A008548 -id:A008548 - cp's OEIS frontend

A380310 Expansion of e.g.f. exp( 1 - 1/(1-5*x)^(1/5) ).

1, -1, -5, -49, -719, -14077, -344909, -10152829, -349045535, -13727327833, -607873987637, -29931556660105, -1622308999459631, -95982568510668373, -6155361624644676989, -425321834949751148053, -31502433469012320013631, -2489898822489054343250737, -209178052238110675644666341

Offset: 0

Seiichi Manyama, Jan 20 2025

sign

Seiichi Manyama, Table of n, a(n) for n = 0..355

Cf. A145561, A380308, A380309.

Cf. A000587, A008548, A028575.

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(1-1/(1-5*x)^(1/5))))

a(n) = Sum_{k=0..n} 5^(n-k) * |Stirling1(n,k)| * A000587(k).
a(n) = e * (-5)^n * n! * Sum_{k>=0} (-1)^k * binomial(-k/5,n)/k!.
a(0) = 1; a(n) = -Sum_{k=1..n} A008548(k) * binomial(n-1,k-1) * a(n-k).

A020040 a(n) = round( Gamma(n+1/5)/Gamma(1/5) ).

Original entry on oeis.org

1, 0, 0, 1, 2, 7, 37, 229, 1647, 13507, 124269, 1267543, 14196477, 173197024, 2286200718, 32464050199, 493453563029, 7993947721071, 137495900802424, 2502425394604114, 48046567576398986, 970540665043259525

Offset: 0

Simon Plouffe

nonn

Gamma(n+1/5)/Gamma(1/5) = 1, 1/5, 6/25, 66/125, 1056/625, 22176/3125, 576576/15625, 17873856/78125, ... - R. J. Mathar, Sep 04 2016

G. C. Greubel, Table of n, a(n) for n = 0..450

Cf. A008548, A020085, A020130.

[Round(Gamma(n+1/5)/Gamma(1/5)): n in [0..30]]; // G. C. Greubel, Dec 06 2019

Digits := 64:f := proc(n,x) round(GAMMA(n+x)/GAMMA(x)); end;
seq( round(pochhammer(1/5, n)), n=0..30); # G. C. Greubel, Dec 06 2019

Table[Round[Pochhammer[1/5,n]], {n,0,30}] (* G. C. Greubel, Dec 06 2019 *)

x=1/5; vector(30, n, round(gamma(n-1+x)/gamma(x)) ) \\ G. C. Greubel, Dec 06 2019

[round(rising_factorial(1/5,n)) for n in (0..30)] # G. C. Greubel, Dec 06 2019

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

A166973 Triangle T(n,k) read by rows: T(n, k) = (mn - mk + 1)T(n - 1, k - 1) + (5k - 4)(mk - (m - 1))*T(n - 1, k) where m = 0.

Original entry on oeis.org

1, 1, 1, 1, 7, 1, 1, 43, 18, 1, 1, 259, 241, 34, 1, 1, 1555, 2910, 785, 55, 1, 1, 9331, 33565, 15470, 1940, 81, 1, 1, 55987, 378546, 281085, 56210, 4046, 112, 1, 1, 335923, 4219993, 4875906, 1461495, 161406, 7518, 148, 1, 1, 2015539, 46755846, 82234489

Offset: 1

Roger L. Bagula, Oct 26 2009

The recursion T(n, k) = (m*n - m*k + 1)*T(n-1, k-1) + (5*k - 4)*(m*k - (m - 1))*T(n-1, k) was intended to range over m values 0 to 4 as given by the original Mathematica code. This sequences is the case for m = 0. - G. C. Greubel, May 29 2016

With offset 0 in the rows and columns this is the Sheffer triangle S2[5,1] = (exp(x), (exp(5*x) - 1)/5). See S2[4,1] = A111578 (with offsets 0), S[3,1] = A111577 (with offsets 0), S2[2,1] = A039755

			Triangle T(n, k) starts:
n\k   1       2        3        4        5       6      7     8   9 10 ...
1:    1
2:    1       1
3:    1       7        1
4:    1      43       18        1
5:    1     259      241       34        1
6:    1    1555     2910      785       55       1
7:    1    9331    33565    15470     1940      81      1
8:    1   55987   378546   281085    56210    4046    112     1
9:    1  335923  4219993  4875906  1461495  161406   7518   148   1
10:   1 2015539 46755846 82234489 35567301 5658051 394464 12846 189  1
... Reformatted, - _Wolfdieter Lang_, Aug 13 2017

G. C. Greubel, Table of n, a(n) for the first 25 rows

Cf. A111577.

S2[4,1] = A111578 (with offsets 0), S2[3,1] = A111577 (with offsets 0), S2[2,1] = A039755. - Wolfdieter Lang, Aug 13 2017

A[n_, 1] := 1; A[n_, n_] := 1; A[n_, k_] := A[n - 1, k - 1] + (5*k - 4)*A[n - 1,k]; Flatten[ Table[A[n, k], {n, 10}, {k, n}]] (* modified by G. C. Greubel, May 29 2016 *)

T(n, k) = T(n - 1, k - 1) + (5*k - 4)*T(n - 1, k).

E.g.f. column k: int(exp(x)*((exp(5*x)-1)/5)^(k-1)/(k-1)!, x) + (-1)^k/A008548(k). - Wolfdieter Lang, Aug 13 2017

A081408 a(n) = (n+1)*a(n-5), with a(0)=a(1)=a(2)=a(3)=a(4)=1.

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 7, 8, 9, 10, 66, 84, 104, 126, 150, 1056, 1428, 1872, 2394, 3000, 22176, 31416, 43056, 57456, 75000, 576576, 848232, 1205568, 1666224, 2250000, 17873856, 27143424, 39783744, 56651616, 78750000, 643458816, 1004306688, 1511782272

Offset: 0

Labos Elemer, Apr 01 2003

nonn

Quintic factorial sequences are generated by single 5-order recursion and appear in unified form.

			A008548, A034323, A034300, A034301, A034325 sequences are combed together as A081408(5n+r) with r=0,1,2,3,4.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

Cf. A001147, A002866, A034001, A007599, A034000, A007696, A000407, A034176, A034177, A008548, A034323, A034300, A034301, A034325 [double, triple, quartic, quintic, factorial subsequences], generated together in A081405-A081408.

a:=[1,1,1,1,1];; for n in [6..40] do a[n]:=n*a[n-5]; od; a; # G. C. Greubel, Aug 15 2019

a081407 n = a081408_list !! n
a081407_list = 1 : 1 : 1 : 1 : zipWith (*) [5..] a081407_list
-- Reinhard Zumkeller, Jan 05 2012

[n le 5 select 1 else n*Self(n-5): n in [1..40]]; // G. C. Greubel, Aug 15 2019

a[0]=a[1]=a[2]=a[3]=a[4]=1; a[x_]:= (x+1)*a[x-5]; Table[a[n], {n, 40}]

m=30; v=concat([1,1,1,1,1], vector(m-5)); for(n=6, m, v[n]=n*v[n-5] ); v \\ G. C. Greubel, Aug 15 2019

def a(n):
    if (n<5): return 1
    else: return (n+1)*a(n-5)
[a(n) for n in (0..40)] # G. C. Greubel, Aug 15 2019

A091545 First column sequence of the array (7,2)-Stirling2 A091747.

Original entry on oeis.org

1, 42, 5544, 1507968, 696681216, 489070213632, 485157651922944, 646229992361361408, 1112808046846264344576, 2405890997281623512973312, 6380422924790865556405223424, 20366309975932442856045473169408, 77025384328976498881563979526701056, 340606249502734078054275917467072069632

Offset: 1

Wolfdieter Lang, Feb 13 2004

Also sixth column (m=5) sequence of triangle A091543.

Pawel Blasiak, Karol A. Penson, and Allan I. Solomon, The general boson normal ordering problem, Physics Letters A, Vol. 309, No. 3-4 (2003), pp. 198-205; arXiv preprint, arXiv:quant-ph/0402027, 2004.

Cf. A008548, A034323, A091543, A091747, A175380, A246745.

a[n_] := 5^(2*n) * Pochhammer[1/5, n] * Pochhammer[2/5, n] / 2; Array[a, 15] (* Amiram Eldar, Sep 01 2025 *)

a(n) = Product_{j=0..n-1} ((5*j+2)*(5*j+1))/2, n>=1. From eq.12 of the Blasiak et al. reference with r=7, s=2, k=1.
a(n) = (5^(2*n))*risefac(1/5, n)*risefac(2/5, n)/2, n>=1, with risefac(x, n) = Pochhammer(x, n).
a(n) = fac5(5*n-3)*fac5(5*n-4)/2, n>=1, with fac5(5*n-4)/2 = A034323(n) and fac5(5*n-3) = A008548(n) (5-factorials).
E.g.f.: (hypergeom([1/5, 2/5], [], 25*x)-1)/2.
a(n) = A091747(n, 2), n>=1.
D-finite with recurrence a(n) - (5*n-3)*(5*n-4)*a(n-1) = 0. - R. J. Mathar, Jul 27 2022
a(n) ~ Pi * (5/e)^(2*n) * n^(2*n-2/5) / (Gamma(1/5) * Gamma(2/5)). - Amiram Eldar, Sep 01 2025
a(n) ~ sqrt(Pi*(1 + sqrt(5))) * 5^(2*n + 1/4) * n^(2*n - 2/5) / (Gamma(1/10) * 2^(7/10) * exp(2*n)). - Vaclav Kotesovec, Sep 01 2025

A113146 Row 5 of table A113143; equal to INVERT of quintic (or 5-fold) factorials shifted one place right.

Original entry on oeis.org

1, 1, 2, 9, 83, 1226, 24727, 627909, 19169758, 682800001, 27776711627, 1270110048234, 64470498348983, 3596569233141701, 218698213338646702, 14395754017090902609, 1019782749198898131883, 77351848007810972904826

Offset: 0

Philippe Deléham and Paul D. Hanna, Oct 28 2005

nonn

			A(x) = 1 + x + 2*x^2 + 9*x^3 + 83*x^4 + 1226*x^5 +...
= 1/(1 - x - x^2 - 6*x^3 - 66*x^4 -...- A008548(n)*x^(n+1)
-...).

Cf. A113143, A008548 (5-fold factorials).

{a(n)=local(x=X+X*O(X^n)); A=1/(1-x-x^2*sum(j=0,n,x^j*prod(i=0,j,5*i+1)));return(polcoeff(A,n,X))}

a(n) = Sum_{j=0..k} 5^(k-j)*A111146(k, j).

a(0) = 1; a(n+1) = Sum_{k=0..n} a(k)*A008548(n-k).

A153189 Triangle T(n,k) = Product_{j=0..k} n*j+1.

Original entry on oeis.org

1, 1, 2, 1, 3, 15, 1, 4, 28, 280, 1, 5, 45, 585, 9945, 1, 6, 66, 1056, 22176, 576576, 1, 7, 91, 1729, 43225, 1339975, 49579075, 1, 8, 120, 2640, 76560, 2756160, 118514880, 5925744000, 1, 9, 153, 3825, 126225, 5175225, 253586025, 14454403425, 939536222625

Offset: 0

Roger L. Bagula, Dec 20 2008

Row sums are: {1, 3, 19, 313, 10581, 599881, 50964103, 6047094369, 954249517513, 193146844030201, 48762935887310811,...}. [Corrected by M. F. Hasler, Oct 28 2014]
This is the lower left triangle of the array A142589. - M. F. Hasler, Oct 28 2014
Row n is a subset of the n-fold factorial sequence for k=0..n. For example, T(8,0..8) is A045755(1..9).  These sequences are listed for n=0..10 in A256268. - Georg Fischer, Feb 15 2020

			Triangle begins as:
  1;
  1, 2;
  1, 3,  15;
  1, 4,  28,  280;
  1, 5,  45,  585,   9945;
  1, 6,  66, 1056,  22176,  576576;
  1, 7,  91, 1729,  43225, 1339975,  49579075;
  1, 8, 120, 2640,  76560, 2756160, 118514880,  5925744000;
  1, 9, 153, 3825, 126225, 5175225, 253586025, 14454403425, 939536222625;

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Cf. A000165, A006882, A007661, A007662, A008544.

Cf. A000142 (row 2), A001147 (3), A007559 (4), A007696 (5), A008548 (6), A008542 (7), A045754 (8), A045755 (9), A045756 (10), A144773 (11), A256268 (combined table).

[(&*[n*j+1: j in [0..k]]): k in [0..n], n in [0..10]]; // G. C. Greubel, Feb 15 2020

seq(seq(mul(n*j+1, j=0..k), k=0..n), n=0..10); # G. C. Greubel, Feb 15 2020

T[n_, k_]= If[n==0 && k==0, 1, Product[n*j+1, {j,0,k}]]; Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 15 2020 *)
T[n_, k_]:= T[n, k]= If[k<2, 1+k*n, ((1+n*k)*T[n, k-1] + (1+n*k)*(1+n*(k-1))* T[n, k-2])/2]; Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* Georg Fischer, Feb 17 2020 *)

T(n,k)=prod(j=0,k,n*j+1) \\ M. F. Hasler, Oct 28 2014

[[ product(n*j+1 for j in (0..k)) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Feb 15 2020

T(n, k) = n^(k+1)*Pochhammer(1/n, k+1).
From Vaclav Kotesovec, Oct 10 2016: (Start)
For fixed n > 0:
T(n, k) ~ sqrt(2*Pi) * n^k * k^(k + 1/2 + 1/n) / (Gamma(1 + 1/n) * exp(k)).
T(n, k) ~ k! * n^k * k^(1/n) / Gamma(1 + 1/n).
(End)
T(n, k) = Sum_{j=0..k+1} (-1)^(k-j+1)*Stirling1(k+1,j)*n^(k-j+1). - G. C. Greubel, Feb 17 2020
T(n, k) = ((1+n*k)*T(n, k-1) + (1+n*k)*(1+n*(k-1))*T(n, k-2))/2. - Georg Fischer, Feb 17 2020

Edited and row 0 added by M. F. Hasler, Oct 28 2014

A276482 a(n) = 5^nGamma(n+1/5)Gamma(n+1)/Gamma(1/5).

Original entry on oeis.org

1, 1, 12, 396, 25344, 2661120, 415134720, 90084234240, 25944259461120, 9573431741153280, 4403778600930508800, 2470519795122015436800, 1660189302321994373529600, 1316530116741341538208972800, 1216473827868999581305090867200, 1295544626680484554089921773568000

Offset: 0

Ilya Gutkovskiy, Sep 05 2016

12-gonal (or dodecagonal) factorial numbers, also polygorial(n, 12).

More generally, the m-gonal factorial numbers (or polygorial(n, m)) is 2^(-n)*(m - 2)^n*Gamma(n+2/(m-2))*Gamma(n+1)/Gamma(2/(m-2)), m>2.

Robert Israel, Table of n, a(n) for n = 0..220
Daniel Dockery, Polygorials, Special "Factorials" of Polygonal Numbers, preprint, 2003.
Index entries for sequences related to factorial numbers

Cf. A000142, A008548, A051624.

Cf. similar sequences of m-gonal factorial numbers (or polygorial(n, m)): A006472 (m=3), A001044 (m=4), A084939 (m=5), A000680 (m=6), A084940 (m=7), A084941 (m=8), A084942 (m=9), A084943 (m=10), A084944 (m=11).

seq(mul(k*(5*k-4),k=1..n), n=0..20); # Robert Israel, Sep 18 2016

FullSimplify[Table[5^n Gamma[n + 1/5] (Gamma[n + 1]/Gamma[1/5]), {n, 0, 15}]]
polygorial[k_, n_] := FullSimplify[ n!/2^n (k -2)^n*Pochhammer[2/(k -2),n]]; Array[polygorial[12, #] &, 16, 0] (* Robert G. Wilson v, Dec 13 2016 *)

a(n) = prod(k=1, n, k*(5*k - 4)); \\ Michel Marcus, Sep 06 2016

a(n) = Product_{k=1..n} k*(5*k - 4), a(0)=1.
a(n) = Product_{k=1..n} A051624(k), a(0)=1.
a(n) = A000142(n)*A008548(n).
a(n) ~ 2*Pi*5^n*n^(2*n+1/5)/(Gamma(1/5)*exp(2*n)).
Sum_{n>=0} 1/a(n) = BesselI(-4/5,2/sqrt(5))*Gamma(1/5)/5^(2/5) = Hypergeometric0F1(1/5, 1/5) = 2.085898421130914...

A368119 Array read by ascending antidiagonals. A(n, k) = Product_{j=0..k-1} (n*j + 1).

Original entry on oeis.org

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

Offset: 0

Peter Luschny, Dec 18 2023

A(n, k) is the number of increasing (n + 1)-ary trees on k vertices. (Following a comment of David Callan in A007559.)

			Array A(n, k) starts:
  [0] 1, 1, 1,   1,    1,      1,       1,         1, ...  A000012
  [1] 1, 1, 2,   6,   24,    120,     720,      5040, ...  A000142
  [2] 1, 1, 3,  15,  105,    945,   10395,    135135, ...  A001147
  [3] 1, 1, 4,  28,  280,   3640,   58240,   1106560, ...  A007559
  [4] 1, 1, 5,  45,  585,   9945,  208845,   5221125, ...  A007696
  [5] 1, 1, 6,  66, 1056,  22176,  576576,  17873856, ...  A008548
  [6] 1, 1, 7,  91, 1729,  43225, 1339975,  49579075, ...  A008542
  [7] 1, 1, 8, 120, 2640,  76560, 2756160, 118514880, ...  A045754
  [8] 1, 1, 9, 153, 3825, 126225, 5175225, 253586025, ...  A045755

Index entries for sequences related to factorial numbers.

Rows: A000012, A000142, A001147, A007559, A007696, A008548, A008542, A045754, A045755.
Columns: A000012, A000027, A000384, A011199, A011245.
Variant: A256268. Main diagonal: A092985.
Cf. A124320, A008279, A326323.

def A(n, k): return n**k * rising_factorial(1/n, k) if n > 0 else 1
for n in range(9): print([A(n, k) for k in range(8)])

Let rf(n, k) denote the rising factorial and ff(n,k) the falling factorial.
A(n, k) = n^k * rf(1/n, k) if n > 0 else 1.
A(n, k) = (-n)^k * ff(-1/n, k) if n > 0 else 1.
A(n, k) = (n^k * Gamma(k + 1/n)) / Gamma(1/n) for n > 0.
A(n, k) = ((-n)^k * Gamma(1 - 1/n)) / Gamma(1 - 1/n - k) for n > 0.
A(n, k) = k! * [x^k](1 - n*x)^(-1/n).
A(n, k) = [x^k] hypergeom([1, 1/n], [], n*x).
Column n + 1 has a linear recurrence with constant coefficients and signature ((-1)^k*binomial(n+1, n-k) for k=0..n).

cp's OEIS Frontend

A380310 Expansion of e.g.f. exp( 1 - 1/(1-5*x)^(1/5) ).

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A020040 a(n) = round( Gamma(n+1/5)/Gamma(1/5) ).

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

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A166973 Triangle T(n,k) read by rows: T(n, k) = (m*n - m*k + 1)*T(n - 1, k - 1) + (5*k - 4)*(m*k - (m - 1))*T(n - 1, k) where m = 0.

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A081408 a(n) = (n+1)*a(n-5), with a(0)=a(1)=a(2)=a(3)=a(4)=1.

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A091545 First column sequence of the array (7,2)-Stirling2 A091747.

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A113146 Row 5 of table A113143; equal to INVERT of quintic (or 5-fold) factorials shifted one place right.

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A166973 Triangle T(n,k) read by rows: T(n, k) = (mn - mk + 1)T(n - 1, k - 1) + (5k - 4)(mk - (m - 1))*T(n - 1, k) where m = 0.

A276482 a(n) = 5^nGamma(n+1/5)Gamma(n+1)/Gamma(1/5).