A008548 -id:A008548 - cp's OEIS frontend

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.

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

A034300 a(n) = n-th quintic factorial number divided by 3.

Original entry on oeis.org

1, 8, 104, 1872, 43056, 1205568, 39783744, 1511782272, 65006637696, 3120318609408, 165376886298624, 9591859405320192, 604287142535172096, 41091525692391702528, 2999681375544594284544, 233975147292478354194432, 19419937225275703398137856, 1708954475824261899036131328

Offset: 1

Wolfdieter Lang

nonn

G. C. Greubel, Table of n, a(n) for n = 1..350

Cf. A008548, A034301, A047056.

List([1..25], n-> Product([1..n], k-> 5*k-2)/3 ); # G. C. Greubel, Aug 23 2019

[(&*[5*k-2: k in [1..n]])/3: n in [1..25]]; // G. C. Greubel, Aug 23 2019

a:= n-> mul(5*k-2, k=1..n)/3; seq(a(n), n=1..25); # G. C. Greubel, Aug 23 2019

Table[Product[5j-2,{j,n}],{n,20}]*1/3 (* Harvey P. Dale, Jul 25 2011 *)

a(n) = prod(k=1,n, 5*k-2)/3;
vector(25, n, a(n)) \\ G. C. Greubel, Aug 23 2019

[5^n*rising_factorial(3/5, n)/3 for n in (1..25)] # G. C. Greubel, Aug 23 2019

3*a(n) = (5*n-2)(!^5) = Product_{j=1..n} (5*j-2) = A047056(n).
E.g.f.: (1-5*x)^(-3/5)/3.
a(n) ~ sqrt(2*Pi) * 5/(3*Gamma(3/5)) * n^(11/10) * (5*n/e)^n * (1 + (169/300)/n - ...). - Joe Keane (jgk(AT)jgk.org), Nov 24 2001
a(n) = 5^n * Pochhammer(3/5, n)/3. - G. C. Greubel, Aug 23 2019
D-finite with recurrence: a(n) +(-5*n+2)*a(n-1)=0. - R. J. Mathar, Feb 20 2020
Sum_{n>=1} 1/a(n) = 3*(e/5^2)^(1/5)*(Gamma(3/5) - Gamma(3/5, 1/5)). - Amiram Eldar, Dec 19 2022

Terms a(17) onward added by G. C. Greubel, Aug 23 2019

A034323 a(n) = n-th quintic factorial number divided by 2.

Original entry on oeis.org

1, 7, 84, 1428, 31416, 848232, 27143424, 1004306688, 42180880896, 1982501402112, 103090072909824, 5876134155859968, 364320317663318016, 24409461283442307072, 1757481212407846109184, 135326053355404150407168, 11096736375143140333387776, 965416064637453209004736512

Offset: 1

Wolfdieter Lang

Robert Israel, Table of n, a(n) for n = 1..355
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], 2014-2020.

Cf. A008548, A034300, A034301, A034325, A047055.

a:=[1];; for n in [2..20] do a[n]:=(5*n-3)*a[n-1]; od; a; # G. C. Greubel, Feb 10 2019

[(&*[5*j-3: j in [1..n]])/2: n in [1..20]]; // G. C. Greubel, Feb 10 2019

f:= gfun:-rectoproc({a(n)=(5*n-3)*a(n-1),a(1)=1},a(n),remember):
map(f, [$1..40]); # Robert Israel, Feb 10 2019

Table[Product[5j-3,{j,n}]/2,{n,20}] (* Harvey P. Dale, Nov 25 2013 *)

vector(20, n, prod(j=1, n, 5*j-3)/2) \\ G. C. Greubel, Feb 10 2019

[product(5*j-3 for j in (1..n))/2 for n in (1..20)] # G. C. Greubel, Feb 10 2019

2*a(n) = (5*n-3)(!^5) = Product_{j=1..n} (5*j-3).
E.g.f.: (-1 + (1-5*x)^(-2/5))/2, with a(0) = 0.
a(n) ~ sqrt(2*Pi) * 5/(2*Gamma(2/5)) * n^(9/10) * (5*n/e)^n * (1 + (109/300)/n - ...). - Joe Keane (jgk(AT)jgk.org), Nov 24 2001
D-finite with recurrence a(n) = (5*n-3)*a(n-1). - Robert Israel, Feb 10 2019
From Amiram Eldar, Dec 19 2022: (Start)
a(n) = A047055(n)/2.
Sum_{n>=1} 1/a(n) = 2*(e/5^3)^(1/5)*(Gamma(2/5) - Gamma(2/5, 1/5)). (End)

A034325 a(n) is the n-th quintic factorial number divided by 5.

Original entry on oeis.org

1, 10, 150, 3000, 75000, 2250000, 78750000, 3150000000, 141750000000, 7087500000000, 389812500000000, 23388750000000000, 1520268750000000000, 106418812500000000000, 7981410937500000000000, 638512875000000000000000

Offset: 1

Wolfdieter Lang

Michael De Vlieger, Table of n, a(n) for n = 1..354
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 594.
Norihiro Nakashima and Shuhei Tsujie, Enumeration of Flats of the Extended Catalan and Shi Arrangements with Species, arXiv:1904.09748 [math.CO], 2019.

Cf. A008548, A034300, A034301, A034323, A052562.

List([1..20], n-> 5^(n-1)*Factorial(n) ); # G. C. Greubel, Aug 23 2019

[5^(n-1)*Factorial(n): n in [1..20]]; // G. C. Greubel, Aug 23 2019

seq(5^(n-1)*n!, n=1..20); # G. C. Greubel, Aug 23 2019

Array[5^(# - 1) #! &, 16] (* Michael De Vlieger, May 30 2019 *)

vector(20, n, 5^(n-1)*n!) \\ G. C. Greubel, Aug 23 2019

[5^(n-1)*factorial(n) for n in (1..20)] # G. C. Greubel, Aug 23 2019

5*a(n) = (5*n)(!^5) = Product_{j=1..n} 5*j = 5^(n-1)*n!.
E.g.f.: (-1 + (1-5*x)^(-1))/5, a(0) = 0.
D-finite with recurrence: a(n) - 5*n*a(n-1) = 0. - R. J. Mathar, Feb 24 2020
From Amiram Eldar, Jan 08 2022: (Start)
Sum_{n>=1} 1/a(n) = 5*(exp(1/5)-1).
Sum_{n>=1} (-1)^(n+1)/a(n) = 5*(1-exp(-1/5)). (End)

A157396 A partition product of Stirling_2 type [parameter k = -6] with biggest-part statistic (triangle read by rows).

Original entry on oeis.org

1, 1, 6, 1, 18, 66, 1, 144, 264, 1056, 1, 600, 4620, 5280, 22176, 1, 4950, 68640, 110880, 133056, 576576, 1, 26586, 639870, 3141600, 3259872, 4036032, 17873856, 1, 234528, 10759056, 69263040, 105557760, 113008896, 142990848

Offset: 1

Peter Luschny, Mar 09 2009

Partition product of prod_{j=0..n-1}((k + 1)*j - 1) and n! at k = -6,
summed over parts with equal biggest part (see the Luschny link).
Underlying partition triangle is A134278.
Same partition product with length statistic is A049385.
Diagonal a(A000217) = A008548.
Row sum is A049412.

Peter Luschny, Counting with Partitions.
Peter Luschny, Generalized Stirling_2 Triangles.

Cf. A157397, A157398, A157399, A157400, A080510, A157401, A157402, A157403, A157404, A157405

T(n,0) = [n = 0] (Iverson notation) and for n > 0 and 1 <= m <= n
T(n,m) = Sum_{a} M(a)|f^a| where a = a_1,..,a_n such that
1*a_1+2*a_2+...+n*a_n = n and max{a_i} = m, M(a) = n!/(a_1!*..*a_n!),
f^a = (f_1/1!)^a_1*..*(f_n/n!)^a_n and f_n = product_{j=0..n-1}(-5*j - 1).

Offset corrected by Peter Luschny, Mar 14 2009

A346984 Expansion of e.g.f. 1 / (6 - 5 * exp(x))^(1/5).

Original entry on oeis.org

1, 1, 7, 85, 1495, 34477, 983983, 33476437, 1322441575, 59492222077, 3002578396255, 168005805229285, 10321907081030167, 690761732852321677, 50015387402165694607, 3895721046926471861365, 324805103526730206129607, 28861947117644330678207389, 2722944810091827410698112959

Offset: 0

Ilya Gutkovskiy, Aug 09 2021

Stirling transform of A008548.

Seiichi Manyama, Table of n, a(n) for n = 0..351

Cf. A000670, A008548, A094418, A305404, A346982, A346983, A346985, A352117, A352118, A352119.

g:= proc(n) option remember; `if`(n<2, 1, (5*n-4)*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/(6 - 5 Exp[x])^(1/5), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS2[n, k] 5^k Pochhammer[1/5, k], {k, 0, n}], {n, 0, 18}]

a(n) = Sum_{k=0..n} Stirling2(n,k) * A008548(k).
a(n) ~ n! / (Gamma(1/5) * 6^(1/5) * n^(4/5) * log(6/5)^(n + 1/5)). - Vaclav Kotesovec, Aug 14 2021
O.g.f. (conjectural): 1/(1 - x/(1 - 6*x/(1 - 6*x/(1 - 12*x/(1 - 11*x/(1 - 18*x/(1 - ... - (5*n-4)*x/(1 - 6*n*x/(1 - ... ))))))))) - a continued fraction of Stieltjes-type. - Peter Bala, Aug 22 2023
a(0) = 1; a(n) = Sum_{k=1..n} (5 - 4*k/n) * binomial(n,k) * a(n-k). - Seiichi Manyama, Sep 09 2023
a(0) = 1; a(n) = a(n-1) - 6*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). - Seiichi Manyama, Nov 16 2023

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

A257623 Triangle read by rows: T(n,k) = t(n-k, k), where t(n,m) = f(m)t(n-1,m) + f(n)t(n,m-1) and f(n) = 5*n + 3.

Original entry on oeis.org

1, 3, 3, 9, 48, 9, 27, 501, 501, 27, 81, 4494, 13026, 4494, 81, 243, 37815, 250230, 250230, 37815, 243, 729, 309324, 4122735, 9008280, 4122735, 309324, 729, 2187, 2498649, 62256627, 256971945, 256971945, 62256627, 2498649, 2187

Offset: 0

Dale Gerdemann, May 10 2015

			Array, t(n,k), begins as:
    1,       3,         9,           27,             81, ... A000244;
    3,      48,       501,         4494,          37815, ...;
    9,     501,     13026,       250230,        4122735, ...;
   27,    4494,    250230,      9008280,      256971945, ...;
   81,   37815,   4122735,    256971945,    11820709470, ...;
  243,  309324,  62256627,   6368680566,   450199373658, ...;
  729, 2498649, 891791568, 144065371932, 15108742867890, ...;
Triangle, T(n,k), begins as:
     1;
     3,       3;
     9,      48,        9;
    27,     501,      501,        27;
    81,    4494,    13026,      4494,        81;
   243,   37815,   250230,    250230,     37815,      243;
   729,  309324,  4122735,   9008280,   4122735,   309324,     729;
  2187, 2498649, 62256627, 256971945, 256971945, 62256627, 2498649, 2187;

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

Cf. A000244, A008548, A142460, A257614.
Cf. A038221, A257180, A257611, A257620, A257621, A257625, A257627.
Similar sequences listed in A256890.

t[n_, k_, p_, q_]:= t[n, k, p, q]= If[n<0 || k<0, 0, If[n==0 && k==0, 1, (p*k+ q)*t[n-1,k,p,q] + (p*n+q)*t[n,k-1,p,q]]];
T[n_, k_, p_, q_]= t[n-k,k,p,q];
Table[T[n,k,5,3], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 27 2022 *)

@CachedFunction
def t(n,k,p,q):
    if (n<0 or k<0): return 0
    elif (n==0 and k==0): return 1
    else: return (p*k+q)*t(n-1,k,p,q) + (p*n+q)*t(n,k-1,p,q)
def A257623(n,k): return t(n-k,k,5,3)
flatten([[A257623(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 27 2022

T(n,k) = t(n-k, k) where 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), and f(n) = 5*n + 3.
Sum_{k=0..n} T(n, k) = A008548(n).
From G. C. Greubel, Feb 27 2022: (Start)
t(k, n) = t(n, k).
T(n, n-k) = T(n, k).
t(0, n) = T(n, 0) = A000244(n). (End)

A051150 Generalized Stirling number triangle of first kind.

Original entry on oeis.org

1, -5, 1, 50, -15, 1, -750, 275, -30, 1, 15000, -6250, 875, -50, 1, -375000, 171250, -28125, 2125, -75, 1, 11250000, -5512500, 1015000, -91875, 4375, -105, 1, -393750000, 204187500, -41037500, 4230625

Offset: 1

Wolfdieter Lang

a(n,m) = R_n^m(a=0, b=5) in the notation of the given 1961 and 1962 references.
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 - 5*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 A008275 with diagonal d >= 0 (main diagonal d = 0) scaled with 5^d.

			Triangle a(n,m) (with rows n >= 1 and columns m = 1..n) begins:
        1;
       -5,      1;
       50,    -15,      1;
     -750,    275,    -30,   1;
    15000,  -6250,    875, -50,    1;
  -375000, 171250, -28125, 2125, -75, 1;
  ...
3rd row o.g.f.: E(3,x) = 50*x - 15*x^2 + x^3.

Wolfdieter Lang, First 10 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. [The numbers R_n^m(a,b) are first introduced.]
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. [Special cases of the numbers R_n^m(a,b) are tabulated.]

First (m=1) column sequence: A052562(n-1).
Row sums (signed triangle): A008546(n-1)*(-1)^(n-1).
Row sums (unsigned triangle): A008548(n).
Cf. A008275 (Stirling1 triangle, b=1), A039683 (b=2), A051141 (b=3), A051142 (b=4).

a(n, m) = a(n-1, m-1) - 5*(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 + 5*x)/5)^m/m!.
a(n, m) = S1(n, m)*5^(n-m), with S1(n, m) := A008275(n, m) (signed Stirling1 triangle).

cp's OEIS Frontend

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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A034300 a(n) = n-th quintic factorial number divided by 3.

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A034323 a(n) = n-th quintic factorial number divided by 2.

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A034325 a(n) is the n-th quintic factorial number divided by 5.

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A157396 A partition product of Stirling_2 type [parameter k = -6] with biggest-part statistic (triangle read by rows).

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A257623 Triangle read by rows: T(n,k) = t(n-k, k), where t(n,m) = f(m)t(n-1,m) + f(n)t(n,m-1) and f(n) = 5*n + 3.