A323540 -id:A323540 - cp's OEIS frontend

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

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.

A323659 a(n) = Product_{k=0..n} (k^10 + (n-k)^10).

Original entry on oeis.org

0, 1, 2097152, 3663302861300625, 7851806399838625464320000, 378407193115560358680820465087890625, 45364420795826382440918950445637925790023680000, 34139918620704541898946466431984113559562219610081631390625

Offset: 0

Vaclav Kotesovec, Jan 23 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..59

Cf. A323540, A323541, A323542, A323543, A323544, A323545, A323546, A320345, A323660, A323661, A323662.

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

a(n) ~ exp((Pi*(1/2 + 2*sqrt(5) - 2*sqrt(1 + 2/sqrt(5))) - 10)*n) * n^(10*n+10).

A323660 a(n) = Product_{k=0..n} (k^11 + (n-k)^11).

Original entry on oeis.org

0, 1, 8388608, 131750272043485209, 2261269183430619234140422144, 1346827225363533058227598667144775390625, 1957831179567376680040414825610884198366949236801536, 23448342360429805388842947812883850305932149345203144459397169329

Offset: 0

Vaclav Kotesovec, Jan 23 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..55

Cf. A323540, A323541, A323542, A323543, A323544, A323545, A323546, A320345, A323659, A323661, A323662.

Table[Product[k^11+(n-k)^11, {k, 0, n}], {n, 0, 10}]

a(n) ~ exp((2*Pi*sqrt((1225 + 504*cos(2*Pi/11) + 1512*sin(Pi/22) - 1264*sin(3*Pi/22) + 240*sin(5*Pi/22))/11)/11 - 10)*n) * n^(11*n+11).

A323661 a(n) = Product_{k=0..n} (k^12 + (n-k)^12).

Original entry on oeis.org

0, 1, 33554432, 4740695283514005729, 651240623131512957219821846528, 4811704081770214536604871809482574462890625, 84537031377296019762303015000377965680906643309559021568, 16210797840416801857079558076889164370156937375891800497483902744790721

Offset: 0

Vaclav Kotesovec, Jan 23 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..51

Cf. A323540, A323541, A323542, A323543, A323544, A323545, A323546, A320345, A323659, A323660, A323662.

Table[Product[k^12+(n-k)^12, {k, 0, n}], {n, 0, 10}]

a(n) ~ exp((Pi*(-5/2 + 2*sqrt(6) + sqrt(2*(5-2*sqrt(6))/3)) - 12)*n) * n^(12*n+12).

A323662 a(n) = Product_{k=0..n} (k^13 + (n-k)^13).

Original entry on oeis.org

0, 1, 134217728, 170623376651175378921, 187556828900191806607614608932864, 17233921359224498311699145473539829254150390625, 3651108402083969086976039852657366429953837378356052425179136

Offset: 0

Vaclav Kotesovec, Jan 23 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..48

Cf. A323540, A323541, A323542, A323543, A323544, A323545, A323546, A320345, A323659, A323660, A323661.

Table[Product[k^13+(n-k)^13, {k, 0, n}], {n, 0, 10}]

a(n) ~ exp((2*Pi*sqrt((2699 - 1920*cos(2*Pi/13) + 4184*cos(3*Pi/13) - 4512*sin(Pi/26) + 4752*sin(3*Pi/26) - 2944*sin(5*Pi/26))/13) / 13 - 12)*n) * n^(13*n+13).

A323751 a(n) = Product_{k=0..n} (k^n + (n-k)^n).

Original entry on oeis.org

2, 1, 32, 59049, 14101250048, 775913238525390625, 13410804447068120796679372800, 112244673425189306235795780017831813874289, 545831702006800417886454373052629612732034857946832699392

Offset: 0

Seiichi Manyama, Jan 26 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..27

Cf. A323540, A323541, A323542, A323543, A323544, A323545, A323546.

Cf. A323575, A323588.

[(&*[k^n +(n-k)^n: k in [0..n]]): n in [0..10]]; // G. C. Greubel, Feb 08 2019

Table[Product[k^n+(n-k)^n, {k,0,n}], {n,0,10}] (* G. C. Greubel, Feb 08 2019 *)

PARI
```
{a(n) = prod(k=0, n, k^n+(n-k)^n)}
```

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

a(n) = n^n * A323588(n). - Vaclav Kotesovec, Feb 08 2019

A375051 Obverse convolution (n^2 - 1)**(n^2 - 1); see Comments.

Original entry on oeis.org

-2, 1, 0, 441, 75264, 14402025, 3451797504, 1043554187025, 392874877255680, 181193143212358641, 100757479882752000000, 66592039534109652160521, 51648427918242896412672000, 46486269540273907302519872025, 48078115878910207012782666153984

Offset: 0

Clark Kimberling, Sep 15 2024

sign

See A374848 for the definition of obverse convolution and a guide to related sequences.

a(2k+1) is a square for k>=0.

Cf. A005563, A323540, A374848, A375052.

s[n_] := n^2 - 1; t[n_] := n^2 - 1;
u[n_] := Product[s[k] + t[n - k], {k, 0, n}];
Table[u[n], {n, 0, 20}]

a(n) ~ n^(2*n+2) / exp(2*n - Pi*n/2). - Vaclav Kotesovec, Sep 19 2024

A376321 Obverse convolution (n^2 + 1)**(n^2 + 1); see Comments.

Original entry on oeis.org

2, 9, 144, 5929, 466560, 59213025, 10958689280, 2771535732849, 915539439919104, 382088350057032025, 196357891384811520000, 121752085389995771825625, 89582478947424173216497664, 77138638421388109999960896369, 76829768389915556918132736000000

Offset: 0

Clark Kimberling, Sep 20 2024

nonn

See A374848 for the definition of obverse convolution and a guide to related sequences.

a(2k+1) is a square for k>=0.

Cf. A323540, A374848, A375051.

s[n_] := n^2 + 1; t[n_] := n^2 + 1;
u[n_] := Product[s[k] + t[n - k], {k, 0, n}];
Table[u[n], {n, 0, 20}]

a(n) ~ n^(2*n+2) / exp(2*n - Pi*n/2). - Vaclav Kotesovec, Sep 20 2024

cp's OEIS Frontend

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

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A323659 a(n) = Product_{k=0..n} (k^10 + (n-k)^10).

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A323660 a(n) = Product_{k=0..n} (k^11 + (n-k)^11).

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A323661 a(n) = Product_{k=0..n} (k^12 + (n-k)^12).

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A323662 a(n) = Product_{k=0..n} (k^13 + (n-k)^13).

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A323751 a(n) = Product_{k=0..n} (k^n + (n-k)^n).

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A375051 Obverse convolution (n^2 - 1)**(n^2 - 1); see Comments.

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A376321 Obverse convolution (n^2 + 1)**(n^2 + 1); see Comments.

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