A034301 -id:A034301 - cp's OEIS frontend

A051690 a(n) = (5n+9)(!^5)/9(!^5), related to A034301 ((5n+2)(!^5) quintic, or 5-factorials).

1, 14, 266, 6384, 185136, 6294624, 245490336, 10801574784, 529277164416, 28580966878464, 1686277045829376, 107921730933080064, 7446599434382524416, 551048358144306806784, 43532820293400237735936, 3656756904645619969818624, 325451364513460177313857536

Offset: 0

Wolfdieter Lang

Row m=9 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)^(14/5))); [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, 13, 5!, 5}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)
With[{nn = 30}, CoefficientList[Series[1/(1 - 5*x)^(14/5), {x, 0, nn}], x]*Range[0, nn]!] (* G. C. Greubel, Aug 15 2018 *)

my(x='x+O('x^30)); Vec(serlaplace(1/(1-5*x)^(14/5))) \\ G. C. Greubel, Aug 15 2018

a(n) = ((5*n+9)(!^5))/9(!^5) = A034301(n+2)/9.

E.g.f.: 1/(1-5*x)^(14/5).

A008546 Quintuple factorial numbers: Product_{k = 0..n-1} (5*k + 4).

Original entry on oeis.org

1, 4, 36, 504, 9576, 229824, 6664896, 226606464, 8837652096, 388856692224, 19053977918976, 1028914807624704, 60705973649857536, 3885182313590882304, 268077579637770878976, 19837740893195045044224, 1567181530562408558493696, 131643248567242318913470464

Offset: 0

Joe Keane (jgk(AT)jgk.org)

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), Article 00.2.4.
Keiichi Shigechi, On the lattice of weighted partitions, arXiv:2212.14666 [math.CO], 2022. See p. 27.

Cf. A011801, A034301, A048994.

Cf. A008548, A047055, A047056, A051150, A052562, A254287.

List([0..20], n-> Product([0..n-1], k-> 5*k+4 )); # G. C. Greubel, Aug 20 2019

[1] cat [(&*[5*k+4: k in [0..n-1]]): n in [1..20]]; // G. C. Greubel, Aug 20 2019

Maple
```
f:= n-> product(5*k+4, k=0..n-1);
```

FoldList[Times, 1, 5Range[0, 20] + 4] (* Vincenzo Librandi, Jun 10 2013 *)
CoefficientList[Series[(1 - 5x)^(-4/5), {x, 0, 20}], x] Range[0, 20]! (* Vaclav Kotesovec, Jan 28 2015 *)
Table[5^n Pochhammer[4/5, n], {n, 0, 20}] (* G. C. Greubel, Aug 20 2019 *)

vector(20, n, n--; prod(j=0,n-1, 5*j+4) ) \\ G. C. Greubel, Aug 20 2019

[5^n*rising_factorial(4/5, n) for n in (0..20)] # G. C. Greubel, Aug 20 2019

a(n) = 4*A034301(n) = (5*n - 1)(!^5), n >= 1, with a(0) = 1.
a(n) = A011801(n + 1, 1) (first column of triangle).
a(n) ~ (sqrt(2*Pi)/Gamma(4/5))*n^(n + 3/10)*(5/e)^n*(1 + 1/(300*n) + ...). - Joe Keane (jgk(AT)jgk.org), Nov 24 2001
G.f.: 1/(1 - 4*x/(1 - 5*x/(1 - 9*x/(1 - 10*x/(1 - 14*x/(1 - 15*x/(1 - 19*x/(1 - 20*x/(1 - 24*x/(1 - ... (continued fraction). - Philippe Deléham, Jan 08 2012
a(n) = (-1)^n*Sum_{k = 0..n} 5^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 - 1/Q(0) )/x where Q(k) = 1 - x*(5*k - 1)/(1 - x*(5*k + 5)/Q(k + 1) ); (continued fraction); e.g.f. (1 - 5*x)^(-4/5). - Sergei N. Gladkovskii, Mar 20 2013
G.f.: 1/x - G(0)/(2*x), where G(k) = 1 + 1/(1 - x*(5*k - 1)/(x*(5*k - 1) + 1/G(k + 1))); (continued fraction). - Sergei N. Gladkovskii, May 27 2013
a(n) = 5^n * Gamma(n + 4/5) / Gamma(4/5). - Vaclav Kotesovec, Jan 28 2015
a(n) + (-5*n + 1)*a(n - 1) = 0. - R. J. Mathar, Sep 04 2016
G.f.: 1/(1 - 4*x - 20*x^2/(1 - 14*x - 90*x^2/(1 - 24*x - 210*x^2/(1 - 34*x - 380*x^2/(1 - 44*x - 600*x^2/(1 - 54*x - 870*x^2/(1 - ...))))))) (Jacobi continued fraction). - Nikolaos Pantelidis, Feb 29 2020
Sum_{n>=0} 1/a(n) = 1 + (e/5)^(1/5)*(Gamma(4/5) - Gamma(4/5, 1/5)). - Amiram Eldar, Dec 19 2022

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)

A035277 One eighth of deca-factorial numbers.

Original entry on oeis.org

1, 18, 504, 19152, 919296, 53319168, 3625703424, 282804867072, 24886828302336, 2438909173628928, 263402190751924224, 31081458508727058432, 3978426689117063479296, 549022883098154760142848, 81255386698526904501141504, 12838351098367250911180357632

Offset: 1

Wolfdieter Lang

G. C. Greubel, Table of n, a(n) for n = 1..320
Index entries for sequences related to factorial numbers.

Cf. A034301, A045757, A035265, A035272, A035273, A035274, A035275, A035276, A035277, A035278, A035279.

List([1..20], n-> Product([1..n], j-> 10*j-2)/8 ); # G. C. Greubel, Nov 11 2019

[(&*[10*j-2: j in [1..n]])/8: n in [1..20]]; // G. C. Greubel, Nov 11 2019

seq( mul(10*j-2, j=1..n)/8, n=1..20); # G. C. Greubel, Nov 11 2019

Table[10^n*Pochhammer[4/5, n]/8, {n,20}] (* G. C. Greubel, Nov 11 2019 *)

vector(20, n, prod(j=1,n, 10*j-2)/8 ) \\ G. C. Greubel, Nov 11 2019

[product( (10*j-2) for j in (1..n))/8 for n in (1..20)] # G. C. Greubel, Nov 11 2019

a(n) = (Pochhammer(8/10,n)*10^n)/8.
8*a(n) = (10*n-2)(!^10)  = Product_{j=1..n} (10*j-2).
a(n) = 2^(n+2)*A034301(n) where 4*A034301(n) = (5*n-1)(!^5).
E.g.f.: (-1 + (1-10*x)^(-4/5))/8.
Sum_{n>=1} 1/a(n) = 8*(e/10^2)^(1/10)*(Gamma(4/5) - Gamma(4/5, 1/10)). - Amiram Eldar, Dec 22 2022

A051687 a(n) = (5n+6)(!^5)/6, related to A008548 ((5n+1)(!^5) quintic, or 5-factorials).

Original entry on oeis.org

1, 11, 176, 3696, 96096, 2978976, 107243136, 4396968576, 202260554496, 10315288279296, 577656143640576, 35237024762075136, 2325643634296958976, 165120698035084087296, 12549173050666390634496

Offset: 0

Wolfdieter Lang

Row m=6 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..353

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)^(11/5))); [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, 10, 5!, 5}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)
With[{nn = 30}, CoefficientList[Series[1/(1 - 5*x)^(11/5), {x, 0, nn}], x]*Range[0, nn]!] (* G. C. Greubel, Aug 15 2018 *)

x='x+O('x^30); Vec(serlaplace(1/(1-5*x)^(11/5))) \\ G. C. Greubel, Aug 15 2018

a(n) = ((5*n+6)(!^5))/6(!^5).

E.g.f.: 1/(1-5*x)^(11/5).

A051691 a(n) = (5n+10)(!^5)/10(!^5), related to A052562 ((5n)(!^5) quintic, or 5-factorials).

Original entry on oeis.org

1, 15, 300, 7500, 225000, 7875000, 315000000, 14175000000, 708750000000, 38981250000000, 2338875000000000, 152026875000000000, 10641881250000000000, 798141093750000000000, 63851287500000000000000

Offset: 0

Wolfdieter Lang

Row m=10 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)^(15/5))); [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, 14, 5!, 5}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)
With[{nn = 30}, CoefficientList[Series[1/(1 - 5*x)^(15/5), {x, 0, nn}], x]*Range[0, nn]!] (* G. C. Greubel, Aug 15 2018 *)

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

a(n) = ((5*n+10)(!^5))/10(!^5) = A052562(n+2)/(5*10).

E.g.f.: 1/(1-5*x)^3.

A025750 5th-order Patalan numbers (generalization of Catalan numbers).

Original entry on oeis.org

1, 1, 10, 150, 2625, 49875, 997500, 20662500, 439078125, 9513359375, 209293906250, 4661546093750, 104884787109375, 2380077861328125, 54401779687500000, 1251240932812500000, 28934946571289062500, 672311993862304687500, 15687279856787109375000, 367412607172119140625000

Offset: 0

Olivier Gérard

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seq., Vol. 3 (2000), Article 00.2.4.
Elżbieta Liszewska and Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.
Thomas M. Richardson, The Super Patalan Numbers, J. Int. Seq. 18 (2015), Article 15.3.3; arXiv preprint, arXiv:1410.5880 [math.CO], 2014.

Cf. A034687, A049393, A340722.

CoefficientList[Series[(6-(1-25x)^(1/5))/5,{x,0,20}],x] (* Harvey P. Dale, Dec 06 2012 *)
a[0] = 1; a[n_] := ((-5)^(n - 1)*Sum[5^(n - k)*StirlingS1[n, k], {k, 1, n}])/n!; Table[a[n], {n, 0, 16}] (* Jean-François Alcover, Mar 19 2013, after Vladimir Kruchinin *)
a[n_] := 25^(n-1) * Pochhammer[4/5, n-1]/n!; a[0] = 1; Array[a, 20, 0] (* Amiram Eldar, Aug 20 2025 *)

a(n):=if n=0 then 1 else (sum((-1)^(n-k-1)*binomial(n+k-1,n-1)*sum(2^j*binomial(k,j)*sum(binomial(j,i-j)*binomial(k-j,n-3*(k-j)-i-1)*5^(3*(k-j)+i),i,j,n-k+j-1),j,0,k),k,0,n-1))/(n); /* Vladimir Kruchinin, Dec 10 2011 */

a(n):=if n=0 then 1 else -binomial(1/5,n)*(-25)^n/5; /* Tani Akinari, Sep 17 2015 */

G.f.: (6-(1-25*x)^(1/5))/5.
a(n) = 5^(n-1)*4*A034301(n-1)/n!, n >= 2, where 4*A034301(n-1) = (5*n-6)(!^5) = Product_{j=2..n} (5*j-6). - Wolfdieter Lang
a(n) = Sum_{k=0..n-1} (-1)^(n-k-1)*binomial(n+k-1,n-1) * Sum_{j=0..k} 2^j*binomial(k,j) * Sum_{i=j..n-k+j-1} binomial(j,i-j)*binomial(k-j,n-3*(k-j)-i-1)*5^(3*(k-j)+i)/n, n > 0, a(0) = 1. - Vladimir Kruchinin, Dec 10 2011
a(n) = ((-5)^(n-1)*Sum_{k=1..n} (5)^(n-k)*stirling1(n,k))/n!, n>0, a(0) = 1. - Vladimir Kruchinin, Mar 19 2013
From Karol A. Penson, Feb 05 2025: (Start)
a(n) without the initial 1 (i.e., a(n) for n >= 1) is given by
a(n+1) = 5^(2*n)*gamma(n + 4/5)/(gamma(4/5)*(n + 1)!), n >= 0.
a(n+1) = Integral_{x=0..25} x^n*W(x) dx, n >= 0,
  where W(x) = sin(Pi/5)*5^(2/5)*(1 - x/25)^(1/5)/(5*Pi*x^(1/5)). W(x) is a positive  function on x = (0, 25), is singular at x = 0 with the singularity (x)^(-1/5), and it goes to zero at x = 25. (End)
a(n) ~ 25^(n-1) / (Gamma(4/5) * n^(6/5)). - Amiram Eldar, Aug 20 2025

A049223 A convolution triangle of numbers obtained from A025750.

Original entry on oeis.org

1, 10, 1, 150, 20, 1, 2625, 400, 30, 1, 49875, 8250, 750, 40, 1, 997500, 174750, 17875, 1200, 50, 1, 20662500, 3780000, 419625, 32500, 1750, 60, 1, 439078125, 83128125, 9810000, 839500, 53125, 2400, 70, 1, 9513359375, 1852500000, 229359375

Offset: 1

Wolfdieter Lang

a(n,1) = A025750(n); a(n,1)= 5^(n-1)*4*A034301(n-1)/n!, n >= 2. G.f. for m-th column: ((1-(1-25*x)^(1/5))/5)^m.

W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

Cf. A048966, A049213. Row sums = A025758.

T(n,m):=(m*sum((-1)^(n-m-3*k)*binomial(n+k-1,n-1)*sum(2^j*binomial(k,j)*sum(binomial(j,i-j)*binomial(k-j,n-m-3*(k-j)-i)*5^(3*(k-j)+i),i,j,n-m-k+j),j,0,k),k,0,n-m))/n; /* Vladimir Kruchinin, Dec 10 2011 */

a(n, m) = 5*(5*(n-1)-m)*a(n-1, m)/n + m*a(n-1, m-1)/n, n >= m >= 1; a(n, m) := 0, n

T(n,m) = (m*sum(k=0..n-m, (-1)^(n-m-3*k)*binomial(n+k-1,n-1)*sum(j=0..k, 2^j*binomial(k,j)*sum(i=j..n-m-k+j, binomial(j,i-j)*binomial(k-j,n-m-3*(k-j)-i)*5^(3*(k-j)+i)))))/n. - Vladimir Kruchinin, Dec 10 2011

Showing 1-10 of 14 results. Next

cp's OEIS Frontend

A051690 a(n) = (5*n+9)(!^5)/9(!^5), related to A034301 ((5*n+2)(!^5) quintic, or 5-factorials).

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A008546 Quintuple factorial numbers: Product_{k = 0..n-1} (5*k + 4).

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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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A035277 One eighth of deca-factorial numbers.

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A051690 a(n) = (5n+9)(!^5)/9(!^5), related to A034301 ((5n+2)(!^5) quintic, or 5-factorials).

A051687 a(n) = (5n+6)(!^5)/6, related to A008548 ((5n+1)(!^5) quintic, or 5-factorials).

A051691 a(n) = (5n+10)(!^5)/10(!^5), related to A052562 ((5n)(!^5) quintic, or 5-factorials).