cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 14 results. Next

A323540 a(n) = Product_{k=0..n} (k^2 + (n-k)^2).

Original entry on oeis.org

0, 1, 32, 2025, 204800, 30525625, 6307891200, 1727713080625, 606076928000000, 265058191985900625, 141409376995328000000, 90403125002859606705625, 68229510086445571768320000, 60026603304487418050791015625, 60893916244529680380723200000000
Offset: 0

Views

Author

Vaclav Kotesovec, Jan 17 2019

Keywords

Links

Crossrefs

Cf. A272244, A323425, A323541, A323542, A323543, A323544, A323545, A323546.
Cf. 2*A000330 and A006331 (with sum instead of product).

Programs

  • Magma
    m:=2; [(&*[k^m + (n-k)^m: k in [0..n]]): n in [0..20]]; // G. C. Greubel, Jan 18 2019
  • Mathematica
    Table[Product[k^2+(n-k)^2, {k, 0, n}], {n, 0, 20}]
  • PARI
    m=2; vector(20, n, n--; prod(k=0,n, k^m + (n-k)^m)) \\ G. C. Greubel, Jan 18 2019
  • Sage
    m=2; [product(k^m +(n-k)^m for k in (0..n)) for n in (0..20)] # G. C. Greubel, Jan 18 2019

Formula

a(n) ~ exp((Pi-4)*n/2) * n^(2*n+2).

A323541 a(n) = Product_{k=0..n} (k^3 + (n-k)^3).

Original entry on oeis.org

0, 1, 128, 59049, 51380224, 80869140625, 207351578198016, 811509810302822449, 4603095542875667038208, 36344623587588604291790241, 386644580358400000000000000000, 5395532942025804980378907333844441, 96578621213529440721046520779140759552
Offset: 0

Views

Author

Vaclav Kotesovec, Jan 17 2019

Keywords

Links

Crossrefs

Cf. A272246, A323496, A323540, A323542, A323543, A323544, A323545, A323546.
Cf. 2*A000537 and A163102 (with sum instead of product).

Programs

  • Magma
    m:=3; [(&*[k^m + (n-k)^m: k in [0..n]]): n in [0..15]]; // G. C. Greubel, Jan 18 2019
  • Mathematica
    Table[Product[k^3+(n-k)^3, {k, 0, n}], {n, 0, 15}]
  • PARI
    m=3; vector(15, n, n--; prod(k=0,n, k^m + (n-k)^m)) \\ G. C. Greubel, Jan 18 2019
  • Sage
    m=3; [product(k^m +(n-k)^m for k in (0..n)) for n in (0..15)] # G. C. Greubel, Jan 18 2019

Formula

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

A323542 a(n) = Product_{k=0..n} (k^4 + (n-k)^4).

Original entry on oeis.org

0, 1, 512, 1896129, 14101250048, 242755875390625, 7888809923487203328, 452522453429009743939201, 42521926771106843499966758912, 6212193882217859346149080691430849, 1350441156698962215630405632000000000000, 421551664651621436548685508587919503984205889
Offset: 0

Views

Author

Vaclav Kotesovec, Jan 17 2019

Keywords

Links

Crossrefs

Cf. A272247, A323497, A323540, A323541, A323543, A323544, A323545, A323546.
Cf. 2*A000538 and A259108 (with sum instead of product).

Programs

  • Magma
    [(&*[(k^4 + (n-k)^4): k in [0..n]]): n in [0..15]]; // Vincenzo Librandi, Jan 18 2019
  • Mathematica
    Table[Product[k^4+(n-k)^4, {k, 0, n}], {n, 0, 15}]
  • PARI
    m=4; vector(15, n, n--; prod(k=0,n, k^m + (n-k)^m)) \\ G. C. Greubel, Jan 18 2019
  • Sage
    m=4; [product(k^m +(n-k)^m for k in (0..n)) for n in (0..15)] # G. C. Greubel, Jan 18 2019

Formula

a(n) ~ exp((Pi*(sqrt(2) - 1/2) - 4)*n) * n^(4*n + 4).

A323543 a(n) = Product_{k=0..n} (k^5 + (n-k)^5).

Original entry on oeis.org

0, 1, 2048, 64304361, 3995393327104, 775913238525390625, 320224500476333990608896, 273342392644434762426370643281, 429621172463958849019228299940855808, 1175198860360296464427314161342724729270241, 5278148679274118560000000000000000000000000000000
Offset: 0

Views

Author

Vaclav Kotesovec, Jan 17 2019

Keywords

Links

Crossrefs

Cf. A272248, A323533, A323540, A323541, A323542, A323544, A323545, A323546.
Cf. 2*A000539 (with sum instead of product).

Programs

  • Magma
    [(&*[(k^5 + (n-k)^5): k in [0..n]]): n in [0..12]]; // Vincenzo Librandi, Jan 18 2019
  • Mathematica
    Table[Product[k^5+(n-k)^5, {k, 0, n}], {n, 0, 12}]
  • PARI
    m=5; vector(12, n, n--; prod(k=0,n, k^m +(n-k)^m)) \\ G. C. Greubel, Jan 18 2019
  • Sage
    m=5; [product(k^m +(n-k)^m for k in (0..n)) for n in (0..12)] # G. C. Greubel, Jan 18 2019

Formula

a(n) ~ exp((2*Pi*sqrt(5 - 2/sqrt(5))/5 - 4)*n) * n^(5*n+5).

A323544 a(n) = Product_{k=0..n} (k^6 + (n-k)^6).

Original entry on oeis.org

0, 1, 8192, 2245338225, 1144394036019200, 2577023355527587890625, 13410804447068120796679372800, 172661401915668867785003701060950625, 4548909593429214367033270472265433088000000, 234845240509381890690238640158397433600579682850625
Offset: 0

Views

Author

Vaclav Kotesovec, Jan 17 2019

Keywords

Links

Crossrefs

Cf. A323534, A323540, A323541, A323542, A323543, A323545, A323546.
Cf. 2*A000540 and A259109 (with sum instead of product).

Programs

  • Magma
    [(&*[(k^6 + (n-k)^6): k in [0..n]]): n in [0..10]]; // Vincenzo Librandi, Jan 18 2019
  • Mathematica
    Table[Product[k^6+(n-k)^6, {k, 0, n}], {n, 0, 10}]
  • PARI
    m=6; vector(10, n, n--; prod(k=0,n, k^m + (n-k)^m)) \\ G. C. Greubel, Jan 18 2019
  • Sage
    m=6; [product(k^m +(n-k)^m for k in (0..n)) for n in (0..10)] # G. C. Greubel, Jan 18 2019

Formula

a(n) ~ exp(((15 - 4*sqrt(3))*Pi/6 - 6)*n) * n^(6*n+6).

A323545 a(n) = Product_{k=0..n} (k^7 + (n-k)^7).

Original entry on oeis.org

0, 1, 32768, 79593387129, 328983774635229184, 8781626117710113525390625, 570409595340477623191338982834176, 112244673425189306235795780017831813874289, 49449149324106963036650868175987491957290049732608, 48527312221741371319651099141827554314119977393170380398241
Offset: 0

Views

Author

Vaclav Kotesovec, Jan 17 2019

Keywords

Links

Crossrefs

Cf. A323535, A323540, A323541, A323542, A323543, A323544, A323546, A320345.
Cf. 2*A000541 (with sum instead of product).

Programs

  • Magma
    m:=7; [(&*[k^m + (n-k)^m: k in [0..n]]): n in [0..10]]; // G. C. Greubel, Jan 18 2019
  • Mathematica
    Table[Product[k^7+(n-k)^7, {k, 0, n}], {n, 0, 10}]
  • PARI
    m=7; vector(10, n, n--; prod(k=0,n, k^m + (n-k)^m)) \\ G. C. Greubel, Jan 18 2019
  • Sage
    m=7; [product(k^m +(n-k)^m for k in (0..n)) for n in (0..10)] # G. C. Greubel, Jan 18 2019

Formula

a(n) ~ exp((8*cos((Pi + arctan(2769*sqrt(3)/239))/6)*Pi/sqrt(21)-6)*n) * n^(7*n+7).
Equivalently, a(n) ~ exp((4*Pi*sqrt(2*(13 + 19*sin(Pi/14) - sin(3*Pi/14))/7)/7 - 6)*n) * n^(7*n+7). - Vaclav Kotesovec, Jan 23 2019

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

Original entry on oeis.org

1, 1, 8, 405, 229376, 1827109375, 257063481999360, 755170478103207873707, 54143353027014803410072371200, 107483342384971486221625795626923693445, 6647872853044955947850033397760000000000000000000, 14166017880429890423491783342799863539312599105433301729629445
Offset: 0

Views

Author

Vaclav Kotesovec, Jan 18 2019

Keywords

Links

Crossrefs

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

Programs

  • Magma
    [1] cat [(&*[k^k +(n-k)^k: k in [1..n]]): n in [1..12]]; // G. C. Greubel, Feb 08 2019
  • Mathematica
    Table[Product[k^k+(n-k)^k, {k, 1, n}], {n, 0, 12}]
  • PARI
    vector(12, n, n--; prod(k=1,n, k^k+(n-k)^k)) \\ G. C. Greubel, Feb 08 2019
  • Sage
    [product(k^k +(n-k)^k for k in (1..n)) for n in (0..12)] # G. C. Greubel, Feb 08 2019

Formula

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

A320345 a(n) = Product_{k=0..n} (k^9 + (n-k)^9).

Original entry on oeis.org

0, 1, 524288, 101957062669641, 27265063485978531856384, 106913106903205778598785400390625, 1052153174824916171740710027027385486934016, 50134911460309737788263856880351450247312603708106641, 6079819706045473432785376698707222441479676085024627875693723648
Offset: 0

Views

Author

Vaclav Kotesovec, Jan 22 2019

Keywords

Links

Crossrefs

Cf. A323538, A323540, A323541, A323542, A323543, A323544, A323545, A323546.

Programs

  • Mathematica
    Table[Product[k^9+(n-k)^9, {k, 0, n}], {n, 0, 10}]

Formula

a(n) ~ exp((4*Pi*(sqrt(3) + 6*sin(Pi/9))/9 - 8)*n) * n^(9*n+9).

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

Views

Author

Vaclav Kotesovec, Jan 18 2019

Keywords

Links

Crossrefs

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

Programs

  • Magma
    [1] cat [(&*[k^n +(n-k)^n: k in [1..n]]): n in [1..10]]; // G. C. Greubel, Feb 08 2019
  • Mathematica
    Table[Product[k^n+(n-k)^n, {k, 1, n}], {n, 0, 10}]
  • PARI
    vector(10, n, n--; prod(k=1,n, k^n+(n-k)^n)) \\ G. C. Greubel, Feb 08 2019
  • Sage
    [product(k^n +(n-k)^n for k in (1..n)) for n in (0..10)] # G. C. Greubel, Feb 08 2019

Formula

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

Views

Author

Vaclav Kotesovec, Jan 23 2019

Keywords

Links

Crossrefs

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

Programs

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

Formula

a(n) ~ exp((Pi*(1/2 + 2*sqrt(5) - 2*sqrt(1 + 2/sqrt(5))) - 10)*n) * n^(10*n+10).
Showing 1-10 of 14 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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