A008548 -id:A008548 - cp's OEIS frontend

A129116 Multifactorial array: A(k,n) = k-tuple factorial of n, for positive n, read by ascending antidiagonals.

1, 1, 2, 1, 2, 6, 1, 2, 3, 24, 1, 2, 3, 8, 120, 1, 2, 3, 4, 15, 720, 1, 2, 3, 4, 10, 48, 5040, 1, 2, 3, 4, 5, 18, 105, 40320, 1, 2, 3, 4, 5, 12, 28, 384, 362880, 1, 2, 3, 4, 5, 6, 21, 80, 945, 3628800, 1, 2, 3, 4, 5, 6, 14, 32, 162, 3840, 39916800, 1, 2, 3, 4, 5, 6, 7, 24, 45, 280, 10395, 479001600

Offset: 1

Jonathan Vos Post, May 24 2007

The term "Quintuple factorial numbers" is also used for the sequences A008546, A008548, A052562, A047055, A047056 which have a different definition. The definition given here is the one commonly used. This problem exists for the other rows as well. "n!2" = n!!, "n!3" = n!!!, "n!4" = n!!!!, etcetera. Main diagonal is A[n,n] = n!n = n.

Similar to A114423 (with rows and columns exchanged). - Georg Fischer, Nov 02 2021

			Table begins:
  k / A(k,n)
  1 | 1 2 6 24 120 720 5040 40320 362880 3628800 ... = A000142.
  2 | 1 2 3  8  15  48  105   384    945    3840 ... = A006882.
  3 | 1 2 3  4  10  18   28    80    162     280 ... = A007661.
  4 | 1 2 3  4   5  12   21    32     45     120 ... = A007662.
  5 | 1 2 3  4   5   6   14    24     36      50 ... = A085157.
  6 | 1 2 3  4   5   6    7    16     27      40 ... = A085158.

Alois P. Heinz, Antidiagonals n = 1..141, flattened
Eric Weisstein's World of Mathematics, Multifactorial.

Cf. A000142 (n!), A006882 (n!!), A007661 (n!!!), A007662(n!4), A085157 (n!5), A085158 (n!6), A114799 (n!7), A114800 (n!8), A114806 (n!9), A288327 (n!10).

Cf. A114423 (transposed).

A:= proc(k,n) option remember; if n >= 1 then n* A(k, n-k) elif n >= 1-k then 1 else 0 fi end: seq(seq(A(1+d-n, n), n=1..d), d=1..16); # Alois P. Heinz, Feb 02 2009

A[k_, n_] := A[k, n] = If[n >= 1, n*A[k, n-k], If[n >= 1-k, 1, 0]]; Table[ A[1+d-n, n], {d, 1, 16}, {n, 1, d}] // Flatten (* Jean-François Alcover, May 27 2016, after Alois P. Heinz *)

A(k,n) = n!k.

A(k,n) = M(n,k) in A114423. - Georg Fischer, Nov 02 2021

Corrected and extended by Alois P. Heinz, Feb 02 2009

A134280 Triangle of numbers obtained from the partition array A134279.

Original entry on oeis.org

1, 6, 1, 66, 6, 1, 1056, 102, 6, 1, 22176, 1452, 102, 6, 1, 576576, 32868, 1668, 102, 6, 1, 17873856, 779328, 35244, 1668, 102, 6, 1, 643458816, 23912064, 843480, 36540, 1668, 102, 6, 1, 26381811456, 812173824, 25416072, 857736, 36540, 1668, 102, 6, 1

Offset: 1

Wolfdieter Lang, Nov 13 2007

This triangle is named S2(6)'.

In the same manner the unsigned Lah triangle A008297 is obtained from the partition array A130561.

			[1]; [6,1]; [66,6,1]; [1056,102,6,1]; [22176,1452,102,6,1]; ...

W. Lang, First 10 rows and more.

Cf. A134275 (S2(5)').
Cf. A134281 (row sums).
Cf. A134282 (alternating row sums).

a(n,m)=sum(product(S2(6;j,1)^e(n,m,k,j),j=1..n),k=1..p(n,m)) if n>=m>=1, else 0. Here p(n,m)=A008284(n,m), the number of m parts partitions of n and e(n,m,k,j) is the exponent of j in the k-th m part partition of n. S2(6;j,1) = A049385(n,1) = A008548(n) = (5*n-4)(!^5)(quintuple- or 5-factorials).

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

A302565 G.f. A(x) satisfies: A(x) = Sum_{n>=0} x^n * A(x)^n * Product_{k=0..n-1} (5*k + 1).

Original entry on oeis.org

1, 1, 7, 85, 1429, 30517, 792007, 24293389, 862902745, 34918162057, 1587910815271, 80217252865861, 4457823231346717, 270261899977497325, 17749585402744292215, 1255201826997862952845, 95083758340337074058545, 7680863233559647281837265, 659040900304099125516970375, 59855299015030039092312638965

Offset: 0

Paul D. Hanna, Apr 09 2018

nonn

			G.f.: A(x) = 1 + x + 7*x^2 + 85*x^3 + 1429*x^4 + 30517*x^5 + 792007*x^6 + 24293389*x^7 + 862902745*x^8 + 34918162057*x^9 + ...
such that
A(x) = 1 + x*A(x) + 6*x^2*A(x)^2 + 66*x^3*A(x)^3 + 1056*x^4*A(x)^4 + 22176*x^5*A(x)^5 + ... + x^n*A(x)^n * Product_{k=0..n-1} (5*k + 1) + ...

Paul D. Hanna, Table of n, a(n) for n = 0..300

Cf. A008548, A088368, A301363, A302100, A302535.

/* Series Reversion of Quintuple Factorials g.f.: */
{a(n) = polcoeff((1/x) * serreverse(x/sum(m=0, n, x^m * prod(k=0, m-1, 5*k + 1)) +x^2*O(x^n)), n)}
for(n=0, 30, print1(a(n), ", "))

/* Differential Equation: */
{a(n) = my(A=1); for(i=0, n, A = 1 + x*A^2*(A + 6*x*A')/(x*A +x^2*O(x^n))'); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

/* Continued fraction: */
{a(n) = my(A=1, CF = 1+x +x*O(x^n)); for(i=1, n, A=CF; for(k=0, n, CF = 1/(1 - floor(5*floor(3*(n-k+1)/2)/3)*x*A*CF ) )); polcoeff(CF, n)}
for(n=0, 30, print1(a(n), ", "))

G.f. A(x) satisfies:
(1) A(x) = Sum_{n>=0} x^n * A(x)^n * Product_{k=0..n-1} (5*k + 1).
(2) A(x) = (1/x)*Series_Reversion( x/F(x) ), where F(x) = Sum_{n>=0} A008548(n)*x^n, the o.g.f. of the quintuple factorials.
(3) A(x) = 1 + x*A(x)^2 * (A(x) + 6*x*A'(x)) / (A(x) + x*A'(x)).
(4) A(x) = 1/(1 - x*A(x)/(1 - 5*x*A(x)/(1 - 6*x*A(x)/(1 - 10*x*A(x)/(1 - 11*x*A(x)/(1 - 15*x*A(x)/(1 - 16*x*A(x)/(1 - ...)))))))), a continued fraction.
a(n) ~ sqrt(2*Pi) * 5^n * n^(n - 3/10) / (Gamma(1/5) * exp(n - 1/5)). - Vaclav Kotesovec, Jun 18 2019

A303488 a(n) = n! * [x^n] 1/(1 - 5*x)^(n/5).

Original entry on oeis.org

1, 1, 14, 312, 9576, 375000, 17873856, 1004306688, 65006637696, 4763494479744, 389812500000000, 35237024762075136, 3487065897634615296, 374960171943074285568, 43532820293400237735936, 5427359437500000000000000, 723181462895975365595529216, 102563963819340862347122245632

Offset: 0

Ilya Gutkovskiy, Apr 24 2018

nonn

			a(1) = 1;
a(2) = 2*7 = 14;
a(3) = 3*8*13 = 312;
a(4) = 4*9*14*19 = 9576;
a(5) = 5*10*15*20*25 = 375000, etc.

Index entries for sequences related to factorial numbers

Column k=5 of A303489.

Cf. A008546, A008548, A034300, A034301, A034323, A034325, A047055, A047056, A051687, A051688, A051689, A051690, A051691, A052562, A113551, A303486, A303487.

Table[n! SeriesCoefficient[1/(1 - 5 x)^(n/5), {x, 0, n}], {n, 0, 17}]
Table[Product[5 k + n, {k, 0, n - 1}], {n, 0, 17}]
Table[5^n Pochhammer[n/5, n], {n, 0, 17}]

a(n) = Product_{k=0..n-1} (5*k + n).
a(n) = 5^n*Gamma(6*n/5)/Gamma(n/5).
a(n) ~ 6^(6*n/5-1/2)*n^n/exp(n).

A051689 a(n) = (5n+8)(!^5)/8(!^5), related to A034300 ((5n+3)(!^5) quintic, or 5-factorials).

Original entry on oeis.org

1, 13, 234, 5382, 150696, 4972968, 188972784, 8125829712, 390039826176, 20672110787328, 1198982425665024, 75535892816896512, 5136440711548962816, 374960171943074285568, 29246893411559794274304

Offset: 0

Wolfdieter Lang

Row m=8 of the array A(6; m,n) := ((5*n+m)(!^5))/m(!^5), m >= 0, n >= 0.

G. C. Greubel, Table of n, a(n) for n = 0..352

Cf. A052562, A008548(n+1), A034323(n+1), A034300(n+1), A034301(n+1), A034325(n+1), A051687-A051691 (rows m=0..10).

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1/(1-5*x)^(13/5))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Aug 15 2018

s=1;lst={s};Do[s+=n*s;AppendTo[lst, s], {n, 12, 5!, 5}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)
With[{nn = 30}, CoefficientList[Series[1/(1 - 5*x)^(13/5), {x, 0, nn}], x]*Range[0, nn]!] (* G. C. Greubel, Aug 15 2018 *)

x='x+O('x^30); Vec(serlaplace(1/(1-5*x)^(13/5))) \\ G. C. Greubel, Aug 15 2018

a(n) = ((5*n+8)(!^5))/8(!^5) = A034300(n+2)/8.

E.g.f.: 1/(1-5*x)^(13/5).

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

A134279 A certain partition array in Abramowitz-Stegun order (A-St order), called M_3(6)/M_3.

Original entry on oeis.org

1, 6, 1, 66, 6, 1, 1056, 66, 36, 6, 1, 22176, 1056, 396, 66, 36, 6, 1, 576576, 22176, 6336, 4356, 1056, 396, 216, 66, 36, 6, 1, 17873856, 576576, 133056, 69696, 22176, 6336, 4356, 2376, 1056, 396, 216, 66, 36, 6, 1, 643458816, 17873856, 3459456, 1463616

Offset: 1

Wolfdieter Lang, Nov 13 2007

Partition number array M_3(6) = A134278 with each entry divided by the corresponding one of the partition number array M_3 = M_3(1) = A036040; in short M_3(6)/M_3.
The sequence of row lengths is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42, ...].
For the A-St order of partitions see the Abramowitz-Stegun reference given in A117506.

			[1]; [6,1]; [66,6,1]; [1056,66,36,6,1]; [22176,1056,396,66,36,6,1]; ...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
W. Lang, First 10 rows and more.

Row sums give A134281 (also of triangle A134280).

Cf. A134274 (M_3(5)/M_3 partition array).

a(n,k) = Product_{j=1..n} S2(6,j,1)^e(n,k,j) with S2(6,n,1) = A049385(n,1) = A008548(n) = (5*n-4)(!^5) (quintuple- or 5-factorials) and with the exponent e(n,k,j) of j in the k-th partition of n in the A-St ordering of the partitions of n.

a(n,k) = A134278(n,k)/A036040(n,k) (division of partition arrays M_3(6) by M_3).

A181122 Decimal expansion of Sum_{k>=0} (-1)^k/(5k+1).

Original entry on oeis.org

8, 8, 8, 3, 1, 3, 5, 7, 2, 6, 5, 1, 7, 8, 8, 6, 3, 8, 0, 4, 0, 7, 5, 5, 2, 2, 7, 0, 2, 0, 3, 7, 9, 3, 4, 6, 2, 7, 8, 1, 1, 0, 8, 3, 0, 7, 7, 5, 4, 5, 8, 1, 7, 1, 2, 0, 5, 9, 7, 0, 6, 8, 2, 0, 8, 4, 7, 6, 9, 9, 0, 6, 9, 6, 4, 0, 4, 2, 3, 8, 0, 4, 1, 5, 8, 1, 9, 7, 3, 6, 7, 1, 9, 2, 4, 2, 0, 4, 5, 9, 7, 0, 7, 6, 6

Offset: 0

Jonathan D. B. Hodgson, Oct 05 2010

			0.88831357265178863804075522702037934627811083077545817120597...

Gheorghe Coserea, Table of n, a(n) for n = 0..1000
H. Wilf, Accelerated series for universal constants, by the WZ method, Discrete Mathematics and Theoretical Computer Science 3(4) (1999), 189-192.

Cf. A002162, A003881, A024396, A113476, A181048, A181049, A181122, A193534, A262246.

(int(1/(1+x^5),x=0..1));
evalf(LerchPhi(-1,1,1/5)/5) ; # R. J. Mathar, Oct 16 2011

(Sqrt[8 + 8/Sqrt[5]]*Pi + 2*Sqrt[5]*ArcCoth[3/Sqrt[5]] + Log[16])/20 // RealDigits[#, 10, 105]& // First (* Jean-François Alcover, Feb 13 2013 *)

default(realprecision, 106);
eval(vecextract(Vec(Str(sumalt(n=0, (-1)^(n)/(5*n+1)))), "3..-2")) \\ Gheorghe Coserea, Oct 06 2015

Sum_{k>=0} (-1)^k/(5k+1) = Integral_{x=0..1}dx/(1+x^5) = (1/10)*sqrt(10-2*sqrt(5))*arctan((3/4)*sqrt(10-2*sqrt(5)) + (1/4)*sqrt(10-2*sqrt(5))*sqrt(5)) + (1/20)*sqrt(10-2*sqrt(5))*arctan(-(1/4)*sqrt(10-2*sqrt(5)) + (1/4)*sqrt(10-2*sqrt(5))*sqrt(5)) + (1/20)*sqrt(10-2*sqrt(5))*sqrt(5)*arctan(-(1/4)*sqrt(10-2*sqrt(5)) + (1/4)*sqrt(10-2*sqrt(5))*sqrt(5)) + (1/20)*log(2)*sqrt(5) + (1/5)*log(2) - (1/20)*log(7-3*sqrt(5))*sqrt(5).
Equals Pi*sqrt(phi)/5^(5/4) + log(phi)/sqrt(5) + log(2)/5, where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Nov 01 2015
From Peter Bala, Feb 19 2024: (Start)
Equals (1/2)*Sum_{n >= 0} n!*(5/2)^n/(Product_{k = 0..n} 5*k + 1) = (1/2)*Sum_{n >= 0} n!*(5/2)^n/A008548(n+1) (apply Euler's series transformation to Sum_{k >= 0} (-1)^k/(5*k + 1)).
Continued fraction: 1/(1 + 1^2/(5 + 6^2/(5 + 11^2/(5 + ... + (5*n + 1)^2/(5 + ... ))))).
The slowly converging series representation Sum_{n >= 0} (-1)^n/(5*n + 1) for the constant can be accelerated to give the following faster converging series:
1/2 + (5/2)*Sum_{n >= 0} (-1)^n/((5*n + 1)(5*n + 6)) and
17/24 + (25/2)*Sum_{n >= 0} (-1)^n/((5*n + 1)(5*n + 6)*(5*n + 11)).
These two series are the cases r = 1 and r = 2 of the general result: for r >= 0, the constant equals
C(r) + ((5/2)^r)*r!*Sum_{n >= 0} (-1)^n/((5*n + 1)*(5*n + 6)*...*(5*n + 5*r + 1)), where C(r) is the rational number (1/2)*Sum_{k = 0..r-1} (5/2)^k*k!/(1*6*11*...*(5*k + 1)). The general result can be proved by the WZ method as described in Wilf. (End)
From Peter Bala, Mar 03 2024: (Start)
Equals hypergeom([1/5, 1], [6/5], -1).
Gauss's continued fraction: 1/(1 + 1^2/(6 + 5^2/(11 + 6^2/(16 + 10^2/(21 + 11^2/(26 + 15^2/(31 + 16^2/(36 + 20^2/(41 + 21^2/(46 + ... )))))))))). (End)

A347013 E.g.f.: exp(x) / (1 - 5 * x)^(1/5).

Original entry on oeis.org

1, 2, 9, 88, 1361, 28182, 726889, 22414988, 803913441, 32867765002, 1508608850249, 76804271962848, 4294870015118641, 261673684619584862, 17252970318529474089, 1223896705010751194068, 92946073511938131386561, 7523666291578066678172562, 646658551118777059833155209

Offset: 0

Ilya Gutkovskiy, Aug 10 2021

nonn

Binomial transform of A008548.

Cf. A000522, A008548, A056546, A084262, A088992, A346258, A347012, A347014.

g:= proc(n) option remember; `if`(n<2, 1, (5*n-4)*g(n-1)) end:
a:= n-> add(binomial(n, k)*g(k), k=0..n):
seq(a(n), n=0..18);  # Alois P. Heinz, Aug 10 2021

nmax = 18; CoefficientList[Series[Exp[x]/(1 - 5 x)^(1/5), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Binomial[n, k] 5^k Pochhammer[1/5, k], {k, 0, n}], {n, 0, 18}]
Table[HypergeometricU[1/5, n + 6/5, 1/5]/5^(1/5), {n, 0, 18}]

a(n) = Sum_{k=0..n} binomial(n,k) * A008548(k).

a(n) ~ n! * exp(1/5) * 5^n / (Gamma(1/5) * n^(4/5)). - Vaclav Kotesovec, Aug 14 2021

cp's OEIS Frontend

A129116 Multifactorial array: A(k,n) = k-tuple factorial of n, for positive n, read by ascending antidiagonals.

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A134280 Triangle of numbers obtained from the partition array A134279.

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

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A302565 G.f. A(x) satisfies: A(x) = Sum_{n>=0} x^n * A(x)^n * Product_{k=0..n-1} (5*k + 1).

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A303488 a(n) = n! * [x^n] 1/(1 - 5*x)^(n/5).

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A051689 a(n) = (5*n+8)(!^5)/8(!^5), related to A034300 ((5*n+3)(!^5) quintic, or 5-factorials).

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

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A051689 a(n) = (5n+8)(!^5)/8(!^5), related to A034300 ((5n+3)(!^5) quintic, or 5-factorials).