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 1-6 of 6 results.

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

Crossrefs

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).

A323496 a(n) = Product_{k=1..n} (binomial(k-1,3) + binomial(n-k,3)).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1600, 1280000, 1225000000, 1487467520000, 2342962705305600, 4757234928058368000, 12302981968140864000000, 39976163552160000000000000, 161025498138224463853824000000, 794000312545927932130993635328000, 4737527580526919896601692054005760000
Offset: 0

Views

Author

Vaclav Kotesovec, Jan 16 2019

Keywords

Crossrefs

Programs

  • Mathematica
    Table[Product[Binomial[k-1,3] + Binomial[n-k,3], {k, 1, n}], {n, 0, 20}]
  • PARI
    a(n) = prod(k=1, n, binomial(k-1,3) + binomial(n-k,3)); \\ Michel Marcus, Jan 17 2019

Formula

a(n) ~ exp(2*Pi*(n-3)/3^(3/2) - 2*n - 3) * n^(3*n) / 6^n.

A323497 a(n) = Product_{k=1..n} (binomial(k-1,4) + binomial(n-k,4)).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 6890625, 67528125000, 771895089000000, 10758502218240000000, 193672800442598400000000, 4520389860871215408000000000, 136445409183108034775390625000000, 5281556250358583667176941845984375000, 259600586924352252185403119405592275390625
Offset: 0

Views

Author

Vaclav Kotesovec, Jan 16 2019

Keywords

Crossrefs

Programs

  • Mathematica
    Table[Product[Binomial[k-1,4] + Binomial[n-k,4], {k, 1, n}], {n, 0, 20}]
  • PARI
    a(n) = prod(k=1, n, binomial(k-1,4) + binomial(n-k,4)); \\ Michel Marcus, Jan 17 2019

Formula

a(n) ~ exp(Pi*(sqrt(2) - 1/2)*(n-4) - 4*n) * n^(4*n) / 24^n.

A323533 a(n) = Product_{k=1..n} (binomial(k-1,5) + binomial(n-k,5)).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 790420571136, 100389735898841088, 14582663231533605863424, 2458550581659926554038239232, 529554691027323329170207744475136, 146980847512952623091566575072055001088, 53003014923687519392206631372837133989462016
Offset: 0

Views

Author

Vaclav Kotesovec, Jan 17 2019

Keywords

Crossrefs

Programs

  • Mathematica
    Table[Product[Binomial[k-1,5] + Binomial[n-k,5], {k, 1, n}], {n, 0, 20}]
  • PARI
    a(n) = prod(k=1, n, binomial(k-1,5) + binomial(n-k,5)); \\ Michel Marcus, Jan 17 2019

Formula

a(n) ~ exp((1 + 2*Pi*sqrt(5 - 2/sqrt(5))/5) * (n-5)) * n^(5*n) / (exp(5*n)*120^n).

A323535 a(n) = Product_{k=1..n} (binomial(k-1,7) + binomial(n-k,7)).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 240248274716697412239360000, 5659588189073370681080838881280000, 148305406398618918682372310424354816000000, 4049882681498254991937037064898924144230400000000, 137651993399006086593846978063252515678682995490816000000
Offset: 0

Views

Author

Vaclav Kotesovec, Jan 17 2019

Keywords

Crossrefs

Programs

  • Mathematica
    Table[Product[Binomial[k-1,7] + Binomial[n-k,7], {k, 1, n}], {n, 0, 20}]
  • PARI
    a(n) = prod(k=1, n, binomial(k-1, 7) + binomial(n-k, 7)); \\ Daniel Suteu, Jan 17 2019

Formula

a(n) ~ exp(-7*n + (n-7)*(1 + c*Pi)) * n^(7*n) / (7!)^n, where c = 8*cos((Pi + arctan(2769*sqrt(3)/239))/6) / sqrt(21) = 1.2446281707164555154936427017... is the root of the equation 823543*c^6 - 3764768*c^4 + 4302592*c^2 - 692224 = 0.

A323538 a(n) = Product_{k=1..n} (binomial(k-1,8) + binomial(n-k,8)).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 676788847291127500630565286687890625, 224202413239751513418389758669186941328125000, 81789054189516490351294844356948943677175390625000000, 29455964980491136378751203264203423123185624125549245000000000
Offset: 0

Views

Author

Vaclav Kotesovec, Jan 17 2019

Keywords

Crossrefs

Programs

  • Mathematica
    Table[Product[Binomial[k-1,8] + Binomial[n-k,8], {k, 1, n}], {n, 0, 20}]
  • PARI
    a(n) = prod(k=1, n, binomial(k-1, 8)+binomial(n-k, 8)) \\ Felix Fröhlich, Jan 17 2019

Formula

a(n) ~ exp(-8*n + (2*sqrt(2+sqrt(2)) - 1/2 - sqrt(2))*Pi*(n-8)) * n^(8*n) / (8!)^n.
Showing 1-6 of 6 results.