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 13 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).

A323546 a(n) = Product_{k=0..n} (k^8 + (n-k)^8).

Original entry on oeis.org

0, 1, 131072, 2843192875329, 94689598336441253888, 30456484467910986480712890625, 24450543078081443740165325452474318848, 74458223912158060479139869222818674648092178561, 545831702006800417886454373052629612732034857946832699392
Offset: 0

Views

Author

Vaclav Kotesovec, Jan 17 2019

Keywords

Links

Crossrefs

Cf. A323538, A323540, A323541, A323542, A323543, A323544, A323545, A320345.
Cf. 2*A000542 (with sum instead of product).

Programs

  • Mathematica
    Table[Product[k^8+(n-k)^8, {k, 0, n}], {n, 0, 10}]
  • PARI
    m=8; vector(10, n, n--; prod(k=0,n, k^m + (n-k)^m)) \\ G. C. Greubel, Jan 18 2019
  • Sage
    m=8; [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(((2*sqrt(2+sqrt(2)) - 1/2 - sqrt(2))*Pi - 8)*n) * n^(8*n+8).

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

A272248 a(n) = Product_{k=0..n} (n^5 + k^5).

Original entry on oeis.org

0, 2, 67584, 7924375800, 2876035930521600, 2693451205324687500000, 5648896640332217707816550400, 23819277009290664033936067933468800, 185754160490281505676324140907107450880000, 2507604631016507710974687639612411760216253760000
Offset: 0

Views

Author

Vaclav Kotesovec, Apr 23 2016

Keywords

Comments

In general, for p>=1, Product_{k=0..n} (n^p + k^p) ~ sqrt(2) * n^(p*(n+1)) * exp(n*Sum_{j>=1} (-1)^(j+1) / (j*(1 + j*p))).

Crossrefs

Cf. A126804, A272244, A272246, A272247, A367823, A367833.
Cf. A255435, A323543, A324438.

Programs

  • Mathematica
    Table[Product[n^5+k^5,{k,0,n}],{n,0,10}]

Formula

a(n) ~ 2^(2*n+1/2) * phi^(sqrt(5)*n) * n^(5*n+5) / exp((5-sqrt(phi)*Pi/5^(1/4))*n), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio.

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).

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).

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

Views

Author

Vaclav Kotesovec, Jan 23 2019

Keywords

Links

Crossrefs

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

Programs

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

Formula

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).
Showing 1-10 of 13 results. Next

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

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