A045756 -id:A045756 - cp's OEIS frontend

A035018 One fifth of 9-factorial numbers.

1, 14, 322, 10304, 422464, 21123200, 1246268800, 84746278400, 6525463436800, 561189855564800, 53313036278656000, 5544555772980224000, 626534802346765312000, 76437245886305368064000, 10013279211106003216384000, 1401859089554840450293760000, 208877004343671227093770240000

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, A034171, A035012, A035013, A035017, A035018.

Cf. A035020, A035021, A035022, A035023, A045756, A147629.

[n le 1 select 1 else (9*n-4)*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, 13, 2*5!, 9}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)
Rest[FoldList[Times,1,9*Range[20]-4]/5] (* Harvey P. Dale, May 22 2013 *)

[9^n*rising_factorial(5/9,n)/5 for n in range(1,40)] # G. C. Greubel, Oct 18 2022

5*a(n) = (9*n-4)(!^9) := Product_{j=1..n} (9*j-4).
E.g.f.: (-1+(1-9*x)^(-5/9))/5.
From G. C. Greubel, Oct 18 2022: (Start)
a(n) = (1/5) * 9^n * Pochhammer(n, 5/9).
a(n) = (9*n-4)*a(n-1). (End)
From Amiram Eldar, Dec 21 2022: (Start)
a(n) = A147629(n+1)/5.
Sum_{n>=1} 1/a(n) = 5*(e/9^4)^(1/9)*(Gamma(5/9) - Gamma(5/9, 1/9)). (End)

A035020 One sixth of 9-factorial numbers.

Original entry on oeis.org

1, 15, 360, 11880, 498960, 25446960, 1526817600, 105350414400, 8217332323200, 714907912118400, 68631159563366400, 7206271754153472000, 821514979973495808000, 101046342536739984384000, 13338117214849677938688000, 1880674527293804589355008000, 282101179094070688403251200000

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, A147630.

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

[n le 1 select 1 else (9*n-3)*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, 14, 2*5!, 9}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)
Table[9^n*Pochhammer[2/3, n]/6, {n, 40}] (* G. C. Greubel, Oct 18 2022 *)

[9^n*rising_factorial(2/3,n)/6 for n in range(1,40)] # G. C. Greubel, Oct 18 2022

6*a(n) = (9*n-3)(!^9) := Product_{j=1..n} (9*j-3) = 3^n*2*A034000(n), where 2*A034000(n) = (3*n-1)(!^3) := Product_{j=1..n} (3*j-1).
E.g.f.: (-1+(1-9*x)^(-2/3))/6.
From G. C. Greubel, Oct 18 2022: (Start)
a(n) = (1/6) * 9^n * Pochhammer(n, 2/3).
a(n) = (9*n - 3)*a(n-1). (End)
From Amiram Eldar, Dec 21 2022: (Start)
a(n) = A147630(n+1)/6.
Sum_{n>=1} 1/a(n) = 6*(e/9^3)^(1/9)*(Gamma(2/3) - Gamma(2/3, 1/9)). (End)

A035021 One seventh of 9-factorial numbers.

Original entry on oeis.org

1, 16, 400, 13600, 584800, 30409600, 1854985600, 129848992000, 10258070368000, 902710192384000, 87562888661248000, 9281666198092288000, 1067391612780613120000, 132356559984796026880000, 17603422477977871575040000, 2499685991872857763655680000, 377452584772801522312007680000

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, A034171, A035012, A035013, A035017, A035018.

Cf. A035020, A035021, A035022, A035023, A045756, A147631.

[n le 1 select 1 else (9*n-2)*Self(n-1): n in [1..40]]; // G. C. Greubel, Oct 19 2022

f := gfun:-rectoproc({(9*n - 2)*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[7/9, n]/7, {n, 40}] (* G. C. Greubel, Oct 19 2022 *)

[9^n*rising_factorial(7/9,n)/7 for n in range(1,40)] # G. C. Greubel, Oct 19 2022

7*a(n) = (9*n-2)(!^9) := Product_{j=1..n} (9*j-2).
E.g.f.: (-1+(1-9*x)^(-7/9))/7.
D-finite with recurrence: a(1) = 1, a(n) = (9*n - 2)*a(n-1) for n > 1. - Georg Fischer, Feb 15 2020
a(n) = (1/7) * 9^n * Pochhammer(n, 7/9). - G. C. Greubel, Oct 19 2022
From Amiram Eldar, Dec 21 2022: (Start)
a(n) = A147631(n+1)/7.
Sum_{n>=1} 1/a(n) = 7*(e/9^2)^(1/9)*(Gamma(7/9) - Gamma(7/9, 1/9)). (End)

Terms a(15) onward added by G. C. Greubel, Oct 19 2022

A256268 Table of k-fold factorials, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 6, 3, 1, 1, 1, 24, 15, 4, 1, 1, 1, 120, 105, 28, 5, 1, 1, 1, 720, 945, 280, 45, 6, 1, 1, 1, 5040, 10395, 3640, 585, 66, 7, 1, 1, 1, 40320, 135135, 58240, 9945, 1056, 91, 8, 1, 1, 1, 362880, 2027025, 1106560, 208845, 22176, 1729, 120, 9, 1, 1

Offset: 0

Philippe Deléham, Jun 01 2015

A variant of A142589.

			1  1   1    1     1       1         1... A000012
1  1   2    6    24     120       720... A000142
1  1   3   15   105     945     10395... A001147
1  1   4   28   280    3640     58240... A007559
1  1   5   45   585    9945    208845... A007696
1  1   6   66  1056   22176    576576... A008548
1  1   7   91  1729   43225   1339975... A008542
1  1   8  120  2640   76560   2756160... A045754
1  1   9  153  3825  126225   5175225... A045755
1  1  10  190  5320  196840   9054640... A045756
1  1  11  231  7161  293601  14977651... A144773

G. C. Greubel, Antidiagonal rows n = 0..100, flattened

Cf. Columns : A000012, A000012, A000384, A011199, A011245.
Cf. Diagonals : A092985, A076111, A158887.
Cf. A048994, A132393.
Cf. A000142 ("1-fold"), A001147 (2-fold), A007559 (3), A007696 (4), A008548 (5), A008542 (6), A045754 (7), A045755 (8), A045756 (9), A144773 (10)

Flat(List([0..12], n-> List([0..n], k-> Product([0..n-k-1], j-> j*k+1) ))); # G. C. Greubel, Mar 04 2020

function T(n,k)
  if k eq 0 or n eq 0 then return 1;
  else return (&*[j*k+1: j in [0..n-1]]);
  end if; return T; end function;
[T(n-k,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 04 2020

seq(seq( mul(j*k+1, j=0..n-k-1), k=0..n), n=0..12); # G. C. Greubel, Mar 04 2020

T[n_, k_]= Product[j*k+1, {j,0,n-1}]; Table[T[n-k,k], {n,0,12}, {k, 0, n}]//Flatten (* G. C. Greubel, Mar 04 2020 *)

T(n,k) = prod(j=0, n-1, j*k+1);
for(n=0,12, for(k=0, n, print1(T(n-k, k), ", "))) \\ G. C. Greubel, Mar 04 2020

[[ product(j*k+1 for j in (0..n-k-1)) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Mar 04 2020

A(n, k) = (-n)^k*FallingFactorial(-1/n, k) for n >= 1. - Peter Luschny, Dec 21 2021

A147629 9-factorial numbers (4).

Original entry on oeis.org

1, 5, 70, 1610, 51520, 2112320, 105616000, 6231344000, 423731392000, 32627317184000, 2805949277824000, 266565181393280000, 27722778864901120000, 3132674011733826560000, 382186229431526840320000, 50066396055530016081920000, 7009295447774202251468800000

Offset: 1

Vladimir Joseph Stephan Orlovsky, Nov 08 2008

nonn

G. C. Greubel, Table of n, a(n) for n = 1..300

Cf. A035012, A035013, A035017, A035018, A035020, A035022, A035023, A045756, A048994, A049211, A051232, A053116, A132393.

[Round(9^(n-1)*Gamma(n-1 +5/9)/Gamma(5/9)): n in [1..20]]; // G. C. Greubel, Dec 03 2019

seq(9^(n-1)*pochhammer(5/9, n-1), n = 1..20); # G. C. Greubel, Dec 03 2019

Table[9^(n-1)*Pochhammer[5/9, n-1], {n,20}] (* G. C. Greubel, Dec 03 2019 *)

vector(20, n, prod(j=0,n-2, 9*j+5) ) \\ G. C. Greubel, Dec 03 2019

[9^(n-1)*rising_factorial(5/9, n-1) for n in (1..20)] # G. C. Greubel, Dec 03 2019

a(n+1) = Sum_{k=0..n} A132393(n,k)*5^k*9^(n-k). - Philippe Deléham, Nov 09 2008
From R. J. Mathar, Nov 09 2008: (Start)
a(n) = a(n-1) + (4 + 9*(n-2))*a(n-1) = (9*n-13)*a(n-1).
a(n) = 9^(n-1)*Gamma(n-4/9)/Gamma(5/9).
G.f.: z*2F0(5/9,1; -; 9*z). (End)
a(n) = (-4)^n*Sum_{k=0..n} (9/4)^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
Sum_{n>=1} 1/a(n) = 1 + (e/9^4)^(1/9)*(Gamma(5/9) - Gamma(5/9, 1/9)). - Amiram Eldar, Dec 21 2022

A144773 10-fold factorials: Product_{k=0..n-1} (10*k+1).

Original entry on oeis.org

1, 1, 11, 231, 7161, 293601, 14973651, 913392711, 64850882481, 5252921480961, 478015854767451, 48279601331512551, 5359035747797893161, 648443325483545072481, 84946075638344404495011, 11977396665006561033796551, 1808586896415990716103279201, 291182490322974505292627951361

Offset: 0

Philippe Deléham, Sep 21 2008

G. C. Greubel, Table of n, a(n) for n = 0..320

Essentially a duplicate of A045757.
Cf. k-fold factorials: A000142 ("1-fold"), A001147 (2-fold), A007559 (3), A007696 (4), A008548 (5), A008542 (6), A045754 (7), A045755 (8), A045756 (9), A256268 (combined table).
Cf. A048994, A132393.

R:=PowerSeriesRing(Rationals(), 15); Coefficients(R!(Laplace( (1-10*x)^(-1/10) ))); // G. C. Greubel, Mar 03 2020

G(x):=(1-10*x)^(-1/10): f[0]:=G(x): for n from 1 to 29 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=0..14); # Zerinvary Lajos, Apr 03 2009

b = 10; Table[FullSimplify[b^n*Gamma[n + 1/b]/Gamma[1/b]], {n, 0, 14}] (* Michael De Vlieger, Sep 14 2016 *)
Join[{1},FoldList[Times,10 Range[0,15]+1]] (* Harvey P. Dale, Oct 24 2022 *)

Vec(serlaplace( (1-10*x)^(-1/10) +O('x^15) )) \\ G. C. Greubel, Mar 03 2020

[10^n*rising_factorial(1/10,n) for n in (0..15)] # G. C. Greubel, Mar 03 2020

a(n) = Sum_{k = 0..n} (-10)^(n - k) * A048994(n, k).
a(n) = Sum_{k = 0..n} 10^(n - k) * A132393(n, k).
E.g.f.: (1 - 10*x)^(-1/10).
a(n) = A045757(n), n>0.
a(n) = (-9)^n * Sum_{k = 0..n} (10/9)^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
G.f.: 1/Q(0), where Q(k) = 1 - (10*k+1)*x/( 1 - 10*x*(k+1)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Jan 09 2014
a(n) = 10^n * Gamma(n + 1/10) / Gamma(1/10). - Artur Jasinski Aug 23 2016
a(n) ~ sqrt(2*Pi)*10^n*n^(n-2/5)/(Gamma(1/10)*exp(n)). - Ilya Gutkovskiy, Sep 11 2016
D-finite with recurrence: a(n) - (10*n-9)*a(n-1) = 0. - R. J. Mathar, Jan 20 2020
Sum_{n>=0} 1/a(n) = 1 + (e/10^9)^(1/10)*(Gamma(1/10) - Gamma(1/10, 1/10)). - Amiram Eldar, Dec 22 2022

A147630 a(1) = 1; for n>1, a(n) = Product_{k = 1..n-1} (9k - 3).

Original entry on oeis.org

1, 6, 90, 2160, 71280, 2993760, 152681760, 9160905600, 632102486400, 49303993939200, 4289447472710400, 411786957380198400, 43237630524920832000, 4929089879840974848000, 606278055220439906304000, 80028703289098067632128000, 11284047163762827536130048000

Offset: 1

Vladimir Joseph Stephan Orlovsky, Nov 08 2008

Original name was: 9-factorial numbers (5).

Cf. A147629, A049211, A051232, A045756, A035012, A035013, A035017, A035018, A035020, A035022, A035023, A053116.

Cf. A048994, A097188, A132393.

[Round(9^n*Gamma(n+6/9)/Gamma(6/9)): n in [0..20]]; // Vincenzo Librandi, Feb 21 2015

s=1;lst={s};Do[s+=n*s;AppendTo[lst,s],{n,5,2*5!,9}];lst
Table[Product[9k-3,{k,1,n-1}],{n,20}] (* Harvey P. Dale, Sep 01 2016 *)

a(n):=n!*sum(binomial(k,n-k-1)*3^k*(-1)^(n-k-1)*binomial(n+k-1,n-1),k,1,n-1)/n; /* Vladimir Kruchinin, Apr 01 2011 */

a(n) = n--; prod(k=1, n, 9*k-3); \\ Michel Marcus, Feb 28 2015

a(n+1) = Sum_{k, 0<=k<=n}A132393(n,k)*6^k*9^(n-k). - Philippe Deléham, Nov 09 2008
a(n) = n!*(Sum_{k=1..n-1} binomial(k,n-k-1)*3^k*(-1)^(n-k-1)*binomial(n+k-1,n-1))/n for n>1, also a(n) = n!*A097188(n-1). - Vladimir Kruchinin, Apr 01 2011
a(n) = (-3)^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
a(n) = round(9^n * Gamma(n+6/9) / Gamma(6/9)). - Vincenzo Librandi, Feb 21 2015
Sum_{n>=1} 1/a(n) = 1 + (e/9^3)^(1/9)*(Gamma(2/3) - Gamma(2/3, 1/9)). - Amiram Eldar, Dec 21 2022

New name from Peter Bala, Feb 20 2015

More terms from Michel Marcus, Feb 28 2015

A051231 Generalized Stirling number triangle of the first kind.

Original entry on oeis.org

1, -9, 1, 162, -27, 1, -4374, 891, -54, 1, 157464, -36450, 2835, -90, 1, -7085880, 1797714, -164025, 6885, -135, 1, 382637520, -104162436, 10655064, -535815, 14175, -189, 1, -24106163760, 6944870988, -775431468, 44411409, -1428840, 26082, -252, 1

Offset: 1

Wolfdieter Lang

T(n,m) = R_n^m(a=0, b=9) in the notation of the given 1962 reference.
T(n,m) is a Jabotinsky matrix, i.e., the monic row polynomials E(n,x) := Sum_{m=1..n} T(n,m)*x^m = Product_{j=0..n-1} (x - 9*j), n >= 1, and E(0,x) := 1 are exponential convolution polynomials (see A039692 for the definition and a Knuth reference).
From Petros Hadjicostas, Jun 06 2020: (Start)
For nonnegative integers n, m and complex numbers a, b (with b <> 0), the numbers R_n^m(a,b) were introduced by Mitrinovic (1961) using slightly different notation. They were further examined by Mitrinovic and Mitrinovic (1962). Special cases were tabulated in this and other related papers.
Special cases of these numbers are related to numbers introduced by Nörlund (1924).
These numbers are defined via the g.f. Product_{r=0..n-1} (x - (a + b*r)) = Sum_{m=0..n} R_n^m(a,b)*x^m for n >= 0. As a result, R_n^m(a,b) = R_{n-1}^{m-1}(a,b) - (a + b*(n-1))*R_{n-1}^m(a,b) for n >= m >= 1 with R_1^0(a,b) = a, R_1^1(a,b) = 1, and R_n^m(a,b) = 0 for n < m. (Because an empty product is by definition 1, we may let R_0^0(a,b) = 1.)
With a = 0 and b = 1, we get the Stirling numbers of the first kind S1(n,m) = R_n^m(a=0, b=1) = A048994(n,m) which satisfy Product_{r=0}^{n-1} (x - r) = Sum_{m=0..n} S1(n,m)*x^m with S1(n,n) = 1 for n >= 0, S1(n,0) = 0 for n >= 1, and S1(n, m) = 0 for m > n. (Array A008275 is the same as array A048994 but with no zero row and no zero column.)
We have R_n^m(a,b) = Sum_{k=0}^{n-m} (-1)^k * a^k * b^(n-m-k) * binomial(m+k, k) * S1(n, m+k) for n >= m >= 0.
For the current array, T(n,m) = R_n^m(a=0, b=9) but with no zero row or column. (End)

			Triangle T(n,m) (with rows n >= 1 and columns m = 1..n) begins:
         1;
        -9,       1;
       162,     -27,       1;
     -4374,     891,     -54,    1;
    157464,  -36450,    2835,  -90,    1;
  -7085880, 1797714, -164025, 6885, -135, 1;
   ...
3rd row o.g.f.: E(3,x) = Product_{j=0..2} (x-9*j) = 162*x - 27*x^2 + x^3. [Edited by _Petros Hadjicostas_, Jun 06 2020]

D. S. Mitrinovic, Sur une relation de récurrence relative aux nombres de Bernoulli, Comptes rendus de l'Académie des sciences de Paris, t. 250 (1960), 4266-4267.
D. S. Mitrinovic, Sur une classe de nombres reliés aux nombres de Stirling, Comptes rendus de l'Académie des sciences de Paris, t. 252 (1961), 2354-2356.
D. S. Mitrinovic and R. S. Mitrinovic, Sur les polynômes de Stirling, Bulletin de la Société des mathématiciens et physiciens de la R. P. de Serbie, t. 10 (1958), 43-49.
D. S. Mitrinovic and R. S. Mitrinovic, Sur les nombres de Stirling et les nombres de Bernoulli d'ordre supérieux, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., No. 43 (1960), 1-63.
D. S. Mitrinovic and R. S. Mitrinovic, Sur une classe de nombres se rattachant aux nombres de Stirling--Appendice: Table des nombres de Stirling, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., No. 60 (1961), 1-15 and 17-62.
D. S. Mitrinovic and R. S. Mitrinovic, Tableaux d'une classe de nombres reliés aux nombres de Stirling, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., No. 77 (1962), 1-77.
D. S. Mitrinovic and R. S. Mitrinovic, Tableaux d'une classe de nombres reliés aux nombres de Stirling, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., No. 77 (1962), 1-77 [jstor stable version].
D. S. Mitrinovic and R. S. Mitrinovic, Tableaux d'une classe de nombres reliés aux nombres de Stirling. V, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., No. 132/142 (1965), 1-22.
Niels Nörlund, Vorlesungen über Differenzenrechnung, Springer, Berlin, 1924.
Wikipedia, Dragoslav Mitrinovic.

First (m=1) column sequence is A051232(n-1).
Row sums (signed triangle): A049211(n-1)*(-1)^(n-1).
Row sums (unsigned triangle): A045756(n).
Cf. A008275 (b=1 triangle), A048994 (b=1 triangle), A051187 (b=8 triangle).

T(n, m) = T(n-1, m-1) - 9*(n-1)*T(n-1, m), n >= m >= 1; T(n, m) := 0, n < m; T(n, 0) := 0 for n >= 1; T(0, 0) = 1.
E.g.f. for the m-th column of the signed triangle: (log(1 + 9*x)/9)^m/m!.
From Petros Hadjicostas, Jun 07 2020: (Start)
T(n,m) = 9^(n-m)*Stirling1(n,m) = 9^(n-m)*A048994(n,m) = 9^(n-m)*A008275(n,m) for n >= m >= 1.
Bivariate e.g.f.-o.g.f.: Sum_{n,m >= 1} T(n,m)*x^n*y^m/n! = exp((y/9)*log(1 + 9*x)) - 1 = (1 + 9*x)^(y/9) - 1. (End)

A153274 Triangle, read by rows, T(n,k) = k^(n+1) * Pochhammer(1/k, n+1).

Original entry on oeis.org

2, 6, 15, 24, 105, 280, 120, 945, 3640, 9945, 720, 10395, 58240, 208845, 576576, 5040, 135135, 1106560, 5221125, 17873856, 49579075, 40320, 2027025, 24344320, 151412625, 643458816, 2131900225, 5925744000, 362880, 34459425, 608608000, 4996616625, 26381811456, 104463111025, 337767408000, 939536222625

Offset: 1

Roger L. Bagula, Dec 22 2008

A Pochhammer function-based triangular sequence.

Row sums are: {2, 21, 409, 14650, 854776, 73920791, 8878927331, 1413788600036, 288152651134776, 73152069870215127, ...}.

			Triangle begins as:
      2;
      6,      15;
     24,     105,      280;
    120,     945,     3640,      9945;
    720,   10395,    58240,    208845,    576576;
   5040,  135135,  1106560,   5221125,  17873856,   49579075;
  40320, 2027025, 24344320, 151412625, 643458816, 2131900225, 5925744000;

G. C. Greubel, Rows n = 1..100 of triangle, flattened

Cf. A000142, A001147, A007559, A007696, A008548, A008542, A045754, A045755, A045756.

Cf. A142589, A256268.

Flat(List([1..12], n-> List([1..n], k-> Product([0..n], j-> j*k+1 )))); # G. C. Greubel, Mar 05 2020

[(&*[j*k+1: j in [0..n]]): k in [1..n], n in [1..12]]; // G. C. Greubel, Mar 05 2020

seq(seq( k^(n+1)*pochhammer(1/k, n+1), k=1..n), n=1..12); # G. C. Greubel, Mar 05 2020

Table[Apply[Plus, CoefficientList[j*k^n*Pochhammer[(j+k)/k, n], j]], {n, 12}, {k,n}]//Flatten (* modified by G. C. Greubel, Mar 05 2020 *)
Table[k^(n+1)*Pochhammer[1/k, n+1], {n,12}, {k,n}]//Flatten (* G. C. Greubel, Mar 05 2020 *)

T(n, k) = prod(j=0, n, j*k+1);
for(n=1, 12, for(k=1, n, print1(T(n, k), ", "))) \\ G. C. Greubel, Mar 05 2020

[[k^(n+1)*rising_factorial(1/k,n+1) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Mar 05 2020

T(n, k) = k^(n+1) * Pochmammer(1/k, n+1).

T(n, k) = Product_{j=0..n} (j*k + 1). - G. C. Greubel, Mar 05 2020

Edited by G. C. Greubel, Mar 05 2020

A147631 9-factorial numbers (6).

Original entry on oeis.org

1, 7, 112, 2800, 95200, 4093600, 212867200, 12984899200, 908942944000, 71806492576000, 6318971346688000, 612940220628736000, 64971663386646016000, 7471741289464291840000, 926495919893572188160000, 123223957345845101025280000, 17497801943110004345589760000

Offset: 1

Vladimir Joseph Stephan Orlovsky, Nov 08 2008

nonn

Cf. A147630, A147629, A049211, A051232, A045756, A035012, A035013, A035017, A035018, A035020, A035022, A035023, A053116.

Cf. A048994, A132393.

s=1;lst={s};Do[s+=n*s;AppendTo[lst,s],{n,6,2*5!,9}];lst

a(n+1) = Sum_{k=0..n} A132393(n,k)*7^k*9^(n-k). - Philippe Deléham, Nov 09 2008
a(n) = (-2)^n*Sum_{k=0..n} (9/2)^k*s(n+1,n+1-k), where s(n,k) are the Stirling numbers of the first kind, A048994. - Mircea Merca, May 03 2012
Sum_{n>=1} 1/a(n) = 1 + (e/9^2)^(1/9)*(Gamma(7/9) - Gamma(7/9, 1/9)). - Amiram Eldar, Dec 21 2022

cp's OEIS Frontend

A035018 One fifth of 9-factorial numbers.

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A035020 One sixth of 9-factorial numbers.

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A035021 One seventh of 9-factorial numbers.

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A256268 Table of k-fold factorials, read by antidiagonals.

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A147629 9-factorial numbers (4).

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A144773 10-fold factorials: Product_{k=0..n-1} (10*k+1).

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A147630 a(1) = 1; for n>1, a(n) = Product_{k = 1..n-1} (9k - 3).

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