A323589 -id:A323589 - cp's OEIS frontend

A323575 a(n) = Product_{k=1..n} (k^k + (n-k)^k).

1, 1, 8, 405, 229376, 1827109375, 257063481999360, 755170478103207873707, 54143353027014803410072371200, 107483342384971486221625795626923693445, 6647872853044955947850033397760000000000000000000, 14166017880429890423491783342799863539312599105433301729629445

Offset: 0

Vaclav Kotesovec, Jan 18 2019

nonn

G. C. Greubel, Table of n, a(n) for n = 0..36

Cf. A323540-A323546, A323588, A323589, A303641.

[1] cat [(&*[k^k +(n-k)^k: k in [1..n]]): n in [1..12]]; // G. C. Greubel, Feb 08 2019

Table[Product[k^k+(n-k)^k, {k, 1, n}], {n, 0, 12}]

vector(12, n, n--; prod(k=1,n, k^k+(n-k)^k)) \\ G. C. Greubel, Feb 08 2019

[product(k^k +(n-k)^k for k in (1..n)) for n in (0..12)] # G. C. Greubel, Feb 08 2019

a(n) ~ c * n^(n*(n+1)/2) * 2^(n^2/2) / exp(n^2/2), where c = A303641 = 2.473655256632129487637893694272428036362097123254579382787777122619864038942...

A323588 a(n) = Product_{k=1..n} (k^n + (n-k)^n).

Original entry on oeis.org

1, 1, 8, 2187, 55083008, 248292236328125, 287440081598682287308800, 136294854579772162759923622710449623, 32534104705262209051040075603284216686012438413312, 5686543339012978225006873713961872387810223003912610672810622880089

Offset: 0

Vaclav Kotesovec, Jan 18 2019

nonn

G. C. Greubel, Table of n, a(n) for n = 0..27

Cf. A323540-A323546, A323575, A323589, A323751.

[1] cat [(&*[k^n +(n-k)^n: k in [1..n]]): n in [1..10]]; // G. C. Greubel, Feb 08 2019

Table[Product[k^n+(n-k)^n, {k, 1, n}], {n, 0, 10}]

vector(10, n, n--; prod(k=1,n, k^n+(n-k)^n)) \\ G. C. Greubel, Feb 08 2019

[product(k^n +(n-k)^n for k in (1..n)) for n in (0..10)] # G. C. Greubel, Feb 08 2019

a(n) ~ c * 2^(n^2) * n^(n^2) / exp(n^2), where
c = 1.7567468186007109703792640049745420817202851050652253469714... if n is even,
c = 1.8080216158688347442204158454365469233524049331246880759722... if n is odd.

A366305 a(n) = Product_{k=1..n} (k^n + (k-1)^n).

Original entry on oeis.org

1, 5, 315, 555713, 47705305725, 305469864195354625, 207095306530955763265880535, 20017329298655447986400838721630926977, 357361761140807273279996172600335233468472149678425, 1481824279740988988264353294673429995981921700740921435758587890625

Offset: 1

Vaclav Kotesovec, Oct 06 2023

nonn

Cf. A036740, A323575, A323588, A323589, A366306.

Table[Product[k^n + (k-1)^n, {k, 1, n}], {n, 1, 10}]

a(n) = (n!)^n * Product_{k=1..n} (1 + (1 - 1/k)^n).
a(n) ~ n!^n * d^n, where d = exp(Integral_{x=0..1} log(1 + exp(-1/x)) dx) = 1.14183186235785012136459060138978468902610644657603999829892450823456733...
a(n) ~ (2*Pi)^(n/2) * d^n * n^(n*(2*n+1)/2) / exp(n^2 - 1/12).

A366306 a(n) = Product_{k=1..n} (k^n - (k-1)^n).

Original entry on oeis.org

1, 3, 133, 170625, 10733002621, 50465283999665535, 25145494699347449245677097, 1787473773567267792523164108726890625, 23480751910878672340765325385856840967957995534681, 71672834655019406921956925590632596034005848922160549420728589375

Offset: 1

Vaclav Kotesovec, Oct 06 2023

nonn

Cf. A036740, A323575, A323588, A323589, A366305.

Table[Product[k^n - (k-1)^n, {k, 1, n}], {n, 1, 10}]

a(n) = (n!)^n * Product_{k=1..n} (1 - (1 - 1/k)^n).
a(n) ~ n!^n * d^n, where d = exp(Integral_{x=0..1} log(1 - exp(-1/x)) dx) = 0.84207793096051704199642805288991601369639823969574423397520945175552718...
a(n) ~ (2*Pi)^(n/2) * d^n * n^(n*(2*n+1)/2) / exp(n^2 - 1/12).

cp's OEIS Frontend

A323575 a(n) = Product_{k=1..n} (k^k + (n-k)^k).

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A323588 a(n) = Product_{k=1..n} (k^n + (n-k)^n).

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Author

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Magma

Mathematica

PARI

Sage

Formula

A366305 a(n) = Product_{k=1..n} (k^n + (k-1)^n).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A366306 a(n) = Product_{k=1..n} (k^n - (k-1)^n).

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Author

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Programs

Mathematica

Formula