A000178 -id:A000178 - cp's OEIS frontend

A203529 a(n) = A203527(n)/A000178(n-1); A000178 = (superfactorials).

1, 5, 175, 44100, 60913125, 427756329000, 20552836095792000, 6952965728817588480000, 11895516181976215338950400000, 99606443887767729350960121600000000, 5830034964946921746536425070101217280000000

Offset: 1

Clark Kimberling, Jan 03 2012

nonn

It is conjectured that every term is an integer.

Cf. A000178, A203527, A203528, A018252, A093883, A203417.

t = Table[If[PrimeQ[k], 0, k], {k, 1, 100}];
nonprime = Rest[Union[t]]              (* A018252 *)
f[j_] := nonprime[[j]]; z = 20;
v[n_] := Product[Product[f[k] + f[j], {j, 1, k - 1}], {k, 2, n}]
d[n_] := Product[(i - 1)!, {i, 1, n}]  (* A000178 *)
Table[v[n], {n, 1, z}]                 (* A203527 *)
Table[v[n + 1]/v[n], {n, 1, z - 1}]    (* A203528 *)
Table[v[n]/d[n], {n, 1, 20}]           (* A203529 *)

A203533 a(n) = A203530(n)/A000178(n-1); A000178 = (superfactorials).

Original entry on oeis.org

1, 10, 840, 464100, 1481407200, 32851686067200, 5186361382800998400, 4436556151786001058816000, 19667253420867342693731328000000, 605862171333980479840975997239296000000, 132207384898194165523202154782408753283072000000

Offset: 1

Clark Kimberling, Jan 03 2012

nonn

It is conjectured that every term is an integer.

Cf. A000178, A203530, A203532, A002808, A093883, A203420.

t = Table[If[PrimeQ[k], 0, k], {k, 1, 100}];
composite = Rest[Rest[Union[t]]]       (* A002808 *)
f[j_] := composite[[j]]; z = 20;
v[n_] := Product[Product[f[k] + f[j], {j, 1, k - 1}], {k, 2, n}]
d[n_] := Product[(i - 1)!, {i, 1, n}]  (* A000178 *)
Table[v[n], {n, 1, z}]                 (* A203530 *)
Table[v[n + 1]/v[n], {n, 1, z - 1}]    (* A203532 *)
Table[v[n]/d[n], {n, 1, 20}]           (* A203533 *)

A219266 Logarithmic derivative of the superfactorials (A000178).

Original entry on oeis.org

1, 3, 31, 1103, 171311, 149089887, 877704854447, 40451674467223423, 16514355739866259408591, 66586047491662065505372477983, 2923692867015618804999172694908629103, 1527767556403309713534536695030930443376591295, 10306227067090276816548435451550663056418226402352755215

Offset: 1

Paul D. Hanna, Nov 16 2012

nonn

Superfactorial A000178(n) equals the product of first n factorials.

			L.g.f.: L(x) = x + 3*x^2/2 + 31*x^3/3 + 1103*x^4/4 + 171311*x^5/5 +...
where
exp(L(x)) = 1 + x + 2*x^2 + 12*x^3 + 288*x^4 + 34560*x^5 + 24883200*x^6 + 125411328000*x^7 +...+ n!*(n-1)!*(n-2)!*...*3!*2!*1!*0!**x^n +...

Cf. A000178, A219267, A219268, A219269.

nmax=15; Rest[CoefficientList[Series[Log[Sum[BarnesG[k+2]*x^k,{k,0,nmax}]],{x,0,nmax}],x] * Range[0,nmax]] (* Vaclav Kotesovec, Jul 10 2015 *)

{a(n)=n*polcoeff(log(sum(k=0,n+1,prod(j=0,k,j!)*x^k)+x*O(x^n)),n)}
for(n=1,21,print1(a(n),", "))

a(n) ~ n^(n^2/2 + n + 17/12) * (2*Pi)^((n+1)/2) / (A * exp(3*n^2/4 + n - 1/12)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Jul 10 2015

A152690 Partial sums of superfactorials (A000178).

Original entry on oeis.org

1, 2, 4, 16, 304, 34864, 24918064, 125436246064, 5056710181206064, 1834938528961266006064, 6658608419043265483506006064, 265790273955000365854215115506006064

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 10 2008

nonn

G. C. Greubel, Table of n, a(n) for n = 1..44

Cf. A152686, A152687, A152688, A152689, A053308, A053309, A053295, A053296.

lst={};p0=1;s0=0;Do[p0*=a[n];s0+=p0;AppendTo[lst,s0],{n,0,4!}];lst
s = 0; lst = {s}; Do[s += BarnesG[n]; AppendTo[lst, s], {n, 2, 13, 1}]; lst (* Zerinvary Lajos, Jul 16 2009 *)
Table[Sum[BarnesG[k+1],{k,1,n}],{n,1,15}] (* Vaclav Kotesovec, Jul 10 2015 *)

G.f.: W(0)/(2-2*x) , where W(k) = 1 + 1/( 1 - x*(k+1)!/( x*(k+1)! + 1/W(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 19 2013
a(n) ~ exp(1/12 - 3*n^2/4) * n^(n^2/2 - 1/12) * (2*Pi)^(n/2) / A, where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Jul 10 2015
a(n) = n! * G(n+1) + a(n-1), where G(z) is the Barnes G-function. - Daniel Suteu, Jul 23 2016

A156584 Triangle T(n,k) = SF(n+1)/(SF(n-k+1)*SF(k+1)) where SF(n) is the superfactorial A000178(n), read by rows.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 12, 12, 1, 1, 60, 240, 60, 1, 1, 360, 7200, 7200, 360, 1, 1, 2520, 302400, 1512000, 302400, 2520, 1, 1, 20160, 16934400, 508032000, 508032000, 16934400, 20160, 1, 1, 181440, 1219276800, 256048128000, 1536288768000, 256048128000, 1219276800, 181440, 1

Offset: 0

Roger L. Bagula, Feb 10 2009

			Triangle begins as:
  1;
  1,     1;
  1,     3,        1;
  1,    12,       12,         1;
  1,    60,      240,        60,         1;
  1,   360,     7200,      7200,       360,        1;
  1,  2520,   302400,   1512000,    302400,     2520,     1;
  1, 20160, 16934400, 508032000, 508032000, 16934400, 20160, 1;

G. C. Greubel, Rows n = 0..30 of the triangle, flattened

Cf. A007318 (m=0), this sequence (m=1), A156764 (m=3).

Cf. A009963.

SF := n -> mul(j!, j=0..n): T := (n,k) -> SF(n-1)/(SF(n-k)*SF(k)):
seq(print(seq(T(n,k),k=1..n-1)),n=0..9); # Peter Luschny, Jan 24 2015

(* First program *)
b[n_, k_]:= If[k==0, n!, Product[Sum[(-1)^(i+j)*(j+1)*StirlingS1[j-1, i]*(k+1)^i, {i, 0, j-1}], {j, 1, n}]];
T[n_, k_, m_] = If[n==0, 1, b[n, m]/(b[k, m]*b[n-k, m])];
Table[T[n, k, 1], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Jun 20 2021 *)
(* Second program *)
f[n_, k_]:= If[k==0, n!, (-1)^n*(n+1)!*BarnesG[n+k+1]/(Gamma[k+1]^n*BarnesG[k+1])];
T[n_, k_, m_]:= If[n==0, 1, f[n,m]/(f[k,m]*f[n-k,m])];
Table[T[n,k,1], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 20 2021 *)

def f(n,k): return factorial(n) if (k==0) else (-1)^n*factorial(n+1)*product( rising_factorial(k+1, j) for j in (0..n-1) )
def T(n,k,m): return 1 if (n==0) else f(n,m)/(f(k,m)*f(n-k,m))
flatten([[T(n,k,1) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 21 2021

From G. C. Greubel, Jun 21 2021: (Start)
T(n, k) = BarnesG(n+3)/(BarnesG(k+3)*BarnesG(n-k+3)).
T(n, k, m) = f(n, m)/(f(k, m)*f(n-k, m)), with T(0, k, m) = 1, f(n, k) = (-1)^n*(n + 1)!*BarnesG(n+k+1)/(Gamma(k+1)^n*BarnesG(k+1)), f(n, 0) = n!, and m = 1. (End)

New name and editing, Peter Luschny, Jan 24 2015

A173345 Number of trailing zeros of the superfactorial of n (A000178).

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 3, 4, 5, 7, 9, 11, 13, 15, 18, 21, 24, 27, 30, 34, 38, 42, 46, 50, 56, 62, 68, 74, 80, 87, 94, 101, 108, 115, 123, 131, 139, 147, 155, 164, 173, 182, 191, 200, 210, 220, 230, 240, 250, 262, 274, 286, 298, 310, 323, 336, 349, 362, 375, 389, 403, 417

Offset: 1

José María Grau Ribas, Feb 16 2010

Andrew Howroyd, Table of n, a(n) for n = 1..1000
A. M. Oller-Marcen and J. Maria Grau, On the Base-b Expansion of the Number of Trailing Zeros of b^k!, J. Int. Seq. 14 (2011), Article 11.6.8.
Eric Weisstein's World of Mathematics, Superfactorial.

Cf. A000178, A027868.

a[n_] := Sum[Z10[i], {i, n}]; Z10[n_]:= Floor[Sum[Floor[n/5^i], {i, 1, Floor[Log[5, n]]}]]; Join[{0},Table[a[n], {n, 2, 200}]]
a[0] := 1; a[1] := 1; a[n_] := n!*a[n - 1];IntegerExponent[Table[a[n], {n, 1, 100}]] (* Stefano Spezia, Jan 26 2023 *)

a(n)=my(t=0);sum(k=5,n,t+=valuation(k,5)) \\ Charles R Greathouse IV, Jun 10 2011

a(n) = Sum_{k=1..n} A027868(k). - Charles R Greathouse IV, Jun 10 2011

A193440 exp( Sum_{n>=1} x^n/G(n) ) = Sum_{n>=0} a(n)*x^n/G(n), where G(n) = Product_{k=0..n} k! = BarnesG(n+2), (see A000178).

Original entry on oeis.org

1, 1, 2, 9, 145, 10489, 4182481, 10893144241, 213590500341121, 35762619247862532481, 57146369032805384396332801, 963199581177063129894232882156801, 187554502919537918586035198740350553881601, 458564976873147078680542618033293809080455988300801

Offset: 0

Paul D. Hanna, Jul 25 2011

nonn

Sum_{n>=0} a(n)/G(n) = 4.88825080515459884947818345139584332...

			A(x) = 1 + x + 2*x^2/2 + 9*x^3/12 + 145*x^4/288 + 10489*x^5/34560 + 4182481*x^6/24883200 + 10893144241*x^7/125411328000 +...+ a(n)*x^n/G(n) +...
where
log(A(x)) = x + x^2/2 + x^3/12 + x^4/288 + x^5/34560 + x^6/24883200 + x^7/125411328000 +...+ x^n/G(n) +...
and G(n) = 0!*1!*2!*3!*...*(n-1)!*n!.

Eric Weisstein's World of Mathematics, Barnes G-Function.

Cf. A000178.

Table[BarnesG[n+2] * SeriesCoefficient[Exp[Sum[x^k/BarnesG[k+2], {k, 1, n}]], {x, 0, n}], {n, 0, 15}] (* Vaclav Kotesovec, Apr 03 2021 *)

{a(n)=prod(k=1,n,k!)*polcoeff(exp(sum(m=1,n+1,x^m/prod(k=1,m,k!)+x*O(x^n))),n)}

Definition corrected by Vaclav Kotesovec, Apr 03 2021

A193521 G.f.: A(x) = ( Sum_{n>=0} x^n/sf(n) )^3 where A(x) = Sum_{n>=0} a(n)*x^n/sf(n), and sf(n) = Product_{k=0..n} k! is the superfactorial of n (A000178).

Original entry on oeis.org

1, 3, 9, 51, 795, 43923, 10372323, 11996843043, 75315947454723, 2788806652875290883, 654625444656522114316803, 1045012738906587147509753740803, 12046169853230117709495421609499289603, 1053916215003128938522329980606467994425804803

Offset: 0

Paul D. Hanna, Jul 29 2011

nonn

			Let F(x) = 1 + x + x^2/(1!*2!) + x^3/(1!*2!*3!) + x^4/(1!*2!*3!*4!) + ... + x^n/sf(n) + ...
then F(x)^3 = 1 + 3*x + 9*x^2/(1!*2!) + 51*x^3/(1!*2!*3!) + 795*x^4/(1!*2!*3!*4!) + 43923*x^5/(1!*2!*3!*4!*5!) + ... + a(n)*x^n/sf(n) + ...

G. C. Greubel, Table of n, a(n) for n = 0..48

Cf. A000178, A009963, A193520.

A193521:= func< n | (&+[ A009963(n,k)*A193520(k): k in [0..n]]) >;
[A193521(n): n in [0..20]]; // G. C. Greubel, Jan 05 2022

a[n_]:= a[n]= Sum[BarnesG[n+2]/(BarnesG[j+2]*BarnesG[k-j+2]*BarnesG[n-k+2]), {k,0,n}, {j,0,k}];
Table[a[n], {n, 0, 20}] (* G. C. Greubel, Jan 05 2022 *)

{a(n) = prod(k=1,n,k!)*polcoeff((sum(m=0, n+1, x^m/prod(k=0, m, k!) + x*O(x^n))^3), n)}

@CachedFunction
def A009963(n,k): return product(factorial(n-j+1)/factorial(j) for j in (1..k))
def A193521(n): return sum(sum(A009963(n,k)*A009963(k,j) for j in (0..k)) for k in (0..n))
[A193521(n) for n in (0..20)] # G. C. Greubel, Jan 05 2022

From G. C. Greubel, Jan 05 2022: (Start)
a(n) = Sum_{k=0..n} Sum_{j=0..k} BarnesG(n+2)/(BarnesG(j+2)*BarnesG(k-j+2 )*BarnesG(n-k+2)).
a(n) = Sum_{k=0..n} A009963(n, k) * Sum_{j=0..k} A009963(k, j).
a(n) = Sum_{j=0..n} A009963(n, j)*A193520(j). (End)
a(n) ~ c(n) * A^2 * 3^(5/4 + n + n^2/6) * n^(-5/6 + n^2/3) / (2*Pi * exp(1/6 + n^2/2)), where c(n) = 1 if mod(n,3) = 0 and c(n) = 3^(4/3) / n^(1/3) if mod(n,3) = 1 or if mod(n,3) = 2, A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Aug 29 2023

A203313 a(n) = v(n)/A000178(n) where v=A203311 and A000178=(superfactorials).

Original entry on oeis.org

1, 1, 1, 4, 105, 46200, 730329600, 976445416826880, 253989513002664748108800, 30302715258078626805231995747942400, 3921367125196579314580337108803595790318851635200, 1315258359298445647817718300301463137710018409451973278413455360000

Offset: 1

Clark Kimberling, Jan 01 2012

nonn

R. Chapman, A polynomial taking integer values, Mathematics Magazine 29 (1996), p. 121.

Cf. A203311.

f[j_] := Fibonacci[j + 1]; z = 15;
v[n_] := Product[Product[f[k] - f[j], {j, 1, k - 1}], {k, 2, n}]
d[n_] := Product[(i - 1)!, {i, 1, n}]
Table[v[n], {n, 1, z}]               (* A203311 *)
Table[v[n + 1]/v[n], {n, 1, z - 1}]  (* A123741 *)
Table[v[n]/d[n], {n, 1, 13}]         (* A203313 *)

from sympy import fibonacci, factorial
from operator import mul
from functools import reduce
def f(j): return fibonacci(j + 1)
def v(n): return 1 if n==1 else reduce(mul, [reduce(mul, [f(k) - f(j) for j in range(1, k)]) for k in range(2, n + 1)])
def d(n): return reduce(mul, [factorial(i - 1) for i in range(1, n + 1)])
print([v(n)//d(n) for n in range(1, 14)]) # Indranil Ghosh, Jul 26 2017

A203467 a(n) = A203309(n)/A000178(n) where A000178 are superfactorials.

Original entry on oeis.org

1, 1, 2, 15, 630, 198450, 589396500, 19912024006875, 8969371213896843750, 61815874928487448987968750, 7358663747680777931818630148437500, 16862758880642741957030086746987589746093750

Offset: 0

Clark Kimberling, Jan 02 2012

nonn

G. C. Greubel, Table of n, a(n) for n = 0..39
R. Chapman, A polynomial taking integer values, Mathematics Magazine, 29 (1996), 121.

Cf. A000178, A203309, A203310.

F:= Factorial; [1] cat [(&*[(F(2*k+2))/(2^k*F(k+2)): k in [0..n]]): n in [0..20]]; // G. C. Greubel, Aug 29 2023

(* First program *)
f[j_]:= j*(j+1)/2; z = 15;
v[n_]:= Product[Product[f[k] - f[j], {j,k-1}], {k,2,n}]
d[n_]:= Product[(i-1)!, {i,n}]
Table[v[n], {n,0,z}]           (* A203309 *)
Table[v[n+1]/v[n], {n,z}]      (* A203310 *)
Table[v[n]/d[n], {n,0,12}]     (* A203467 *)
(* Second program *)
Table[Product[(2*k+2)!/(2^k*(k+2)!), {k,n-1}], {n,0,20}] (* G. C. Greubel, Aug 29 2023 *)

f=factorial; [product((f(2*j+2))/(2^j*f(j+2)) for j in range(n)) for n in range(21)] # G. C. Greubel, Aug 29 2023

From G. C. Greubel, Aug 29 2023: (Start)
a(n) = (2^(n+3)/Pi)^(n/2)*BarnesG(n+3/2)/(Gamma(n+ 2)*BarnesG(3/2)).
a(n) = (1/2)^binomial(n,2)*BarnesG(n+1)*Product_{k=2..n} binomial(2*k, k+1).
a(n) = Product_{k=1..n-1} (2*k+2)!/(2^k*(k+2)!). (End)
a(n) ~ sqrt(A/Pi) * 2^(n^2/2 + 2*n - 7/24) * n^(n^2/2 - n/2 - 35/24) / exp(3*n^2/4 - n/2 + 1/24), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Nov 19 2023

Name edited by Michel Marcus, May 17 2019

a(0) = 1 prepended by G. C. Greubel, Aug 29 2023

cp's OEIS Frontend

A203529 a(n) = A203527(n)/A000178(n-1); A000178 = (superfactorials).

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A203533 a(n) = A203530(n)/A000178(n-1); A000178 = (superfactorials).

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A219266 Logarithmic derivative of the superfactorials (A000178).

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A152690 Partial sums of superfactorials (A000178).

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A156584 Triangle T(n,k) = SF(n+1)/(SF(n-k+1)*SF(k+1)) where SF(n) is the superfactorial A000178(n), read by rows.

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A173345 Number of trailing zeros of the superfactorial of n (A000178).

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A193440 exp( Sum_{n>=1} x^n/G(n) ) = Sum_{n>=0} a(n)*x^n/G(n), where G(n) = Product_{k=0..n} k! = BarnesG(n+2), (see A000178).

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A193521 G.f.: A(x) = ( Sum_{n>=0} x^n/sf(n) )^3 where A(x) = Sum_{n>=0} a(n)*x^n/sf(n), and sf(n) = Product_{k=0..n} k! is the superfactorial of n (A000178).

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