742400, 5, 65, 1365, 39585, 1464645, 65909025, 3493178325, 213083877825, 14702787569925

Offset: 0

Roger L. Bagula, Dec 22 2008

Row sums are {5, 35, 355, 6645, 219425, 11640805, 917404295, 101177741765, 14919432040765, 2839006665525525, 677815000136926955, ...}.

			Triangle begins as:
  5;
  5, 30;
  5, 35, 315;
  5, 40, 440,  6160;
  5, 45, 585,  9945, 208845;
  5, 50, 750, 15000, 375000, 11250000;
  5, 55, 935, 21505, 623645, 21827575, 894930575;

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Cf. A153271 (m=2), this sequence (m=3), A153272 (m=4).
Sequences related to m values:
m=2: A001147, A001710, A008545, A032031, A047056, A144739, A144758.
m=3: A001720, A007696, A008543, A034000, A051577, A052562, A147585, A147625, A147629.

m:=3;
function T(n,k)
  if k eq 0 then return NthPrime(m);
  else return (&*[j*n + NthPrime(m): j in [0..k]]);
  end if; return T; end function;
[T(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Dec 03 2019

m:=3; seq(seq(`if`(k=0, ithprime(m), mul(j*n + ithprime(m), j=0..k)), k=0..n), n=0..10); # G. C. Greubel, Dec 03 2019

T[n_, k_, m_]:= If[k==0, Prime[m], Product[j*n + Prime[m], {j,0,k}]];
Table[T[n,k,3], {n,0,10}, {k,0,n}]//Flatten

T(n,k) = my(m=3); if(k==0, prime(m), prod(j=0,k, j*n + prime(m)) ); \\ G. C. Greubel, Dec 03 2019

def T(n, k):
    m=3
    if (k==0): return nth_prime(m)
    else: return product(j*n + nth_prime(m) for j in (0..k))
[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Dec 03 2019

T(n, k) = Product_{j=0..k} (j*n + prime(m)), with T(n, 0) = prime(m) and m = 3.

Edited by G. C. Greubel, Dec 03 2019

cp's OEIS Frontend

A352069 Expansion of e.g.f. 1 / (1 + log(1 - 3*x) / 3).

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

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A145448 a(n) = 12^n*n!.

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A203412 Triangle read by rows, a(n,k), n>=k>=1, which represent the s=3, h=1 case of a two-parameter generalization of Stirling numbers arising in conjunction with normal ordering.

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A304117 If n = Product (p_j^k_j) then a(n) = Product (pi(p_j)*k_j), where pi() = A000720.

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A091535 First column (k=2) of array A091534 ((5,2)-Stirling2).

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A091539 Second column (k=3) of array A091534 ((5,2)-Stirling2) divided by 10.

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A091540 Rescaled second column A091539 of array A091534 ((5,2)-Stirling2).

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