A008544 -id:A008544 - cp's OEIS frontend

A034000 One half of triple factorial numbers.

1, 5, 40, 440, 6160, 104720, 2094400, 48171200, 1252451200, 36321084800, 1162274713600, 40679614976000, 1545825369088000, 63378840132608000, 2788668965834752000, 131067441394233344000, 6553372069711667200000, 347328719694718361600000, 19450408302904228249600000

Offset: 1

Wolfdieter Lang

Preface the series with a 1, then the next term = (1, 4, 7, 10, ...) dot (1, 1, 5, 40, ...). E.g., a(5) = 6160 = (1, 4, 7, 10, 13) dot (1, 1, 5, 40, 440) = (1 + 4 + 35 + 400 + 5720). - Gary W. Adamson, May 17 2010

In other words, a(n) = Sum_{i=0..n-1} b(i)*A016777(i) where b(0)=1 and b(n)=a(n). - Michel Marcus, Dec 18 2022

G. C. Greubel, Table of n, a(n) for n = 1..375
J.-C. Novelli and J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv:1403.5962 [math.CO], 2014.

Cf. A007559, A034001, A025748, A034724, A016777.

a:=[1];; for n in [2..20] do a[n]:=(3*n-1)*a[n-1]; od; a; # G. C. Greubel, Aug 15 2019

[n le 1 select 1 else (3*n-1)*Self(n-1): n in [1..20]]; // G. C. Greubel, Aug 15 2019

A034000:=n->`if`(n=1, 1, (3*n-1)*A034000(n-1)); seq(A034000(n), n=1..20); # G. C. Greubel, Aug 15 2019

nxt[{n_,a_}]:={n+1,(3(n+1)-1)*a}; Transpose[NestList[nxt,{1,1},20]][[2]] (* Harvey P. Dale, Aug 22 2015 *)
Table[3^(n-1)*Pochhammer[5/3, n-1], {n,20}] (* G. C. Greubel, Aug 15 2019 *)

m=20; v=concat([1], vector(m-1)); for(n=2, m, v[n]=(3*n-1)*v[n-1]); v \\ G. C. Greubel, Aug 15 2019

def a(n):
    if n==1: return 1
    else: return (3*n-1)*a(n-1)
[a(n) for n in (1..20)] # G. C. Greubel, Aug 15 2019

a(n) = A007661(3n-1)/2 = A008544(n)/2.
2*a(n+1) = (3*n+2)!!! = Product_{j=0..n} (3*j+2), n >= 0.
E.g.f.: (-1 + (1-3*x)^(-2/3))/2.
a(n) = (3*n-1)!/(2*3^(n-1)*(n-1)!*A007559(n)).
a(n) ~ 3/2*2^(1/2)*Pi^(1/2)*Gamma(2/3)^-1*n^(7/6)*3^n*e^-n*n^n*{1 + 23/36*n^-1 + ...}. - Joe Keane (jgk(AT)jgk.org), Nov 23 2001
a(n) = 3^n*(n+2/3)!/(2/3)!, with offset 0. - Paul Barry, Sep 04 2005
D-finite with recurrence a(n) + (1-3*n)*a(n-1) = 0. - R. J. Mathar, Dec 03 2012
Sum_{n>=1} 1/a(n) = 2*(e/3)^(1/3)*(Gamma(2/3) - Gamma(2/3, 1/3)). - Amiram Eldar, Dec 18 2022

A084939 Pentagorials: n-th polygorial for k=5.

Original entry on oeis.org

1, 1, 5, 60, 1320, 46200, 2356200, 164934000, 15173928000, 1775349576000, 257425688520000, 45306921179520000, 9514453447699200000, 2350070001581702400000, 674470090453948588800000, 222575129849803034304000000

Offset: 0

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

Robert Israel, Table of n, a(n) for n = 0..243
M. A. Asiru, Sequence factorial of g-gonal numbers, Int. J. Math. Educ. Sci. Technol., 44(4) (2012), 579-586.
Daniel Dockery, Polygorials, Special "Factorials" of Polygonal Numbers, preprint, 2003.

Cf. A000326, A006472, A001044, A000680, A084940, A084941, A084942, A084943, A084944, A085356.

a := n->(n!/2^n)*mul(3*i+2,i=0..n-1); [seq(a(j),j=0..30)];

Table[k! Pochhammer[2/3, k] (3/2)^k, {k, 0, 20}] (* Jan Mangaldan, Mar 20 2013 *)
polygorial[k_, n_] := FullSimplify[ n!/2^n (k -2)^n*Pochhammer[2/(k -2), n]]; Array[polygorial[5, #] &, 17, 0] (* Robert G. Wilson v, Dec 17 2016 *)

a(n)=n!/2^n*prod(i=1,n,3*i-1) \\ Charles R Greathouse IV, Dec 13 2016

a(n) = polygorial(n, 5) = (A000142(n)/A000079(n))*A008544(n) = (n!/2^n)*Product_{i=0..n-1} (3*i+2) = (n!/2^n)*3^n*Pochhammer(2/3, n) = (n!/2^n)*3^n*GAMMA(n+2/3)/GAMMA(2/3).
a(n) ~ Gamma(1/3) * 3^(n+1/2) * n^(2*n+2/3) / (2^n * exp(2*n)). - Vaclav Kotesovec, Jul 17 2015
D-finite with recurrence a(n+1) = ((n+1)*(3*n+2)/2)*a(n) = A000326(n+1)*a(n). - Muniru A Asiru, Apr 05 2016
E.g.f.: hypergeom([2/3, 1], [], (3/2)*x). - Robert Israel, Apr 05 2016

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

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

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

Original entry on oeis.org

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

Offset: 0

Wolfdieter Lang

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

Sequences of the form m^n*Pochhammer((m-1)/m, n): A000007 (m=1), A001147 (m=2), A008544 (m=3), A008545 (m=4), A008546 (m=5), A008543 (m=6), A049209 (m=7), A049210 (m=8), this sequence (m=9), A049212 (m=10), A254322 (m=11), A346896 (m=12).

Cf. A035022, A048994.

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

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

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

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

a(n) = 8*A035022(n) = (9*n-1)(!^9), n >= 1, a(0) = 1.
a(n) = (-1)^n*Sum_{k=0..n} 9^k*s(n+1,n+1-k), where s(n,k) are the Stirling numbers of the first kind, A048994. - Mircea Merca, May 03 2012
a(n) = 9^n * Gamma(n+8/9) / Gamma(8/9). - Vaclav Kotesovec, Jan 28 2015
E.g.f: (1-9*x)^(-8/9). - Vaclav Kotesovec, Jan 28 2015
From Nikolaos Pantelidis, Dec 09 2020: (Start)
G.f.: 1/(1-8*x-72*x^2/(1-26*x-306*x^2/(1-44*x-702*x^2/(1-62*x-1260*x^2/(1-80*x-1980*x^2/(1-...)))))) (Jacobi continued fraction).
G.f.: 1/(1-8*x/(1-9*x/(1-17*x/(1-18*x/(1-26*x/(1-27*x/(1-35*x/(1-36*x/(1-44*x/(1-45*x/(1-...))))))))))) (Stieltjes continued fraction). (End)
From Nikolaos Pantelidis, Dec 19 2020: (Start)
G.f.: 1/G(0) where G(k) = 1 - (18*k+8)*x - 9*(k+1)*(9*k+8)*x^2/G(k+1) (continued fraction).
G.f.: 1/Q(0) where Q(k) = 1 - x*(9*k+8)/(1 - x*(9*k+9)/Q(k+1) ) (continued fraction). (End)
G.f.: hypergeometric2F0([1, 8/9], [--], 9*x). - G. C. Greubel, Feb 08 2022
Sum_{n>=0} 1/a(n) = 1 + (e/9)^(1/9)*(Gamma(8/9) - Gamma(8/9, 1/9)). - Amiram Eldar, Dec 21 2022

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

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.

Original entry on oeis.org

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

A029768 Number of increasing mobiles with n elements.

Original entry on oeis.org

0, 1, 1, 2, 7, 36, 245, 2076, 21059, 248836, 3356609, 50896380, 856958911, 15864014388, 320245960333, 7001257954796, 164792092647355, 4154906594518116, 111719929072986521, 3191216673497748444

Offset: 0

N. J. A. Sloane, Dec 11 1999

A labeled tree of size n is a rooted tree on n nodes that are labeled by distinct integers from the set {1,...,n}. An increasing tree is a labeled tree such that the sequence of labels along any branch starting at the root is increasing.
a(n) counts increasing trees with cyclically ordered branches.
a(n+1) counts the non-plane (where the subtrees stemming from a node are not ordered between themselves) increasing trees on n nodes where the nodes of outdegree k come in k+1 colors. An example is given below. The number of plane increasing trees on n nodes where the nodes of outdegree k come in k+1 colors is given by the triple factorial numbers A008544. - Peter Bala, Aug 30 2011
a(n+1)/a(n)/n tends to 1/A073003 = 1.676875... . - Vaclav Kotesovec, Mar 11 2014

			a(4) = 7: D^2[(1+x)*exp(x)] = exp(2*x)*(2*x^2+8*x+7). Evaluated at x = 0 this gives a(4) = 7. Denote the colors of the nodes by the letters a,b,c,.... The 7 possible trees on 3 nodes with nodes of outdegree k coming in k+1 colors are:
........................................................
...1a....1b....1a....1b........1a.......1b........1c....
...|.....|.....|.....|......../.\....../..\....../..\...
...2a....2b....2b....2a......2...3....2....3....2....3..
...|.....|.....|.....|..................................
...3.....3.....3.....3..................................
G.f. = x + x^2 + 2*x^3 + 7*x^4 + 36*x^5 + 245*x^6 + 2076*x^7 + 21059*x^8 + ...

F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 392.

Vaclav Kotesovec, Table of n, a(n) for n = 0..400
C. G. Bower, Transforms
C. G. Bower, Transforms (2)
F. Bergeron, Ph. Flajolet and B. Salvy, Varieties of Increasing Trees, Lecture Notes in Computer Science vol. 581, ed. J.-C. Raoult, Springer 1992, pp. 24-48.
D. Dominici, Nested derivatives: A simple method for computing series expansions of inverse functions. arXiv:math/0501052v2 [math.CA], 2015.
Sergey Fomin and Grigory Mikhalkin, Labeled floor diagrams for plane curves, arXiv:0906.3828 [math.AG], 2009-2010.
Index entries for sequences related to mobiles

Cf. A032220, A038037, A055356, A008544, A145271.

S:= rhs(dsolve({diff(a(x),x) = log(1/(1-a(x)))+1,a(0)=0},a(x),series,order=101)):
seq(coeff(S,x,j)*j!,j=0..100); # Robert Israel, Apr 17 2015

Multinomial1[list_] := Apply[Plus, list]!/Apply[Times, (#1! & ) /@ list]; a[1]=1; a[n_]/;n>=2 := a[n] = Sum[Map[Multinomial1[ # ]Product[Map[a,# ]]/Length[ # ]&,Compositions[n-1]]]; Table[a[n],{n,8}] (* David Callan, Nov 29 2007 *)
nmax=20; b = ConstantArray[0,nmax]; b[[1]]=0; b[[2]]=1; Do[b[[n+1]] = b[[n]] + Sum[Binomial[n-2,i]*b[[i+1]]*b[[n-i+1]],{i,1,n-2}],{n,2,nmax-1}]; b (* Vaclav Kotesovec after Vladimir Kruchinin, Mar 11 2014 *)
terms = 20; A[x_] := x; Do[A[x_] = Integrate[(1 + A[x])*Exp[A[x] + O[x]^j], x] + O[x]^j // Normal // Simplify, {j, 1, terms - 1}]; Join[{0, 1}, CoefficientList[A[x], x]*Range[0, terms - 2]! // Rest] (* Jean-François Alcover, May 22 2014, updated Jan 12 2018 (after PARI script by Michael Somos) *)

{a(n) = my(A = x + O(x^2)); if( n<2, n==1, n--; for(k=1, n-1, A = intformal( (1 + A) * exp(A)));  n! * polcoeff(A, n))}; /* Michael Somos, Apr 17 2015 */
for(n=1,30,print1(a(n),", "))

seq(N) = {
  my(a = vector(N)); a[1] = 1;
  for (n = 2, N, a[n] = a[n-1] + sum(k=1, n-2, binomial(n-2, k)*a[k]*a[n-k]));
  concat(0, a);
};
seq(19)
\\ test: N=200; y=serconvol(Ser(seq(N),'x), exp('x+O('x^N))); y' == y''*(1-y)
\\ Gheorghe Coserea, Jun 26 2018

Bergeron et al. give several formulas. Shifts left under "CIJ" (necklace, indistinct, labeled) transform.
E.g.f.: A(x) =
  x + (1/2)*x^2 + (1/3)*x^3 + (7/24)*x^4 + (3/10)*x^5 + (49/144)*x^6 + (173/420)*x^7 + (21059/40320)*x^8 + (8887/12960)*x^9 + ...
and satisfies the differential equation A'(x)=log(1/(1-A(x)))+1. - Vladimir Kruchinin, Jan 22 2011
E.g.f. A(x) satisfies: A''(x) = A'(x) * exp(A'(x)-1). - Paul D. Hanna, Apr 17 2015
From Robert Israel, Apr 17 2015 (Start):
E.g.f. A(x) satisfies e*(Ei(1,A'(x)) - Ei(1,1)) = integral(s = 1 .. A'(x), exp(1-s)/s ds) = -x.
a(n) = e^(1-n)*limit(w -> 1, (d^(n-2)/dw^(n-2))(((w-1)/(Ei(1,1)-Ei(1,w)))^(n-1))) for n >= 2. (End)
a(n) = sum(i=1..n-2,binomial(n-2,i)*a(i)*a(n-i))+a(n-1), a(0)=0, a(1)=1. - Vladimir Kruchinin, Jan 24 2011
The following remarks refer to the interpretation of this sequence as counting increasing trees where the nodes of outdegree k come in k+1 colors. Thus we work with the generating function B(x) = A'(x)-1 = x + 2*x^2/2!+7*x^3/3!+36*x^4/4!+.... The degree function phi(x) (see [Bergeron et al.] for definition) for this variety of trees is phi(x) = 1+2*x+3*x^2/2!+4*x^3/3!+5*x^4/4!+... = (1+x)*exp(x). The generating function B(x) satisfies the autonomous differential equation B' = phi(B(x)) with initial condition B(0) = 0. It follows that the inverse function B(x)^(-1) may be expressed as an integral B(x)^(-1) = int {t = 0..x} 1/phi(t) dt = int {t = 0..x} exp(-t)/(1+t) dt. Applying [Dominici, Theorem 4.1] to invert the integral produces the result B(x) = sum {n>=1} D^(n-1)[(1+x)*exp(x)](0)*x^n/n!, where the nested derivative D^n[f](x) of a function f(x) is defined recursively as D^0[f](x) = 1 and D^(n+1)[f](x) = d/dx(f(x)*D^n[f](x)) for n >= 0. Thus a(n+1) = D^(n-1)[(1+x)*exp(x)](0). - Peter Bala, Aug 30 2011

More terms from Christian G. Bower

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

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

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

Original entry on oeis.org

1, 7, 105, 2415, 74865, 2919735, 137227545, 7547514975, 475493443425, 33760034483175, 2667042724170825, 232032717002861775, 22043108115271868625, 2270440135873002468375, 252018855081903273989625, 29990243754746489604765375, 3808760956852804179805202625

Offset: 0

Wolfdieter Lang

Seiichi Manyama, Table of n, a(n) for n = 0..333
Index entries for sequences related to factorial numbers.

Cf. A034975, A048994, A051187.

Sequences of the form m^n*Pochhammer((m-1)/m, n): A000007 (m=1), A001147 (m=2), A008544 (m=3), A008545 (m=4), A008546 (m=5), A008543 (m=6), A049209 (m=7), this sequence (m=8), A049211 (m=9), A049212 (m=10), A254322 (m=11), A346896 (m=12).

m:=8; [Round(m^n*Gamma(n +(m-1)/m)/Gamma((m-1)/m)): n in [0..30]]; // G. C. Greubel, Feb 16 2022

FoldList[Times,1,8*Range[20]-1] (* Harvey P. Dale, Aug 03 2014 *)
CoefficientList[Series[(1-8*x)^(-7/8),{x,0,20}],x] * Range[0,20]! (* Vaclav Kotesovec, Jan 28 2015 *)

a(n) = -prod(k=0, n, 8*k-1); \\ Michel Marcus, Jan 08 2015

m=8; [m^n*rising_factorial((m-1)/m, n) for n in (0..30)] # G. C. Greubel, Feb 16 2022

a(n) = 7*A034975(n) = (8*n-1)(!^8), n >= 1, a(0) = 1.
G.f.: 1/(1-7*x/(1-8*x/(1-15*x/(1-16*x/(1-23*x/(1-24*x/(1-31*x/(1-32*x/(1-... (continued fraction). - Philippe Deléham, Jan 07 2012
a(n) = (-1)^n*Sum_{k=0..n} 8^k*s(n+1,n+1-k), where s(n,k) are the Stirling numbers of the first kind, A048994. - Mircea Merca, May 03 2012
G.f.: ( 1 - 1/Q(0) )/x where Q(k) = 1 - x*(8*k-1)/(1 - x*(8*k+8)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 20 2013
a(n) = 8^n*Gamma(n+7/8)/Gamma(7/8). - R. J. Mathar, Mar 20 2013
E.g.f: (1-8*x)^(-7/8). - Vaclav Kotesovec, Jan 28 2015
G.f.: 1/(1-7*x-56*x^2/(1-23*x-240*x^2/(1-39*x-552*x^2/(1-55*x-992*x^2/(1-71*x-1560*x^2/(1-... )))))) (Jacobi continued fraction). - Nikolaos Pantelidis, Dec 09 2020
G.f.: 1/G(0) where G(k) = 1 - (16*k+7)*x - 8*(k+1)*(8*k+7)*x^2/G(k+1); (continued fraction). - Nikolaos Pantelidis, Dec 19 2020
Sum_{n>=0} 1/a(n) = 1 + (e/8)^(1/8)*(Gamma(7/8) - Gamma(7/8, 1/8)). - Amiram Eldar, Dec 20 2022

cp's OEIS Frontend

A034000 One half of triple factorial numbers.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

A084939 Pentagorials: n-th polygorial for k=5.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Sage

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Formula

Extensions

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.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

A029768 Number of increasing mobiles with n elements.

Views

Author

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.