A008548 -id:A008548 - cp's OEIS frontend

A318180 Expansion of e.g.f. exp((1 - exp(-5*x))/5).

1, 1, -4, 11, 21, -674, 6551, -33479, -174114, 7478121, -117699599, 1090997976, 865365421, -302755297739, 7922094623596, -127940743443649, 974028543402401, 21377262410290446, -1179125036786673989, 31760741865879345821, -552216474702144564074, 2814873629049018241701

Offset: 0

Ilya Gutkovskiy, Aug 20 2018

sign

Robert Israel, Table of n, a(n) for n = 0..459
Eric Weisstein's World of Mathematics, Bell Polynomial

Cf. A005011, A008548, A009235, A014182, A317996, A318179, A318181.

seq((-5)^n*BellB(n,-1/5),n=0..30); # Robert Israel, Nov 11 2020

nmax = 21; CoefficientList[Series[Exp[(1 - Exp[-5 x])/5], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[(-5)^(n - k) StirlingS2[n, k], {k, 0, n}], {n, 0, 21}]
a[n_] := a[n] = Sum[(-5)^(k - 1) Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 21}]
Table[(-5)^n BellB[n, -1/5], {n, 0, 21}] (* Peter Luschny, Aug 20 2018 *)

my(x = 'x + O('x^25)); Vec(serlaplace(exp((1 - exp(-5*x))/5))) \\ Michel Marcus, Nov 11 2020

a(n) = Sum_{k=0..n} (-5)^(n-k)*Stirling2(n,k).
a(0) = 1; a(n) = Sum_{k=1..n} (-5)^(k-1)*binomial(n-1,k-1)*a(n-k).
a(n) = (-5)^n*BellPolynomial_n(-1/5). - Peter Luschny, Aug 20 2018

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.

A346987 Expansion of e.g.f. 1 / (1 + 5 * log(1 - x))^(1/5).

Original entry on oeis.org

1, 1, 7, 86, 1524, 35370, 1015590, 34757400, 1381147440, 62498177880, 3172764322680, 178566159846480, 11034757650750960, 742773843654742080, 54094804600076176320, 4238009228531321452800, 355400361455423327193600, 31764402860426288679456000, 3014207878695233997923193600

Offset: 0

Ilya Gutkovskiy, Aug 11 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..350

Cf. A007840, A008548, A346978, A346984, A347015, A347016, A347019.

nmax = 18; CoefficientList[Series[1/(1 + 5 Log[1 - x])^(1/5), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Abs[StirlingS1[n, k]] 5^k Pochhammer[1/5, k], {k, 0, n}], {n, 0, 18}]

a[n]:=if n=0 then 1 else sum(n!/(n-k)!*(5/k-4/n)*a[n-k],k,1,n);
makelist(a[n],n,0,50); /* Tani Akinari, Aug 27 2023 */

a(n) = Sum_{k=0..n} |Stirling1(n,k)| * A008548(k).
a(n) ~ n! * exp(n/5) / (Gamma(1/5) * 5^(1/5) * n^(4/5) * (exp(1/5) - 1)^(n + 1/5)). - Vaclav Kotesovec, Aug 14 2021
For n > 0, a(n) = Sum_{k=1..n} (n!/(n-k)!)*(5/k-4/n)*a(n-k). - Tani Akinari, Aug 27 2023

A347022 Expansion of e.g.f. 1 / (1 - 5 * log(1 + x))^(1/5).

Original entry on oeis.org

1, 1, 5, 50, 720, 13650, 320370, 8967720, 291538080, 10795026840, 448484788680, 20658543923280, 1044915105622800, 57572197848878400, 3432143603792520000, 220109018869587398400, 15110184224165199667200, 1105545474191480800492800, 85881534014930659599571200

Offset: 0

Ilya Gutkovskiy, Aug 11 2021

nonn

Cf. A006252, A008548, A320343, A346984, A346987, A347020, A347021, A347023.

nmax = 18; CoefficientList[Series[1/(1 - 5 Log[1 + x])^(1/5), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS1[n, k] 5^k Pochhammer[1/5, k], {k, 0, n}], {n, 0, 18}]

a(n) = Sum_{k=0..n} Stirling1(n,k) * A008548(k).
a(n) ~ n! * exp(1/25) / (Gamma(1/5) * 5^(1/5) * n^(4/5) * (exp(1/5) - 1)^(n + 1/5)). - Vaclav Kotesovec, Aug 14 2021
a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k-1) * (5 - 4*k/n) * (k-1)! * binomial(n,k) * a(n-k). - Seiichi Manyama, Sep 11 2023

A113132 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 = 5.

Original entry on oeis.org

1, 1, 5, 50, 775, 16250, 426750, 13402500, 488566875, 20249281250, 939823431250, 48278138937500, 2719288331093750, 166652371531562500, 11040797013538437500, 786338134640203125000, 59916445436152444921875

Offset: 0

Philippe Deléham and Paul D. Hanna, Oct 28 2005

nonn

			a(2) = 5.
a(3) = 2*5^2 = 50.
a(4) = 5*3*50 + 1*5*5 = 775.
a(5) = 5*4*775 + 1*5*50 + 2*50*5 = 16250.
a(6) = 5*5*16250 + 1*5*775 + 2*50*50 + 3*775*5 = 426750.
G.f.: A(x) = 1 + x + 5*x^2 + 50*x^3 + 775*x^4 + 16250*x^5 +...
= x/series_reversion(x + x^2 + 6*x^3 + 66*x^4 + 1056*x^5
+...).

Cf. A008548, A075834(x=1), A111088(x=2), A113130(x=3), A113131(x=4), A113133(x=6), A113134(x=7), A113135(x=8).

x=5;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, 17}](Robert G. Wilson v)

a(n)=Vec(x/serreverse(x*Ser(vector(n+1,k,if(k==1,1, prod(j=0,k-2,5*j+1))))))[n+1]

a(n,x=5)=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} 5^k*A113129(n, k).
G.f.: A(x) = x/series_reversion(x*G(x)) where G(x) = g.f. of quintic factorials (A008548).
G.f. satisfies: A(x*G(x)) = G(x) = g.f. of quintic factorials (A008548).

A230134 Expansion of e.g.f. 1/(1 - sin(5*x))^(1/5).

Original entry on oeis.org

1, 1, 6, 41, 456, 6301, 108576, 2207981, 52012416, 1390239481, 41593598976, 1376769180401, 49955931795456, 1971671764875541, 84095262825824256, 3854514200269774901, 188942180401957502976, 9863099585213327293681, 546266997049408050364416, 31993839349571172423492281

Offset: 0

Paul D. Hanna, Dec 20 2013

nonn

			E.g.f.: A(x) = 1 + x + 6*x^2/2! + 41*x^3/3! + 456*x^4/4! + 6301*x^5/5! +...
O.g.f.: 1/(1-x - 5*1*2/2*x^2/(1-6*x - 5*2*7/2*x^2/(1-11*x - 5*3*12/2*x^2/(1-16*x - 5*4*17/2*x^2/(1-21*x - 5*5*22/2*x^2/(1-...)))))), a continued fraction.

Cf. A001586, A007788, A144015, A227544, A230114.

Cf. A008548, A136630.

CoefficientList[Series[1/(1-Sin[5*x])^(1/5), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jan 03 2014 *)

{a(n)=local(X=x+x*O(x^n)); n!*polcoeff((1-sin(5*X))^(-1/5), n)}
for(n=0, 20, print1(a(n), ", "))

{a(n)=local(A=1+x+x*O(x^n)); for(i=0, n, A=exp(intformal(A^(5/2)/subst(A^(5/2), x, -x)))); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))

a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a008548(n) = prod(k=0, n-1, 5*k+1);
a(n) = sum(k=0, n, a008548(k)*(5*I)^(n-k)*a136630(n, k)); \\ Seiichi Manyama, Jun 24 2025

E.g.f. A(x) satisfies: A(x) = (cos(5*x/2) - sin(5*x/2))^(-2/5).
O.g.f.: 1/G(0) where G(k) = 1 - (5*k+1)*x - 5*(k+1)*(5*k+2)/2*x^2/G(k+1) [continued fraction formula from A144015 due to Sergei N. Gladkovskii].
a(n) ~ n! * sqrt(5+sqrt(5)) * Gamma(3/5) * 2^(n-9/10) * 5^n / (n^(3/5) * Pi^(n+7/5)). - Vaclav Kotesovec, Jan 03 2014
a(n) = Sum_{k=0..n} A008548(k) * (5*i)^(n-k) * A136630(n,k), where i is the imaginary unit. - Seiichi Manyama, Jun 24 2025

A035265 One half of deca-factorial numbers.

Original entry on oeis.org

1, 12, 264, 8448, 354816, 18450432, 1143926784, 82362728448, 6753743732736, 621344423411712, 63377131187994624, 7098238693055397888, 865985120552758542336, 114310035912964127588352, 16232025099640906117545984, 2467267815145417729866989568, 399697386053557672238452310016

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. A008548, A045757, A035272, A035273, A035274, A035275, A035276, A035277, A035278, A035279.

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

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

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

Table[10^n*Pochhammer[2/10, n]/2, {n,20}] (* G. C. Greubel, Nov 11 2019 *)

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

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

2*a(n) = (10*n-8)(!^10) = Product_{j=1..n} (10*j-8).
a(n) = 2^n*A008548(n) where A008548(n) = (5*n-4)(!^5) = Product_{j=1..n} (5*j-4).
E.g.f.: (-1 + (1-10*x)^(-1/5))/2.
a(n) = (Pochhammer(2/10,n)*10^n)/2.
Sum_{n>=1} 1/a(n) = 2*(e/10^8)^(1/10)*(Gamma(1/5) - Gamma(1/5, 1/10)). - Amiram Eldar, Dec 22 2022

A051688 a(n) = (5n+7)(!^5)/7(!^5), related to A034323 ((5n+2)(!^5) quintic, or 5-factorials).

Original entry on oeis.org

1, 12, 204, 4488, 121176, 3877632, 143472384, 6025840128, 283214486016, 14727153272832, 839447736551424, 52045759666188288, 3487065897634615296, 251068744629692301312, 19332293336486307201024

Offset: 0

Wolfdieter Lang

Row m=7 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)^(12/5))); [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, 11, 5!, 5}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)
With[{nn = 30}, CoefficientList[Series[1/(1 - 5*x)^(12/5), {x, 0, nn}], x]*Range[0, nn]!] (* G. C. Greubel, Aug 15 2018 *)

x='x+O('x^30); Vec(serlaplace(1/(1-5*x)^(12/5))) \\ G. C. Greubel, Aug 15 2018

a(n) = ((5*n+7)(!^5))/7(!^5) = A034323(n+2)/7.

E.g.f.: 1/(1-5*x)^(12/5).

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

Original entry on oeis.org

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).

A088992 Derangement numbers d(n,5) where d(n,k) = k(n-1)(d(n-1,k) + d(n-2,k)), with d(0,k) = 1 and d(1,k) = 0.

Original entry on oeis.org

1, 0, 5, 50, 825, 17500, 458125, 14268750, 515440625, 21188375000, 976671703125, 49893003906250, 2797832158515625, 170863509745312500, 11287987223748828125, 802119551344589843750, 61005565392625400390625, 4944614795517599218750000

Offset: 0

N. J. A. Sloane, Nov 02 2003

In general, d(n,k) is asymptotic to sqrt(2*Pi) * k^n * n^(n + 1/2) / (Gamma(1/k) * exp((n*k+1)/k) * n^((k-1)/k)), for k>0. - Vaclav Kotesovec, Oct 31 2017

Cf. A000166, A053871, A033030, A088991.

Inverse binomial transform of A008548. E.g.f.: exp(-x)/(1-5*x)^(1/5). - Vladeta Jovovic, Dec 17 2003
a(n) ~ Pi * sqrt(2) * n^(n-3/10) * 5^n / (sqrt(Pi) * Gamma(1/5) * exp(n + 1/5)). - Vaclav Kotesovec, Oct 31 2017
a(n) = (-1)^n * n! * Sum_{k=0..n} 5^k * binomial(-1/5,k)/(n-k)!. - Seiichi Manyama, Apr 23 2025

cp's OEIS Frontend

A318180 Expansion of e.g.f. exp((1 - exp(-5*x))/5).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A346987 Expansion of e.g.f. 1 / (1 + 5 * log(1 - x))^(1/5).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Maxima

Formula

A347022 Expansion of e.g.f. 1 / (1 - 5 * log(1 + x))^(1/5).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A113132 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 = 5.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A230134 Expansion of e.g.f. 1/(1 - sin(5*x))^(1/5).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

Formula

A035265 One half of deca-factorial numbers.

Views

Author

Keywords

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

A051688 a(n) = (5*n+7)(!^5)/7(!^5), related to A034323 ((5*n+2)(!^5) quintic, or 5-factorials).

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

A113132 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 = 5.

A051688 a(n) = (5n+7)(!^5)/7(!^5), related to A034323 ((5n+2)(!^5) quintic, or 5-factorials).

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