A045754 -id:A045754 - cp's OEIS frontend

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

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

A034832 a(n) = n-th sept-factorial number divided by 5.

Original entry on oeis.org

1, 12, 228, 5928, 195624, 7824960, 367773120, 19859748480, 1211444657280, 82378236695040, 6178367752128000, 506626155674496000, 45089727855030144000, 4328613874082893824000, 445847229030538063872000, 49043195193359187025920000, 5738053837623024882032640000

Offset: 1

Wolfdieter Lang

G. C. Greubel, Table of n, a(n) for n = 1..339
Index entries for sequences related to factorial numbers.

Cf. A045754, A034829, A034830, A034831, A034833, A034834, A084947, A147585.

Rest[FoldList[Times,1,7*Range[20]-2]/5] (* Harvey P. Dale, May 30 2013 *)
Drop[With[{nn = 50}, CoefficientList[Series[(-1 + (1 - 7*x)^(-5/7))/5, {x, 0, nn}], x]*Range[0, nn]!], 1] (* G. C. Greubel, Feb 22 2018 *)

my(x='x+O('x^30)); Vec(serlaplace((-1 + (1-7*x)^(-5/7))/5)) \\ G. C. Greubel, Feb 22 2018

5*a(n) = (7*n-2)(!^7) = Product_{j=1..n} (7*j-2).
E.g.f.: (-1 + (1-7*x)^(-5/7))/5.
From Amiram Eldar, Dec 20 2022: (Start)
a(n) = A147585(n+1)/5.
Sum_{n>=1} 1/a(n) = 5*(e/7^2)^(1/7)*(Gamma(5/7) - Gamma(5/7, 1/7)). (End)

More terms from G. C. Greubel, Feb 22 2018

A034831 a(n) = n-th sept-factorial number divided by 4.

Original entry on oeis.org

1, 11, 198, 4950, 158400, 6177600, 284169600, 15060988800, 903659328000, 60545174976000, 4480342948224000, 362907778806144000, 31935884534940672000, 3033909030819363840000, 309458721143575111680000, 33731000604649687173120000, 3912796070139363712081920000

Offset: 1

Wolfdieter Lang

G. C. Greubel, Table of n, a(n) for n = 1..339
Index entries for sequences related to factorial numbers.

Cf. A049209, A045754, A034829, A034830, A034832, A034833, A034834, A144827.

[(&*[(7*k-3): k in [1..n]])/4: n in [1..30]]; // G. C. Greubel, Feb 24 2018

Drop[With[{nn = 40}, CoefficientList[Series[(-1 + (1 - 7*x)^(-4/7))/4, {x, 0, nn}], x]*Range[0, nn]!], 1] (* G. C. Greubel, Feb 22 2018 *)
Table[Product[7 j - 3, {j, n}], {n, 30}]/4 (* Vincenzo Librandi, Feb 24 2018 *)

my(x='x+O('x^30)); Vec(serlaplace((-1 + (1-7*x)^(-4/7))/4)) \\ G. C. Greubel, Feb 22 2018

4*a(n) = (7*n-3)(!^7) = Product_{j=1..n} (7*j-3).
E.g.f.: (-1 + (1-7*x)^(-4/7))/4.
From Amiram Eldar, Dec 20 2022: (Start)
a(n) = A144827(n)/4.
Sum_{n>=1} 1/a(n) = 4*(e/7^3)^(1/7)*(Gamma(4/7) - Gamma(4/7, 1/7)). (End)

More terms added by G. C. Greubel, Feb 23 2018

A113134 a(0) = a(1) = 1, a(2) = x, a(3) = 2x^2, a(n) = x(n-1)a(n-1) + Sum_{j=2..n-2} (j-1)a(j)a(n-j), n>=4 and for x = 7.

Original entry on oeis.org

1, 1, 7, 98, 2107, 61054, 2215094, 96203268, 4856212179, 279081882086, 17981777803682, 1283631249683804, 100557420457355358, 8577121056958121836, 791318123914138366924, 78521346319092948749576

Offset: 0

Philippe Deléham and Paul D. Hanna, Oct 28 2005

nonn

			a(2) = 7.
a(3) = 2*7^2 = 98.
a(4) = 7*3*98 + 1*7*7 = 2107.
a(5) = 7*4*2107 + 1*7*98 + 2*98*7 = 61054.
a(6) = 7*5*61054 + 1*7*2107 + 2*98*98 + 3*2107*7 = 2215094.
G.f.: A(x) = 1 + x + 7*x^2 + 98*x^3 + 2107*x^4 + 61054*x^5
+...
= x/series_reversion(x + x^2 + 8*x^3 + 120*x^4 + 2640*x^5
+...).

Cf. A045754, A075834(x=1), A111088(x=2), A113130(x=3), A113131(x=4), A113132(x=5), A113133(x=6), A113135(x=8).

x=7;a[0]=a[1]=1;a[2]=x;a[3]=2x^2;a[n_]:=a[n]=x*(n-1)*a[n-1]+Sum[(j-1)*a[j ]*a[n-j], {j, 2, n-2}];Table[a[n], {n, 0, 16}](Robert G. Wilson v)

a(n)=Vec(x/serreverse(x*Ser(vector(n+1,k,if(k==1,1, prod(j=0,k-2,7*j+1))))))[n+1]

a(n,x=7)=if(n<0,0,if(n==0 || n==1,1,if(n==2,x,if(n==3,2*x^2,x*(n-1)*a(n-1)+sum(j=2,n-2,(j-1)*a(j)*a(n-j))))))

a(n+1) = Sum{k, 0<=k<=n} 7^k*A113129(n, k).
G.f.: A(x) = x/series_reversion(x*G(x)) where G(x) = g.f. of 7-fold factorials.
G.f. satisfies: A(x*G(x)) = G(x) = g.f. of 7-fold factorials.

A051186 Generalized Stirling number triangle of first kind.

Original entry on oeis.org

1, -7, 1, 98, -21, 1, -2058, 539, -42, 1, 57624, -17150, 1715, -70, 1, -2016840, 657874, -77175, 4165, -105, 1, 84707280, -29647548, 3899224, -252105, 8575, -147, 1, -4150656720, 1537437132, -220709524, 16252369, -672280, 15778, -196, 1

Offset: 1

Wolfdieter Lang

T(n,m) = R_n^m(a=0, b=7) 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-7*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 07 2020: (Start)
For integers n, m >= 0 and complex numbers a, b (with b <> 0), the numbers R_n^m(a,b) were introduced by Mitrinovic (1961) and further examined by Mitrinovic and Mitrinovic (1962).
They are defined via 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, R_n^m(a,b) = 0 for n < m, and 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).
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=7) but with no zero row or column. (End)

			Triangle T(n,m) (with rows n >= 1 and columns m = 1..n) begins:
         1;
        -7,      1;
        98,    -21,      1;
     -2058,    539,    -42,    1;
     57624, -17150,   1715,  -70,    1;
  -2016840, 657874, -77175, 4165, -105, 1;
  ...
3rd row o.g.f.: E(3,x) = Product_{j=0..2} (x - 7*j) = 98*x - 21*x^2 + x^3.

G. C. Greubel, Rows n = 1..50 of the triangle, flattened
Wolfdieter Lang, First ten rows.
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 nombres de Stirling et les nombres de Bernoulli d'ordre supérieur, 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].

Cf. A000142, A045754 (unsigned row sums), A049209 (row sums), A051188.

The b=1..6 triangles are: A008275 (Stirling1 triangle), A039683, A051141, A051142, A051150, A051151.

[7^(n-k)*StirlingFirst(n,k): k in [1..n], n in [1..12]]; // G. C. Greubel, Feb 22 2022

Table[7^(n-k)*StirlingS1[n, k], {n,12}, {k,n}]//Flatten (* G. C. Greubel, Feb 22 2022 *)

flatten([[(-7)^(n-k)*stirling_number1(n,k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Feb 22 2022

T(n, m) = T(n-1, m-1) - 7*(n-1)*T(n-1, m) for n >= m >= 1, T(n, m) = 0 for n < m, T(n, 0) = 0 for n >= 1, and T(0, 0) = 1.
T(n, 1) = A051188(n-1).
Sum_{k=0..n} T(n, k) = (-1)^(n-1)*A049209(n-1).
Sum_{k=0..n} (-1)^(n-k)*T(n, k) = A045754(n).
E.g.f. for m-th column of signed triangle: (log(1 + 7*x)/7)^m/m!.
T(n,m) = 7^(n-m)*S1(n,m) with the (signed) Stirling1 triangle S1(n,m) = A008275(n,m).
Bivariate e.g.f.-o.g.f.: Sum_{n,m >= 1} T(n,m)*x^n*y^m/n! = exp((y/7)*log(1 + 7*x)) - 1 = (1 + 7*x)^(y/7) - 1. - Petros Hadjicostas, Jun 07 2020
T(n, 0) = (-7)^(n-1)*A000142(n-1). - G. C. Greubel, Feb 22 2022

A053106 a(n) = ((7n+10)(!^7))/10(1^7), related to A034830 (((7n+3)(!^7))/3 sept-, or 7-factorials).

Original entry on oeis.org

1, 17, 408, 12648, 480624, 21628080, 1124660160, 66354949440, 4379426663040, 319698146401920, 25575851712153600, 2225099098957363200, 209159315301992140800, 21125090845501206220800

Offset: 0

Wolfdieter Lang

Row m=10 of the array A(8; m,n) := ((7*n+m)(!^7))/m(!^7), m >= 0, n >= 0.

G. C. Greubel, Table of n, a(n) for n = 0..337

Cf. A051188, A045754(n+1), A034829-A034834(n+1), A053104-A053106 (rows m=0..10).

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1/(1-7*x)^(17/7))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Aug 16 2018

s=1;lst={s};Do[s+=n*s;AppendTo[lst, s], {n, 16, 5!, 7}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)
With[{nn = 30}, CoefficientList[Series[1/(1 - 7*x)^(17/7), {x, 0, nn}], x]*Range[0, nn]!] (* G. C. Greubel, Aug 16 2018 *)

x='x+O('x^30); Vec(serlaplace(1/(1-7*x)^(17/7))) \\ G. C. Greubel, Aug 16 2018

a(n) = ((7*n+10)(!^7))/10(!^7) = A034830(n+2)/10.

E.g.f.: 1/(1-7*x)^(17/7).

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

A235128 Expansion of e.g.f. 1/(1 - sin(7*x))^(1/7).

Original entry on oeis.org

1, 1, 8, 71, 1072, 20161, 476288, 13315751, 432387712, 15959926081, 660372282368, 30265936565831, 1522069164439552, 83327826089289601, 4933286107483701248, 314052936209639958311, 21392225375507849838592, 1552501782546292090638721, 119588747474281844162428928

Offset: 0

Vaclav Kotesovec, Jan 03 2014

Generally, for e.g.f. 1/(1-sin(p*x))^(1/p) we have a(n) ~ n! * 2^(n+3/p) * p^n / (Gamma(2/p) * n^(1-2/p) * Pi^(n+2/p)).

Cf. A001586 (p=2), A007788 (p=3), A144015 (p=4), A230134 (p=5), A227544 (p=6), A230114 (p=8).

Cf. A045754, A136630.

CoefficientList[Series[1/(1-Sin[7*x])^(1/7), {x, 0, 20}], x] * Range[0, 20]!

a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a045754(n) = prod(k=0, n-1, 7*k+1);
a(n) = sum(k=0, n, a045754(k)*(7*I)^(n-k)*a136630(n, k)); \\ Seiichi Manyama, Jun 24 2025

a(n) ~ n! * 2^(n+3/7) * 7^n / (Gamma(2/7) * n^(5/7) * Pi^(n+2/7)).

a(n) = Sum_{k=0..n} A045754(k) * (7*i)^(n-k) * A136630(n,k), where i is the imaginary unit. - Seiichi Manyama, Jun 24 2025

A053105 a(n) = ((7n+9)(!^7))/9(!^7), related to A034829 (((7n+2)(!^7))/2 sept-, or 7-factorials).

Original entry on oeis.org

1, 16, 368, 11040, 408480, 17973120, 916629120, 53164488960, 3455691782400, 248809808332800, 19655974858291200, 1690413837813043200, 157208486916613017600, 15720848691661301760000

Offset: 0

Wolfdieter Lang

Row m=9 of the array A(8; m,n) := ((7*n+m)(!^7))/m(!^7), m >= 0, n >= 0.

G. C. Greubel, Table of n, a(n) for n = 0..337

Cf. A051188, A045754(n+1), A034829-34(n+1), A053104-A053106 (rows m=0..10).

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1/(1-7*x)^(16/7))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Aug 16 2018

s=1;lst={s};Do[s+=n*s;AppendTo[lst, s], {n, 15, 5!, 7}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)
CoefficientList[Series[1/(1-7x)^(16/7),{x,0,20}],x]Range[0,20]! (* Harvey P. Dale, Sep 11 2011 *)

x='x+O('x^30); Vec(serlaplace(1/(1-7*x)^(16/7))) \\ G. C. Greubel, Aug 16 2018

a(n) = ((7*n+9)(!^7))/9(!^7)= A034829(n+2)/9.

E.g.f.: 1/(1-7*x)^(16/7).

A088996 Triangle T(n, k) read by rows: T(n, k) = Sum_{j=0..n} binomial(j, n-k) * |Stirling1(n, n-j)|.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 2, 7, 6, 0, 6, 29, 46, 24, 0, 24, 146, 329, 326, 120, 0, 120, 874, 2521, 3604, 2556, 720, 0, 720, 6084, 21244, 39271, 40564, 22212, 5040, 0, 5040, 48348, 197380, 444849, 598116, 479996, 212976, 40320

Offset: 0

Philippe Deléham, Dec 01 2003, Aug 17 2007

			Triangle begins:
  1;
  0,    1;
  0,    1,     2;
  0,    2,     7,      6;
  0,    6,    29,     46,     24;
  0,   24,   146,    329,    326,    120;
  0,  120,   874,   2521,   3604,   2556,    720;
  0,  720,  6084,  21244,  39271,  40564,  22212,   5040;
  0, 5040, 48348, 197380, 444849, 598116, 479996, 212976, 40320;
  ...

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Trevor Hyde, Liminal reciprocity and factorization statistics, arXiv:1803.08438 [math.NT], 2018.

Variant: A059364, diagonals give A000007, A000142, A067318.

Cf. A001147 (row sums), A048994, A084938.

A088996:= func< n,k | (&+[(-1)^j*Binomial(j,n-k)*StirlingFirst(n,n-j): j in [0..n]]) >;
[A088996(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Feb 23 2022

A059364 := (n, k) -> add(abs(Stirling1(n, n - j))*binomial(j, n - k), j = 0..n);
seq(seq(A059364(n, k), k = 0..n), n = 0..8);  # Peter Luschny, Aug 27 2025

T[n_, k_]:= T[n, k]= Sum[(-1)^(n-i)*Binomial[i, k] StirlingS1[n+1, n+1-i], {i, 0, n}]; {{1}}~Join~Table[Abs@ T[n, k], {n,0,10}, {k,n+1,0,-1}] (* Michael De Vlieger, Jun 19 2018 *)

def A088996(n,k): return add((-1)^(n-i)*binomial(i,k)*stirling_number1(n+1,n+1-i) for i in (0..n))
for n in (0..10): [A088996(n,k) for k in (0..n)]  # Peter Luschny, May 12 2013

T(n, k) given by [0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, ...] DELTA [1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, ...] where DELTA is the operator defined in A084938. [Original name.]
Sum_{k=0..n} (-1)^k*T(n,k) = (-1)^n.
From Vladeta Jovovic, Dec 15 2004: (Start)
E.g.f.: (1-y-y*x)^(-1/(1+x)).
Sum_{k=0..n} T(n, k)*x^k = Product_{k=1..n} (k*x+k-1). (End)
T(n, k) = n*T(n-1, k-1) + (n-1)*T(n-1, k); T(0, 0) = 1, T(0, k) = 0 if k > 0, T(n, k) = 0 if k < 0. - Philippe Deléham, May 22 2005
Sum_{k=0..n} T(n,k)*x^(n-k) = A019590(n+1), A000012(n), A000142(n), A001147(n), A007559(n), A007696(n), A008548(n), A008542(n), A045754(n), A045755(n) for x = -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, respectively. Sum_{k=0..n} T(n,k)*x^k = A033999(n), A000007(n), A001147(n), A008544(n), A008545(n), A008546(n), A008543(n), A049209(n), A049210(n), A049211(n), A049212(n) for x = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, respectively. - Philippe Deléham, Aug 10 2007
T(n, k) = Sum_{j=0..n} (-1)^j*binomial(j, n-k)*StirlingS1(n, n-j). - G. C. Greubel, Feb 23 2022

New name using a formula of G. C. Greubel by Peter Luschny, Aug 27 2025

cp's OEIS Frontend

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

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A034832 a(n) = n-th sept-factorial number divided by 5.

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A034831 a(n) = n-th sept-factorial number divided by 4.

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A113134 a(0) = a(1) = 1, a(2) = x, a(3) = 2x^2, a(n) = x*(n-1)*a(n-1) + Sum_{j=2..n-2} (j-1)*a(j)*a(n-j), n>=4 and for x = 7.

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A051186 Generalized Stirling number triangle of first kind.

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A053106 a(n) = ((7*n+10)(!^7))/10(1^7), related to A034830 (((7*n+3)(!^7))/3 sept-, or 7-factorials).

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

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A113134 a(0) = a(1) = 1, a(2) = x, a(3) = 2x^2, a(n) = x(n-1)a(n-1) + Sum_{j=2..n-2} (j-1)a(j)a(n-j), n>=4 and for x = 7.

A053106 a(n) = ((7n+10)(!^7))/10(1^7), related to A034830 (((7n+3)(!^7))/3 sept-, or 7-factorials).

A053105 a(n) = ((7n+9)(!^7))/9(!^7), related to A034829 (((7n+2)(!^7))/2 sept-, or 7-factorials).