A036740 -id:A036740 - cp's OEIS frontend

A323717 a(n) = Product_{k=0..n} (n! + k!).

2, 4, 36, 4704, 23400000, 7778123781120, 245221791787632844800, 980866487456532919096049664000, 647456833933936977045736601678008811520000, 89423837415458106416291101560480526982914768896000000000

Offset: 0

Vaclav Kotesovec, Jan 25 2019

nonn

G. C. Greubel, Table of n, a(n) for n = 0..29

Cf. A036740, A203308, A203483, A323702.

F:= Factorial; [(&*[F(n) + F(j): j in [0..n]]): n in [0..20]]; // G. C. Greubel, Aug 30 2023

Table[Product[n!+k!, {k, 0, n}], {n, 0, 10}]

f=factorial; [product(f(n) + f(k) for k in range(n+1)) for n in range(21)] # G. C. Greubel, Aug 30 2023

a(n) ~ 2^((n+3)/2) * Pi^((n+1)/2) * n^((n+1)*(2*n+1)/2) / exp(n^2 + n - 1/12).

a(n) ~ 2 * (n!)^(n+1). - Vaclav Kotesovec, Mar 28 2019

A336247 a(n) = (n!)^n * Sum_{k=0..n} 1 / (k!)^n.

Original entry on oeis.org

1, 2, 9, 460, 684545, 50547203126, 280807908057046657, 165858480204085842350156792, 13997217669604247492958380810030809089, 218434494471443385260764665498960241287478619115850, 792268399795067334328715213043856435592857850955707257780000000001

Offset: 0

Ilya Gutkovskiy, Jul 14 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..30

Cf. A000522, A006040, A036740, A060943, A217284, A336248.

Main diagonal of A343863.

Table[(n!)^n Sum[1/(k!)^n, {k, 0, n}], {n, 0, 10}]

a(n) = (n!)^n * sum(k=0, n, 1/(k!)^n); \\ Michel Marcus, Jul 14 2020

A060943 a(n) = n!^n * Sum_{k=1..n} 1/k^n.

Original entry on oeis.org

1, 5, 251, 357904, 25795462624, 141727869124448256, 83296040059942781485105152, 7013444132843374500928464765799366656, 109329825340451764123791003609208862665771818418176, 396334659032531033249146049131230887376087800711479296000000000000

Offset: 1

Leroy Quet, May 07 2001

			a(3) = 6^3 *(1 + 1/2^3 + 1/3^3) = 251.

Harry J. Smith, Table of n, a(n) for n = 1..30

Cf. A036740.

Main diagonal of A291556.

[(Factorial(n))^n*(&+[1/j^n: j in [1..n]]): n in [1..15]]; // G. C. Greubel, Apr 09 2021

A060943:= n-> (n!)^n*add(1/j^n, j=1..n); seq(A060943(n), n=1..15); # G. C. Greubel, Apr 09 2021

Table[(n!)^n * Sum[1/i^n, {i, 1, n}], {n, 1, 10}] (* Vaclav Kotesovec, Aug 27 2017 *)

{ default(realprecision, 100); for (n=1, 30, write("b060943.txt", n, " ", n!^n * sum(k=1, n, 1/k^n)); ) } \\ Harry J. Smith, Jul 14 2009

[(factorial(n))^n*sum(1/j^n for j in (1..n)) for n in (1..15)] # G. C. Greubel, Apr 09 2021

a(n) ~ 2^(n/2) * Pi^(n/2) * n^(n*(2*n+1)/2) / exp(n^2 - 1/12). - Vaclav Kotesovec, Aug 27 2017

a(n) = (n!)^n * [x^n] PolyLog(n,x)/(1 - x), where PolyLog() is the polylogarithm function. - Ilya Gutkovskiy, Nov 27 2017

A134373 a(n) = ((2n)!)^3.

Original entry on oeis.org

1, 8, 13824, 373248000, 65548320768000, 47784725839872000000, 109903340320478724096000000, 662559760549147780765974528000000, 9159226129831418921308831875072000000000, 262435789155225791087396177124997988352000000000

Offset: 0

Artur Jasinski, Oct 22 2007

nonn

Christian Elsholtz, Golomb’s Conjecture on Prime Gaps, arXiv:1604.06903 [math.NT], 2016.
Christian Elsholtz, Golomb’s Conjecture on Prime Gaps, The American Mathematical Monthly 124.4 (2017): 365-368.

Cf. A000142, A001044, A000442, A036740, A010050, A134366, A134367, A134368, A134369, A134371, A134373, A134374, A134375.

Table[((2n)!)^(3), {n, 0, 10}]
((2*Range[0, 10])!)^3 (* Harvey P. Dale, Jul 25 2016 *)

[factorial(2*n)**3 for n in range(0,9)] # Stefano Spezia, Apr 22 2025

Definition corrected by Harvey P. Dale, Jul 25 2016

A261490 The total element sum of the n-fold f transform applied to the length n sequence of 1's. And f returns a sorted result after multiplying the elements in its input sequence with 1, 2, 3,... in descending size order.

Original entry on oeis.org

0, 1, 4, 19, 100, 633, 4626, 37878, 348224, 3542952, 39339852, 478962252, 6289532928, 89038853856, 1346224983936, 21729308136720, 371924399416320, 6740200653419520, 128878557725067264, 2598800542616444724, 54986036469506668800, 1217069235297874269792

Offset: 0

Alois P. Heinz, Aug 21 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..250

Main diagonal of A166278.

Cf. A031971 (for ascending sort), A036740 (when sum is replaced by product).

f:= l-> sort([seq(sort(l, `>`)[i]*i, i=1..nops(l))]):
a:= n-> add(i, i=(f@@n)([1$n])):
seq(a(n), n=0..35);

A343899 a(n) = Sum_{k=0..n} (k!)^k * binomial(n,k).

Original entry on oeis.org

1, 2, 7, 232, 332669, 24884861086, 139314218808181027, 82606412229102532926819812, 6984964247802365417561163907914436537, 109110688415634181158572146813823590758078301022074, 395940866122426284350759726810156652343313286283891529199276099071

Offset: 0

Seiichi Manyama, May 03 2021

nonn

Binomial transform of (n!)^n.

Seiichi Manyama, Table of n, a(n) for n = 0..30

Cf. A000522, A036740, A046662, A086331, A343898, A343900, A343928.

a[n_] := Sum[(k!)^k * Binomial[n, k], {k, 0, n}]; Array[a, 11, 0] (* Amiram Eldar, May 05 2021 *)

a(n) = sum(k=0, n, k!^k*binomial(n, k));

my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (k!*x)^k/(1-x)^(k+1)))

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x)*sum(k=0, N, k!^(k-1)*x^k)))

G.f.: Sum_{k>=0} (k! * x)^k/(1 - x)^(k+1).

E.g.f.: exp(x) * Sum_{k>=0} (k!)^(k-1) * x^k.

A343900 a(n) = Sum_{k=0..n} (k!)^(k+1) * binomial(n,k).

Original entry on oeis.org

1, 2, 11, 1324, 7967861, 2986023826166, 100306147958903465407, 416336313421816733159702737376, 281633758448076539969292901914477101456489, 39594086612245054028213574779019294652734771094507399786

Offset: 0

Seiichi Manyama, May 03 2021

nonn

Binomial transform of (n!)^(n+1).

Seiichi Manyama, Table of n, a(n) for n = 0..29

Cf. A000522, A036740, A046662, A091868, A343898, A343899, A343929.

a[n_] := Sum[(k!)^(k+1) * Binomial[n, k], {k, 0, n}]; Array[a, 10, 0] (* Amiram Eldar, May 05 2021 *)

a(n) = sum(k=0, n, k!^(k+1)*binomial(n, k));

my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, k!^(k+1)*x^k/(1-x)^(k+1)))

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x)*sum(k=0, N, (k!*x)^k)))

G.f.: Sum_{k>=0} (k!)^(k+1) * x^k/(1 - x)^(k+1).

E.g.f.: exp(x) * Sum_{k>=0} (k! * x)^k.

A036739 a(n) = (n!)^n+1.

Original entry on oeis.org

2, 2, 5, 217, 331777, 24883200001, 139314069504000001, 82606411253903523840000001, 6984964247141514123629140377600000001, 109110688415571316480344899355894085582848000000001, 395940866122425193243875570782668457763038822400000000000000000001

Offset: 0

G. L. Honaker, Jr.

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..30

Cf. A019514, A020549, A036740, A038507.

List([0..11],n->Factorial(n)^n+1); # Muniru A Asiru, Mar 19 2018

seq(factorial(n)^n+1,n=0..11); # Muniru A Asiru, Mar 19 2018

Table[(n!)^n+1,{n,0,10}] (* Harvey P. Dale, Apr 10 2012 *)

a(n) = (n!)^n + 1; \\ Altug Alkan, Mar 19 2018

a(n) ~ (2*Pi)^(n/2) * n^(n^2 + n/2) / exp(n^2 - 1/12). - Vaclav Kotesovec, Mar 19 2018

One more term from Harvey P. Dale, Apr 10 2012

A299036 a(n) = [x^n] Product_{k=1..n} 1/(1-k!*x).

Original entry on oeis.org

1, 1, 7, 381, 502789, 33572762781, 175123095782787181, 99374457734129265819664221, 8158897372191288496224413025490409437, 124778468912108975502836576328262294089846582756189

Offset: 0

Seiichi Manyama, Feb 01 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..30

Cf. A000178, A036740.

Cf. A007820, A298851, A299035.

Table[SeriesCoefficient[Product[1/(1 - k!*x), {k, 1, n}], {x, 0, n}], {n, 0, 12}] (* Vaclav Kotesovec, Feb 02 2018 *)

From Vaclav Kotesovec, Feb 02 2018: (Start)
a(n) ~ (n!)^n.
a(n) ~ 2^(n/2) * Pi^(n/2) * n^(n*(2*n+1)/2) / exp(n^2 - 1/12). (End)

A100685 Powers of factorials A000142.

Original entry on oeis.org

1, 2, 4, 6, 8, 16, 24, 32, 36, 64, 120, 128, 216, 256, 512, 576, 720, 1024, 1296, 2048, 4096, 5040, 7776, 8192, 13824, 14400, 16384, 32768, 40320, 46656, 65536, 131072, 262144, 279936, 331776, 362880, 518400, 524288, 1048576, 1679616, 1728000

Offset: 1

Kyle Schalm and Jonathan Sondow, Dec 08 2004

Subsequence of A001013. Supersequence of A036740 without its first term.

Supersequence also of A046882 and A055209 without their first terms. - Jonathan Sondow and Robert G. Wilson v, Dec 19 2004

			24 = (4!)^1 and 36 = (3!)^2.

G. C. Greubel, Table of n, a(n) for n = 1..9660
Index entries for sequences related to factorial numbers.

Cf. A000142, A001013, A036740, A331373.
Cf. also A046882 and A055209.
Subsequences: A000079, A000400, A009968.

With[{ln = Log[10!]}, Table[With[{f = m!}, Table[f^j, {j, 0, Floor[ln/Log[f]]}]], {m, 2, 10}]] //Flatten //Union

Sum_{n>=1} 1/a(n) = 1 + A331373. - Amiram Eldar, Nov 21 2021

cp's OEIS Frontend

A323717 a(n) = Product_{k=0..n} (n! + k!).

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A336247 a(n) = (n!)^n * Sum_{k=0..n} 1 / (k!)^n.

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A060943 a(n) = n!^n * Sum_{k=1..n} 1/k^n.

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A134373 a(n) = ((2n)!)^3.

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A261490 The total element sum of the n-fold f transform applied to the length n sequence of 1's. And f returns a sorted result after multiplying the elements in its input sequence with 1, 2, 3,... in descending size order.

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Maple

A343899 a(n) = Sum_{k=0..n} (k!)^k * binomial(n,k).

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A343900 a(n) = Sum_{k=0..n} (k!)^(k+1) * binomial(n,k).

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A036739 a(n) = (n!)^n+1.