A007559 -id:A007559 - cp's OEIS frontend

A286718 Triangle read by rows: T(n, k) is the Sheffer triangle ((1 - 3x)^(-1/3), (-1/3)log(1 - 3*x)). A generalized Stirling1 triangle.

1, 1, 1, 4, 5, 1, 28, 39, 12, 1, 280, 418, 159, 22, 1, 3640, 5714, 2485, 445, 35, 1, 58240, 95064, 45474, 9605, 1005, 51, 1, 1106560, 1864456, 959070, 227969, 28700, 1974, 70, 1, 24344320, 42124592, 22963996, 5974388, 859369, 72128, 3514, 92, 1, 608608000, 1077459120, 616224492, 172323696, 27458613, 2662569, 159978, 5814, 117, 1, 17041024000, 30777463360, 18331744896, 5441287980, 941164860, 102010545, 7141953, 322770, 9090, 145, 1

Offset: 0

Wolfdieter Lang, May 18 2017

This is a generalization of the unsigned Stirling1 triangle A132393.
In general the lower triangular Sheffer matrix ((1 - d*x)^(-a/d), (-1/d)*log(1 - d*x)) is called here |S1hat[d,a]|. The signed matrix S1hat[d,a] with elements (-1)^(n-k)*|S1hat[d,a]|(n, k) is the inverse of the generalized Stirling2 Sheffer matrix S2hat[d,a] with elements S2[d,a](n, k)/d^k, where S2[d,a] is Sheffer (exp(a*x), exp(d*x) - 1).
In the Bala link the signed S1hat[d,a] (with row scaled elements S1[d,a](n,k)/d^n where S1[d,a] is the inverse matrix of S2[d,a]) is denoted by s_{(d,0,a)}, and there the notion exponential Riordan array is used for Sheffer array.
In the Luschny link the elements of |S1hat[m,m-1]| are called Stirling-Frobenius cycle numbers SF-C with parameter m.
From Wolfdieter Lang, Aug 09 2017: (Start)
The general row polynomials R(d,a;n,x) = Sum_{k=0..n} T(d,a;n,k)*x^k of the Sheffer triangle |S1hat[d,a]| satisfy, as special polynomials of the Boas-Buck class (see the reference), the identity (we use the notation of Rainville, Theorem 50, p. 141, adapted to an exponential generating function)
  (E_x - n*1)*R(d,a;n,x) = -n!*Sum_{k=0..n-1} d^k*(a*1 + d*beta(k)*E_x)*R(d,a;n-1-k,x)/(n-1-k)!, for n >= 0, with E_x = x*d/dx (Euler operator), and beta(k) = A002208(k+1)/A002209(k+1).
This entails a recurrence for the sequence of column k, for n > k >= 0: T(d,a;n,k) = (n!/(n - k))*Sum_{p=k..n-1} d^(n-1-p)*(a + d*k*beta(n-1-p))*T(d,a;p,k)/p!, with input T(d,a;k,k) = 1. For the present [d,a] = [3,1] case see the formula and example sections below. (End)
The inverse of the Sheffer triangular matrix S2[3,1] = A282629 is the Sheffer matrix S1[3,1] = (1/(1 + x)^(1/3), log(1 + x)/3) with rational elements S1[3,1](n, k) = (-1)^(n-m)*T(n, k)/3^n. - Wolfdieter Lang, Nov 15 2018

			The triangle T(n, k) begins:
n\k        0        1        2       3      4     5    6  7 8 ...
O:         1
1:         1        1
2:         4        5        1
3:        28       39       12       1
4:       280      418      159      22      1
5:      3640     5714     2485     445     35     1
6:     58240    95064    45474    9605   1005    51    1
7:   1106560  1864456   959070  227969  28700  1974   70  1
8:  24344320 42124592 22963996 5974388 859369 72128 3514 92 1
...
From _Wolfdieter Lang_, Aug 09 2017: (Start)
Recurrence: T(3, 1) = T(2, 0) + (3*3-2)*T(2, 1) = 4 + 7*5 = 39.
Boas-Buck recurrence for column k = 2 and n = 5:
T(5, 2) = (5!/3)*(3^2*(1 + 6*(3/8))*T(2,2)/2! + 3*(1 + 6*(5/12)*T(3, 2)/3! + (1 + 6*(1/2))* T(4, 2)/4!)) = (5!/3)*(9*(1 + 9/4)/2 + 3*(1 + 15/6)*12/6 + (1 + 3)*159/24) = 2485.
The beta sequence begins: {1/2, 5/12, 3/8, 251/720, 95/288, 19087/60480, ...}.
(End)

Ralph P. Boas, jr. and R. Creighton Buck, Polynomial Expansions of analytic functions, Springer, 1958, pp. 17 - 21, (last sign in eq. (6.11) should be -).
Earl D. Rainville, Special Functions, The Macmillan Company, New York, 1960, ch. 8, sect. 76, 140 - 146.

P. Bala, A 3 parameter family of generalized Stirling numbers
Wolfdieter Lang, On Sums of Powers of Arithmetic Progressions, and Generalized Stirling, Eulerian and Bernoulli Numbers, arXiv:math/1707.04451 [math.NT], July 2017.
Wolfdieter Lang, On Generating functions of Diagonals Sequences of Sheffer and Riordan Number Triangles, arXiv:1708.01421 [math.NT], August 2017.
Peter Luschny, The Stirling-Frobenius numbers.

S2[d,a] for [d,a] = [1,0], [2,1], [3,1], [3,2], [4,1] and [4,3] is A048993, A154537, A282629, A225466, A285061 and A225467, respectively.
S2hat[d,a] for these [d,a] values is A048993, A039755, A111577 (offset 0), A225468, A111578 (offset 0) and A225469, respectively.
|S1hat[d,a]| for [d,a] = [1,0], [2,1], [3,2], [4,1] and [4,3] is A132393, A028338, A225470, A290317 and A225471, respectively.
Column sequences for k = 0..4: A007559, A024216(n-1), A286721(n-2), A382984, A382985.
Diagonal sequences: A000012, A000326(n+1), A024212(n+1), A024213(n+1).
Row sums: A008544. Alternating row sums: A000007.
Beta sequence: A002208(n+1)/A002209(n+1).

T[n_ /; n >= 1, k_] /; 0 <= k <= n := T[n, k] = T[n-1, k-1] + (3*n-2)* T[n-1, k]; T[, -1] = 0; T[0, 0] = 1; T[n, k_] /; nJean-François Alcover, Jun 20 2018 *)

Recurrence: T(n, k) = T(n-1, k-1) + (3*n-2)*T(n-1, k), for n >= 1, k = 0..n, and T(n, -1) = 0, T(0, 0) = 1 and T(n, k) = 0 for n < k.
E.g.f. of row polynomials R(n, x) = Sum_{k=0..n} T(n, k)*x^k (i.e., e.g.f. of the triangle) is (1 - 3*z)^{-(x+1)/3}.
E.g.f. of column k is (1 - 3*x)^(-1/3)*((-1/3)*log(1 - 3*x))^k/k!.
Recurrence for row polynomials is R(n, x) = (x+1)*R(n-1, x+3), with R(0, x) = 1.
Row polynomial R(n, x) = risefac(3,1;x,n) with the rising factorial
risefac(d,a;x,n) := Product_{j=0..n-1} (x + (a + j*d)). (For the signed case see the Bala link, eq. (16)).
T(n, k) = sigma^{(n)}{n-k}(a_0,a_1,...,a{n-1}) with the elementary symmetric functions with indeterminates a_j = 1 + 3*j.
T(n, k) = Sum_{j=0..n-k} binomial(n-j, k)*|S1|(n, n-j)*3^j, with the unsigned Stirling1 triangle |S1| = A132393.
Boas-Buck column recurrence (see a comment above): T(n, k) =
(n!/(n - k))*Sum_{p=k..n-1} 3^(n-1-p)*(1 + 3*k*beta(n-1-p))*T(p, k)/p!, for n > k >= 0, with input T(k, k) = 1, with beta(k) = A002208(k+1)/A002209(k+1). See an example below. - Wolfdieter Lang, Aug 09 2017

A034689 a(n) = n-th sextic factorial number divided by 2.

Original entry on oeis.org

1, 8, 112, 2240, 58240, 1863680, 70819840, 3116072960, 155803648000, 8725004288000, 540950265856000, 36784618078208000, 2722061737787392000, 217764939022991360000, 18727784755977256960000, 1722956197549907640320000, 168849707359890948751360000

Offset: 1

Wolfdieter Lang

G. C. Greubel, Table of n, a(n) for n = 1..325
Index entries for sequences related to factorial numbers.

Cf. A007559, A008542, A034723, A034724, A034787, A034788, A047657, A047058.

[n le 1 select 1 else (6*n-4)*Self(n-1): n in [1..40]]; // G. C. Greubel, Oct 21 2022

Table[6^n*Pochhammer[1/3, n]/2, {n, 40}] (* G. C. Greubel, Oct 21 2022 *)

[6^n*rising_factorial(1/3,n)/2 for n in range(1,40)] # G. C. Greubel, Oct 21 2022

2*a(n) = (6*n-4)(!^6) = Product_{j=1..n} (6*j-4) = 2^n*A007559(n), A007559(n) = (3*n-2)(!^3) = Product_{j=1..n} (3*j-2).
E.g.f.: (-1 + (1-6*x)^(-1/3))/2.
D-finite with recurrence: a(n) = 2*(3*n-2)*a(n-1). - R. J. Mathar, Feb 24 2020
a(n) = 3*6^(n-1)*Pochhammer(n, 1/3). - G. C. Greubel, Oct 21 2022
From Amiram Eldar, Dec 18 2022: (Start)
a(n) = A047657(n)/2.
Sum_{n>=1} 1/a(n) = 2*(e/6^4)^(1/6)*(Gamma(1/3, 1/6) - Gamma(1/3)). (End)

A035022 One eighth of 9-factorial numbers.

Original entry on oeis.org

1, 17, 442, 15470, 680680, 36076040, 2236714480, 158806728080, 12704538246400, 1130703903929600, 110808982585100800, 11856561136605785600, 1375361091846271129600, 171920136480783891200000, 23037298288425041420800000, 3294333655244780923174400000, 500738715597206700322508800000

Offset: 1

Wolfdieter Lang

G. C. Greubel, Table of n, a(n) for n = 1..325
Index entries for sequences related to factorial numbers.

Cf. A007559, A034000, A034171, A035012, A035013, A035017, A049211.

Cf. A035018, A035020, A035021, A035022, A035023, A045756.

[n le 1 select 1 else (9*n-1)*Self(n-1): n in [1..40]]; // G. C. Greubel, Oct 19 2022

f := gfun:-rectoproc({(9*n - 1)*a(n - 1) - a(n) = 0, a(1) = 1}, a(n), remember);
map(f, [$ (1 .. 20)]); # Georg Fischer, Feb 15 2020

Table[9^n*Pochhammer[8/9, n]/8, {n, 40}] (* G. C. Greubel, Oct 19 2022 *)

[9^n*rising_factorial(8/9,n)/8 for n in range(1,40)] # G. C. Greubel, Oct 19 2022

8*a(n) = (9*n-1)(!^9) := Product_{j=1..n} (9*j - 1).
a(n) = (9*n)!/(n!*2^4*3^(4*n)*5*7*A045756(n)*A035012(n)*A007559(n)*A035017(n) *A035018(n)*A034000(n) *A035021(n)).
E.g.f.: (-1+(1-9*x)^(-8/9))/8.
D-finite with recurrence: a(1) = 1, a(n) = (9*n - 1)*a(n-1) for n > 1. - Georg Fischer, Feb 15 2020
a(n) = (1/8) * 9^n * Pochhammer(n, 8/9). - G. C. Greubel, Oct 19 2022
From Amiram Eldar, Dec 21 2022: (Start)
a(n) = A049211(n)/8.
Sum_{n>=1} 1/a(n) = 8*(e/9)^(1/9)*(Gamma(8/9) - Gamma(8/9, 1/9)). (End)

A035023 One ninth of 9-factorial numbers.

Original entry on oeis.org

1, 18, 486, 17496, 787320, 42515280, 2678462640, 192849310080, 15620794116480, 1405871470483200, 139181275577836800, 15031577762406374400, 1758694598201545804800, 221595519373394771404800, 29915395115408294139648000, 4307816896618794356109312000

Offset: 1

Wolfdieter Lang

E.g.f. is g.f. for A001019(n-1) (powers of nine).

G. C. Greubel, Table of n, a(n) for n = 1..325
Index entries for sequences related to factorial numbers.

Cf. A000142, A007559, A001019, A034171, A035012, A035013.

Cf. A035017, A035018, A035020, A035021, A035022, A045756.

[9^(n-1)*Factorial(n): n in [1..40]]; // G. C. Greubel, Oct 19 2022

With[{nn=20},Rest[CoefficientList[Series[(-1+1/(1-9*x))/9,{x,0,nn}],x] Range[ 0,nn]!]] (* Harvey P. Dale, Apr 07 2019 *)
Table[9^(n-1)*n!, {n, 40}] (* G. C. Greubel, Oct 19 2022 *)

[9^(n-1)*factorial(n) for n in range(1,40)] # G. C. Greubel, Oct 19 2022

9*a(n) = (9*n)(!^9) = Product_{j=1..n} 9*j = 9^n*n!.
E.g.f.: (-1+1/(1-9*x))/9.
D-finite with recurrence: a(n) - 9*n*a(n-1) = 0. - R. J. Mathar, Jan 28 2020
From Amiram Eldar, Jan 08 2022: (Start)
Sum_{n>=1} 1/a(n) = 9*(exp(1/9)-1).
Sum_{n>=1} (-1)^(n+1)/a(n) = 9*(1-exp(-1/9)). (End)
a(n) = A001019(n-1) * A000142(n). - G. C. Greubel, Oct 19 2022

A051141 Triangle read by rows: a(n, m) = S1(n, m)*3^(n-m), where S1 are the signed Stirling numbers of first kind A008275 (n >= 1, 1 <= m <= n).

Original entry on oeis.org

1, -3, 1, 18, -9, 1, -162, 99, -18, 1, 1944, -1350, 315, -30, 1, -29160, 22194, -6075, 765, -45, 1, 524880, -428652, 131544, -19845, 1575, -63, 1, -11022480, 9526572, -3191076, 548289, -52920, 2898, -84, 1, 264539520, -239660208

Offset: 1

Wolfdieter Lang

Previous name was: Generalized Stirling number triangle of first kind.
a(n,m) = R_n^m(a=0,b=3) in the notation of the given reference.
a(n,m) is a Jabotinsky matrix, i.e., the monic row polynomials E(n,x) := Sum_{m=1..n} a(n,m)*x^m = Product_{j=0..n-1} (x - 3*j), n >= 1 and E(0,x) := 1 are exponential convolution polynomials (see A039692 for the definition and a Knuth reference).
This is the signed Stirling1 triangle with diagonals d>=0 (main diagonal d=0) scaled with 3^d.
Exponential Riordan array [1/(1 + 3*x), log(1 + 3*x)/3]. The unsigned triangle is [1/(1 - 3*x), log(1/(1 - 3*x)^(1/3))]. - Paul Barry, Apr 29 2009
Also the Bell transform of the triple factorial numbers A032031 which adds a first column (1, 0, 0 ...) on the left side of the triangle and computes the unsigned values. For the definition of the Bell transform, see A264428. See A004747 for the triple factorial numbers A008544 and A203412 for the triple factorial numbers A007559 as well as A039683 and A132062 for the case of double factorial numbers. - Peter Luschny, Dec 21 2015

			Triangle starts:
       1;
      -3,       1;
      18,      -9,      1;
    -162,      99,    -18,      1;
    1944,   -1350,    315,    -30,    1;
  -29160,   22194,  -6075,    765,  -45,   1;
  524880, -428652, 131544, -19845, 1575, -63, 1;
---
Row polynomial E(3,x) = 18*x-9*x^2+x^3.
From _Paul Barry_, Apr 29 2009: (Start)
The unsigned array [1/(1 - 3*x), log(1/(1 - 3*x)^(1/3))] has production matrix
    3,    1;
    9,    6,    1;
   27,   27,    9,   1;
   81,  108,   54,  12,   1;
  243,  405,  270,  90,  15,  1;
  729, 1458, 1215, 540, 135, 18, 1;
  ...
which is A007318^{3} beheaded (by viewing A007318 as a lower triangular matrix). See the comment above. (End)

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
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.
Wolfdieter Lang, First 10 rows.
D. S. Mitrinovic and M. S. Mitrinovic, Tableaux d'une classe de nombres reliés aux nombres de Stirling, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. 77 (1962), 1-77.

First (m=1) column sequence is: A032031(n-1).
Row sums (signed triangle): A008544(n-1)*(-1)^(n-1).
Row sums (unsigned triangle): A007559(n).
Cf. A008275 (Stirling1 triangle, b=1), A039683 (b=2), A051142 (b=4).
Cf. A039683, A132062, A264428.

a[n_, m_] /; n >= m >= 1 := a[n, m] = a[n-1, m-1] - 3(n-1)*a[n-1, m]; a[n_, m_] /; n < m = 0; a[, 0] = 0; a[1, 1] = 1; Flatten[Table[a[n, m], {n, 1, 9}, {m, 1, n}]][[1 ;; 38]] (* _Jean-François Alcover, Jun 01 2011, after formula *)
Table[StirlingS1[n, m]*3^(n - m), {n, 1, 10}, {m, 1, n}]//Flatten (* G. C. Greubel, Oct 24 2017 *)

for(n=1,10, for(m=1,n, print1(stirling(n,m,1)*3^(n-m), ", "))) \\ G. C. Greubel, Oct 24 2017

# uses[bell_transform from A264428]
triplefactorial = lambda n: 3^n*factorial(n)
def A051141_row(n):
    trifact = [triplefactorial(k) for k in (0..n)]
    return bell_transform(n, trifact)
[A051141_row(n) for n in (0..8)] # Peter Luschny, Dec 21 2015

a(n, m) = a(n-1, m-1) - 3*(n-1)*a(n-1, m) for n >= m >= 1; a(n, m) = 0 for n < m; a(n, 0) = 0 for n >= 1; a(0, 0) = 1.
E.g.f. for the m-th column of the signed triangle: (log(1 + 3*x)/3)^m/m!.
|a(n,1)| = A032031(n-1). - Peter Luschny, Dec 23 2015

Name clarified using a formula of the author by Peter Luschny, Dec 23 2015

A111146 Triangle T(n,k), read by rows, given by [0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, ...] DELTA [1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, ...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 0, 1, 0, 0, 2, 0, 0, 1, 4, 0, 0, 2, 5, 8, 0, 0, 6, 15, 17, 16, 0, 0, 24, 62, 68, 49, 32, 0, 0, 120, 322, 359, 243, 129, 64, 0, 0, 720, 2004, 2308, 1553, 756, 321, 128, 0, 0, 5040, 14508, 17332, 11903, 5622, 2151, 769, 256, 0, 0, 40320, 119664

Offset: 0

Philippe Deléham, Oct 19 2005

Let R(m,n,k), 0<=k<=n, the Riordan array (1, x*g(x)) where g(x) is g.f. of the m-fold factorials . Then Sum_{k, 0<=k<=n} = R(m,n,k) = Sum_{k, 0<=k<=n} T(n,k)*m^(n-k).
For m = -1, R(-1,n,k) is A026729(n,k).
For m = 0, R(0,n,k) is A097805(n,k).
For m = 1, R(1,n,k) is A084938(n,k).
For m = 2, R(2,n,k) is A111106(n,k).

			Triangle begins:
.1;
.0, 1;
.0, 0, 2;
.0, 0, 1, 4;
.0, 0, 2, 5, 8;
.0, 0, 6, 15, 17, 16;
.0, 0, 24, 62, 68, 49, 32;
.0, 0, 120, 322, 359, 243, 129, 64;
.0, 0, 720, 2004, 2308, 1553, 756, 321, 128;
.0, 0, 5040, 14508, 17332, 11903, 5622, 2151, 769, 256;
.0, 0, 40320, 119664, 148232, 105048, 49840, 18066, 5756, 1793, 512;
....................................................................
At y=2: Sum_{k=0..n} 2^k*T(n,k) = A113327(n) where (1 + 2*x + 8*x^2 + 36*x^3 +...+ A113327(n)*x^n +..) = 1/(1 - 2/1!*x*(1! + 2!*x + 3!*x^2 + 4!*x^3 +..) ).
At y=3: Sum_{k=0..n} 3^k*T(n,k) = A113328(n) where (1 + 3*x + 18*x^2 + 117*x^3 +...+ A113328(n)*x^n +..) = 1/(1 - 3/2!*x*(2! + 3!*x + 4!*x^2 + 5!*x^3 +..) ).
At y=4: Sum_{k=0..n} 4^k*T(n,k) = A113329(n) where (1 + 4*x + 32*x^2 + 272*x^3 +...+ A113329(n)*x^n +..) = 1/(1 - 4/3!*x*(3! + 4!*x + 5!*x^2 + 6!*x^3 +..) ).

Alois P. Heinz, Rows n = 0..140, flattened

Cf. A000045, A026729, A051295, A084938, A097805, A111106, A112934.
Cf. m-fold factorials : A000142, A001147, A007559, A007696, A008548, A008542.
Cf. A113326, A113327 (y=2), A113328 (y=3), A113329 (y=4), A113330 (y=5), A113331 (y=6).

T[n_, k_] := Module[{x = X + X*O[X]^n, y = Y + Y*O[Y]^k}, A = 1/(1 - x*y*Sum[x^j*Product[y + i, {i, 0, j - 1}], {j, 0, n}]); Coefficient[ Coefficient[A, X, n], Y, k]];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 26 2019, from PARI *)

{T(n,k)=local(x=X+X*O(X^n),y=Y+Y*O(Y^k)); A=1/(1-x*y*sum(j=0,n,x^j*prod(i=0,j-1,y+i))); return(polcoeff(polcoeff(A,n,X),k,Y))} (Hanna)

Sum_{k, 0<=k<=n} (-1)^(n-k)*T(n, k) = A000045(n+1), Fibonacci numbers.
Sum_{k, 0<=k<=n} T(n, k) = A051295(n).
Sum_{k, 0<=k<=n} 2^(n-k)*T(n, k) = A112934(n).
T(0, 0) = 1, T(n, n) = 2^(n-1).
G.f.: A(x, y) = 1/(1 - x*y*Sum_{j>=0} (y-1+j)!/(y-1)!*x^j ). - Paul D. Hanna, Oct 26 2005

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

Original entry on oeis.org

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

Offset: 0

Jonathan Vos Post, Feb 18 2006

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

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

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

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

Cf. A045754, A084947, A288094.

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

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

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

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

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

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

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

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

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

Edited by M. F. Hasler, Feb 23 2018

A257610 Triangle read by rows: 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) = 3*x + 2.

Original entry on oeis.org

1, 2, 2, 4, 20, 4, 8, 132, 132, 8, 16, 748, 2112, 748, 16, 32, 3964, 25124, 25124, 3964, 32, 64, 20364, 256488, 552728, 256488, 20364, 64, 128, 103100, 2398092, 9670840, 9670840, 2398092, 103100, 128, 256, 518444, 21246736, 147146804, 270783520, 147146804, 21246736, 518444, 256

Offset: 0

Dale Gerdemann, May 03 2015

			Triangle begins as:
    1;
    2,      2;
    4,     20,        4;
    8,    132,      132,         8;
   16,    748,     2112,       748,        16;
   32,   3964,    25124,     25124,      3964,        32;
   64,  20364,   256488,    552728,    256488,     20364,       64;
  128, 103100,  2398092,   9670840,   9670840,   2398092,   103100,    128;
  256, 518444, 21246736, 147146804, 270783520, 147146804, 21246736, 518444, 256;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A007559 (row sums), A038208, A142458, A257620, A257622, A257624, A257626.
Cf. A256890, A257609, A257610, A257612, A257614, A257616, A257617, A257618, A257619.
See similar sequences listed in A256890.

T[n_, k_, a_, b_]:= T[n, k, a, b]= If[k<0 || k>n, 0, If[n==0, 1, (a*(n-k)+b)*T[n-1, k-1, a, b] + (a*k+b)*T[n-1, k, a, b]]];
Table[T[n,k,3,2], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 20 2022 *)

def T(n,k,a,b): # A257610
    if (k<0 or k>n): return 0
    elif (n==0): return 1
    else: return  (a*k+b)*T(n-1,k,a,b) + (a*(n-k)+b)*T(n-1,k-1,a,b)
flatten([[T(n,k,3,2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 20 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) = 3*x + 2.
Sum_{k=0..n} T(n, k) = A007559(n).
T(n, k) = (a*k + b)*T(n-1, k) + (a*(n-k) + b)*T(n-1, k-1), with T(n, 0) = 1, a = 3, and b = 2. - G. C. Greubel, Mar 20 2022

A346982 Expansion of e.g.f. 1 / (4 - 3 * exp(x))^(1/3).

Original entry on oeis.org

1, 1, 5, 41, 477, 7201, 133685, 2945881, 75145677, 2177900241, 70687244965, 2539879312521, 100086803174077, 4291845333310081, 198954892070938645, 9914294755149067961, 528504758009562261677, 30010032597449931644721, 1808359960001658961070725

Offset: 0

Ilya Gutkovskiy, Aug 09 2021

Stirling transform of A007559.

Seiichi Manyama, Table of n, a(n) for n = 0..373

Cf. A000670, A007559, A305404, A346983, A346984, A346985, A352117, A352118, A352119.

Cf. A032033, A365558.

g:= proc(n) option remember; `if`(n<2, 1, (3*n-2)*g(n-1)) end:
b:= proc(n, m) option remember;
     `if`(n=0, g(m), m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..18);  # Alois P. Heinz, Aug 09 2021

nmax = 18; CoefficientList[Series[1/(4 - 3 Exp[x])^(1/3), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS2[n, k] 3^k Pochhammer[1/3, k], {k, 0, n}], {n, 0, 18}]

a(n) = Sum_{k=0..n} Stirling2(n,k) * A007559(k).
a(n) ~ n! / (Gamma(1/3) * 2^(2/3) * n^(2/3) * log(4/3)^(n + 1/3)). - Vaclav Kotesovec, Aug 14 2021
From  Peter Bala, Aug 22 2023: (Start)
O.g.f. (conjectural): 1/(1 - x/(1 - 4*x/(1 - 4*x/(1 - 8*x/(1 - 7*x/(1 - 12*x/(1 - ... - (3*n-2)*x/(1 - 4*n*x/(1 - ... ))))))))) - a continued fraction of Stieltjes-type (S-fraction).
More generally, it appears that the o.g.f. of the sequence whose e.g.f. is equal to 1/(r+1 - r*exp(s*x))^(m/s) corresponds to the S-fraction 1/(1 - r*m*x/(1 - s*(r+1)*x/(1 - r*(m+s)*x/(1 - 2*s(r+1)*x/(1 - r*(m+2*s)*x/(1 - 3*s(r+1)*x/( 1 - ... ))))))). This is the case r = 3, s = 1, m = 1/3. (End)
a(0) = 1; a(n) = Sum_{k=1..n} (3 - 2*k/n) * binomial(n,k) * a(n-k). - Seiichi Manyama, Sep 09 2023
a(0) = 1; a(n) = a(n-1) - 4*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). - Seiichi Manyama, Nov 16 2023

A035013 One third of 9-factorial numbers.

Original entry on oeis.org

1, 12, 252, 7560, 294840, 14152320, 806682240, 53241027840, 3993077088000, 335418475392000, 31193918211456000, 3181779657568512000, 353177541990104832000, 42381305038812579840000, 5467188350006822799360000, 754471992300941546311680000, 110907382868238407307816960000, 17301551727445191540019445760000

Offset: 1

Wolfdieter Lang

E.g.f. is g.f. for A034171(n-1).

G. C. Greubel, Table of n, a(n) for n = 1..325
Index entries for sequences related to factorial numbers.

Cf. A007559, A034171, A035012, A035013, A035017, A035018.

Cf. A035020, A035021, A035022, A035023, A045756, A144758.

[n le 1 select 1 else (9*n-6)*Self(n-1): n in [1..40]]; // G. C. Greubel, Oct 18 2022

s=1;lst={s};Do[s+=n*s;AppendTo[lst, s], {n, 11, 2*5!, 9}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)
Table[9^n*Pochhammer[1/3, n]/3, {n, 40}] (* G. C. Greubel, Oct 18 2022 *)

[9^n*rising_factorial(1/3,n)/3 for n in range(1,40)] # G. C. Greubel, Oct 18 2022

3*a(n) = (9*n-6)(!^9) := Product_{j=1..n} (9*j-6) = 3^n*A007559(n).
E.g.f.: (-1+(1-9*x)^(-1/3))/3.
From G. C. Greubel, Oct 18 2022: (Start)
a(n) = (1/3) * 9^n * Pochhammer(n, 1/3).
a(n) = (9*n-6)*a(n-1). (End)
From Amiram Eldar, Dec 21 2022: (Start)
a(n) = A144758(n)/3.
Sum_{n>=1} 1/a(n) = 3*(e/9^6)^(1/9)*(Gamma(1/3) - Gamma(1/3, 1/9)). (End)

Terms a(15) onward added by G. C. Greubel, Oct 18 2022

cp's OEIS Frontend

A286718 Triangle read by rows: T(n, k) is the Sheffer triangle ((1 - 3*x)^(-1/3), (-1/3)*log(1 - 3*x)). A generalized Stirling1 triangle.

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A034689 a(n) = n-th sextic factorial number divided by 2.

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A035022 One eighth of 9-factorial numbers.

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A035023 One ninth of 9-factorial numbers.

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A051141 Triangle read by rows: a(n, m) = S1(n, m)*3^(n-m), where S1 are the signed Stirling numbers of first kind A008275 (n >= 1, 1 <= m <= n).

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A111146 Triangle T(n,k), read by rows, given by [0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, ...] DELTA [1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, ...] where DELTA is the operator defined in A084938.

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

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A286718 Triangle read by rows: T(n, k) is the Sheffer triangle ((1 - 3x)^(-1/3), (-1/3)log(1 - 3*x)). A generalized Stirling1 triangle.

A257610 Triangle read by rows: 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) = 3*x + 2.