A100734 -id:A100734 - cp's OEIS frontend

A100733 a(n) = (4*n)!.

1, 24, 40320, 479001600, 20922789888000, 2432902008176640000, 620448401733239439360000, 304888344611713860501504000000, 263130836933693530167218012160000000, 371993326789901217467999448150835200000000, 815915283247897734345611269596115894272000000000

Offset: 0

Ralf Stephan, Dec 08 2004

From Karol A. Penson, Jun 11 2009: (Start)
Integral representation of a(n) as n-th moment of a positive function W(x) = (1/4)*exp(-x^(1/4))/x^(3/4) on the positive axis:
a(n) = Integral_{x=0..oo} x^n*W(x) dx = Integral_{x=0..oo} x^n*(1/4)*exp(-x^(1/4))/x^(3/4) dx, n >= 0.
This is the solution of the Stieltjes moment problem with the moments a(n), n >= 0.
As the moments a(n) grow very rapidly this suggests, but does not prove, that this solution may not be unique.
This is indeed the case as by construction the following "doubly" infinite family:
V(k,a,x) = (1/4)*exp(-x^(1/4))*(a*sin((3/4)*Pi*k + tan((1/4)*Pi*k)*x^(1/4)) + 1)/x^(3/4),
with the restrictions k=+-1,+-2,..., abs(a) < 1 is still positive on 0 <= x < infinity and has moments a(n).
(End)

Vincenzo Librandi, Table of n, a(n) for n = 0..100

Cf. A000142, A010050, A100732, A100734, A268505, A332890.

[Factorial(4*n): n in [0..10]]; // Vincenzo Librandi, Sep 24 2011

(4*Range[0,20])! (* Harvey P. Dale, Oct 03 2014 *)

E.g.f.: 1/(1-x^4).
From Ilya Gutkovskiy, Jan 20 2017: (Start)
a(n) ~ sqrt(Pi)*2^(8*n+3/2)*n^(4*n+1/2)/exp(4*n).
Sum_{n>=0} 1/a(n) = (cos(1) + cosh(1))/2 = 1.04169147034169174... = A332890. (End)
Sum_{n>=0} (-1)^n/a(n) = cos(1/sqrt(2))*cosh(1/sqrt(2)). - Amiram Eldar, Feb 14 2021

More terms from Harvey P. Dale, Oct 03 2014

A268506 a(n) = Product_{k=0..n} (5*k)!.

Original entry on oeis.org

1, 120, 435456000, 569434649591808000000, 1385378702517271000054360965120000000000, 21488900044302744250061179567064173417691432878080000000000000000

Offset: 0

Vaclav Kotesovec, Apr 16 2016

nonn

Partial products of A100734. - Michel Marcus, Jul 06 2019

Seiichi Manyama, Table of n, a(n) for n = 0..16

Cf. A000178, A098694, A100734, A268504, A268505, A271946, A271947.

Mathematica
```
Table[Product[(5*k)!,{k,0,n}],{n,0,8}]
```

{a(n) = prod(k=1, n, (5*k)!)} \\ Seiichi Manyama, Jul 06 2019

a(n) ~ Gamma(1/5)^(3/5) * Gamma(2/5)^(2/5) * Gamma(3/5)^(1/5) * 2^(n/2 - 1/10) * Pi^(n/2 - 1/10) * 5^(23/60 + 3*n + 5*n^2/2) * exp(1/60 - 3*n - 15*n^2/4) * n^(41/60 + 3*n + 5*n^2/2) / A^(1/5), where A = A074962 is the Glaisher-Kinkelin constant.

A100732 a(n) = (3*n)!.

Original entry on oeis.org

1, 6, 720, 362880, 479001600, 1307674368000, 6402373705728000, 51090942171709440000, 620448401733239439360000, 10888869450418352160768000000, 265252859812191058636308480000000

Offset: 0

Ralf Stephan, Dec 08 2004

a(n) equals (-1)^n times the determinant of the (3n+1) X (3n+1) matrix with consecutive integers from 1 to 3n+1 along the main diagonal, consecutive integers from 2 to 3n+1 along the superdiagonal, consecutive integers from 1 to 3n along the subdiagonal, and 1's everywhere else (see Mathematica code below). - John M. Campbell, Jul 12 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..100

Cf. A000142, A010050, A100733, A100734, A143084, A143819, A268504.

a100732 = a000142 . a008585  -- Reinhard Zumkeller, Feb 19 2013

[Factorial(3*n): n in [0..15]]; // Vincenzo Librandi, Sep 24 2011

Table[(-1)^n*Det[Array[KroneckerDelta[#1, #2]*(#1 - 1) + KroneckerDelta[#1, #2 - 1]*(#1) + KroneckerDelta[#1, #2 + 1]*(#1 - 2) + 1 &, {3*n + 1, 3*n + 1}]], {n, 0, 24}] (* John M. Campbell, Jul 12 2011 *)
(3Range[0,10])! (* Harvey P. Dale, Sep 23 2011 *)

[factorial(3*n) for n in range(0, 11)] # Peter Luschny, Jun 06 2016

a(n) = A000142(A008585(n)).
E.g.f.: 1/(1-x^3).
From Ilya Gutkovskiy, Jan 20 2017: (Start)
a(n) ~ sqrt(2*Pi)*3^(3*n+1/2)*n^(3*n+1/2)/exp(3*n).
Sum_{n>=0} 1/a(n) = (exp(3/2) + 2*cos(sqrt(3)/2))/(3*exp(1/2)) = A143819. (End)
Sum_{n>=0} (-1)^n/a(n) = (1 + 2*exp(3/2)*cos(sqrt(3)/2))/(3*e). - Amiram Eldar, Feb 14 2021
a(n) = A143084(2n,n). - Alois P. Heinz, Jul 12 2024

A195390 a(n) = (6*n)!.

Original entry on oeis.org

1, 720, 479001600, 6402373705728000, 620448401733239439360000, 265252859812191058636308480000000, 371993326789901217467999448150835200000000, 1405006117752879898543142606244511569936384000000000, 12413915592536072670862289047373375038521486354677760000000000

Offset: 0

Vincenzo Librandi, Sep 24 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..75

Cf. A000142, A008588, A010050, A100732, A100733, A100734, A271946, A332892.

Magma
```
[Factorial(6*n): n in [0..10]];
```

(6*Range[0,10])! (* Harvey P. Dale, Dec 16 2013 *)

From Amiram Eldar, Apr 03 2021: (Start)
a(n) = A000142(A008588(n)).
Sum_{n>=0} 1/a(n) = A332892. (End)

A195391 a(n) = (7*n)!.

Original entry on oeis.org

1, 5040, 87178291200, 51090942171709440000, 304888344611713860501504000000, 10333147966386144929666651337523200000000, 1405006117752879898543142606244511569936384000000000, 608281864034267560872252163321295376887552831379210240000000000, 710998587804863451854045647463724949736497978881168458687447040000000000000

Offset: 0

Vincenzo Librandi, Sep 24 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..60

Cf. A000152, A010050, A100732, A100733, A100734, A195390, A271947.

Magma
```
[Factorial(7*n): n in [0..10]];
```

(7*Range[0, 8])! (* Paolo Xausa, Aug 12 2025 *)

a(n)=(7*n)! \\ Charles R Greathouse IV, Dec 27 2011

A322252 a(0) = 1 and a(n) = (5n)!/(5!n!^5) for n > 0.

Original entry on oeis.org

1, 1, 945, 1401400, 2546168625, 5194672859376, 11423951396577720, 26478825654361766400, 63805953776276649848625, 158421985022100255941485000, 402789797982510165934296910320, 1044048983553856888083223814102400, 2749848597736878877579660426025283000

Offset: 0

Seiichi Manyama, Nov 30 2018

nonn

Row 5 of A060540.

Cf. A000142, A100734, A008978, A060542, A082368.

[1] cat [Factorial(5*n)/(120*Factorial(n)^5):n in [1..12]]; // Marius A. Burtea, Feb 18 2020

a[n_]:=(5*n)!/(5!*n!^5); Array[a, 20] (* or *) CoefficientList[Series[HypergeometricPFQ[{1/5, 2/5, 3/5, 4/5}, {1, 1, 1}, 3125 x]/(120 x) , {x, 0, 20}], x] (* Stefano Spezia, Dec 01 2018 *)

O.g.f.: F({1/5, 2/5, 3/5, 4/5}, {1, 1, 1}, 3125*x)/(120*x), where F is the generalized hypergeometric function. - Stefano Spezia, Dec 01 2018

a(n) = (1/5!)*A008978(n) for n >= 1. - Peter Bala, Feb 18 2020

A195392 a(n) = (8*n)!.

Original entry on oeis.org

1, 40320, 20922789888000, 620448401733239439360000, 263130836933693530167218012160000000, 815915283247897734345611269596115894272000000000, 12413915592536072670862289047373375038521486354677760000000000, 710998587804863451854045647463724949736497978881168458687447040000000000000

Offset: 0

Vincenzo Librandi, Sep 24 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..55

Cf. A000152, A010050, A100732, A100733, A100734, A195390, A195391

Magma
```
[Factorial(8*n): n in [0..10]];
```

(8*Range[0, 8])! (* Paolo Xausa, Aug 12 2025 *)

A269296 Decimal expansion of Sum_{k>=0} 1/(5k)!.

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Feb 21 2016

From Vaclav Kotesovec, Feb 24 2016: (Start)
Sum_{k>=0} 1/k! = A001113 = exp(1).
Sum_{k>=0} 1/(2k)! = A073743 = cosh(1).
Sum_{k>=0} 1/(3k)! = A143819 = (2*cos(sqrt(3)/2)*exp(-1/2) + exp(1))/3.
Sum_{k>=0} 1/(4k)! = (cos(1) + cosh(1))/2 = 1.0416914703416917479394211141...
(End)
For q integer >= 1, Sum_{m>=0} 1/(q*m)! = (1/q) * Sum_{k=1..q} exp(X_k) where X_1, X_2, ..., X_q are the q-th roots of unity. - Bernard Schott, Mar 02 2020
Continued fraction: 1 + 1/(120 - 120/(30241 - 30240/(360361 - ... - P(n-1)/((P(n) + 1) - ... )))), where P(n) = (5*n)*(5*n - 1)*(5*n - 2)*(5*n - 3)*(5*n - 4) for n >= 1. Cf. A346441. - Peter Bala, Feb 22 2024

			1 + 1/5! + 1/10! + 1/15! + ... = 1.008333608907290289976453667354838786...

Eric Weisstein's World of Mathematics, Factorial Sums

Cf. A100734.

Cf. A001113 (Sum 1/k!), A073743 (Sum 1/(2k)!), A143819 (Sum 1/(3k)!), A332890 (Sum 1/(4k)!), this sequence (Sum 1/(5k)!), A332892 (Sum 1/(6k)!), A346441.

evalf((exp(1) + 2*exp(-(sqrt(5) + 1)/4) * cos(sqrt((5 - sqrt(5))/2)/2) + 2*exp((sqrt(5) - 1)/4) * cos(sqrt((5 + sqrt(5))/2)/2))/5, 120); # Vaclav Kotesovec, Feb 24 2016

RealDigits[HypergeometricPFQ[{}, {1/5, 2/5, 3/5, 4/5}, 1/3125], 10, 120][[1]]

suminf(k=0, 1/(5*k)!) \\ Michel Marcus, Feb 21 2016

Equals Sum_{k>=0} 1/A100734(k).
Equals (exp(1) + exp(-(-1)^(1/5)) + exp((-1)^(2/5)) + exp(-(-1)^(3/5)) + exp((-1)^(4/5)))/5.
Equals (exp(1) + 2*exp(-(sqrt(5) + 1)/4) * cos(sqrt((5 - sqrt(5))/2)/2) + 2*exp((sqrt(5) - 1)/4) * cos(sqrt((5 + sqrt(5))/2)/2))/5. - Vaclav Kotesovec, Feb 24 2016
Sum_{k>=0} (-1)^k / (5*k)! = (exp(-1) + 2*cos(5^(1/4)/(2*sqrt(phi))) * exp(phi/2) + 2*cos(5^(1/4)*sqrt(phi)/2) / exp(1/(2*phi)))/5 = 0.99166694223909419..., where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Mar 02 2020

A381161 a(n) = (10n)!/((n!)^3(2n)!(5*n)!).

Original entry on oeis.org

1, 15120, 3491888400, 1304290155168000, 601680868708529610000, 312696069714024464473125120, 175460887238127057573116837126400, 103865765423748548466734695459219968000, 63958974275578307119821712720619705931210000, 40596987692554701292235753375257230410967703200000

Offset: 0

Stefano Spezia, Feb 15 2025

nonn

Calabi-Yau series number 2.

S. Hassani, J-M. Maillard, and N. Zenine, On the diagonals of rational functions: the minimal number of variables (unabridged version), arXiv:2502.05543 [math-ph], 2025. See p. 14.

Cf. A000442, A010050, A100734, A195394.

Cf. A381162, A381163, A381164, A381165.

a[n_]:=(10n)!/((n!)^3*(2n)!*(5n)!); Array[a,10,0]

G.f.: hypergeom([1/10, 3/10, 7/10, 9/10], [1, 1, 1], 2^8*5^5*x).

a(n) ~ 9*2^(3+8*n)*5^(1+5*n)/((1 + 24*n)*(1 + 60*n)*Pi^2).

A381164 a(n) = Sum_{k=0..n} binomial(n,k)(5k)!/(k!)^5.

Original entry on oeis.org

1, 121, 113641, 168508561, 306213587881, 624890127114721, 1374618918516663841, 3187068298971939367561, 7682172545187676630759081, 19079663136489248380982551201, 48525227073661262262248690661841, 125818607409307965748858681991235961, 331488456546076036761442657285875590881

Offset: 0

Stefano Spezia, Feb 15 2025

nonn

Calabi-Yau series number 79.

S. Hassani, J.-M. Maillard, and N. Zenine, On the diagonals of rational functions: the minimal number of variables (unabridged version), arXiv:2502.05543 [math-ph], 2025. See p. 16.

Cf. A007318, A008978, A100734.

Cf. A381161, A381162, A381163, A381165.

a[n_]:=Sum[Binomial[n, k](5k)!/k!^5, {k, 0, n}]; Array[a, 13, 0]

G.f.: hypergeom([1/5, 2/5, 3/5, 4/5], [1, 1, 1], 5^5*x/(1-x))/(1-x).
a(n) = hypergeom([1/5, 2/5, 3/5, 4/5, -n], [1, 1, 1, 1], -5^5).
a(n) == 1 (mod 120).
a(n) ~ 2^n * 3^(n+2) * 521^(n+2) / (5^(19/2) * Pi^2 * n^2). - Vaclav Kotesovec, May 29 2025

cp's OEIS Frontend

A100733 a(n) = (4*n)!.

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A268506 a(n) = Product_{k=0..n} (5*k)!.

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A100732 a(n) = (3*n)!.

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A195390 a(n) = (6*n)!.

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A195391 a(n) = (7*n)!.

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Magma

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A322252 a(0) = 1 and a(n) = (5*n)!/(5!*n!^5) for n > 0.

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Magma

Mathematica

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A195392 a(n) = (8*n)!.

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Magma

Mathematica

A269296 Decimal expansion of Sum_{k>=0} 1/(5k)!.

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Comments

A322252 a(0) = 1 and a(n) = (5n)!/(5!n!^5) for n > 0.

A381161 a(n) = (10n)!/((n!)^3(2n)!(5*n)!).

A381164 a(n) = Sum_{k=0..n} binomial(n,k)(5k)!/(k!)^5.