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 1-7 of 7 results.

A084939 Pentagorials: n-th polygorial for k=5.

Original entry on oeis.org

1, 1, 5, 60, 1320, 46200, 2356200, 164934000, 15173928000, 1775349576000, 257425688520000, 45306921179520000, 9514453447699200000, 2350070001581702400000, 674470090453948588800000, 222575129849803034304000000
Offset: 0

Views

Author

Daniel Dockery (peritus(AT)gmail.com), Jun 13 2003

Keywords

Links

Crossrefs

Cf. A000326, A006472, A001044, A000680, A084940, A084941, A084942, A084943, A084944, A085356.

Programs

  • Maple
    a := n->(n!/2^n)*mul(3*i+2,i=0..n-1); [seq(a(j),j=0..30)];
  • Mathematica
    Table[k! Pochhammer[2/3, k] (3/2)^k, {k, 0, 20}] (* Jan Mangaldan, Mar 20 2013 *)
polygorial[k_, n_] := FullSimplify[ n!/2^n (k -2)^n*Pochhammer[2/(k -2), n]]; Array[polygorial[5, #] &, 17, 0] (* Robert G. Wilson v, Dec 17 2016 *)
  • PARI
    a(n)=n!/2^n*prod(i=1,n,3*i-1) \\ Charles R Greathouse IV, Dec 13 2016

Formula

a(n) = polygorial(n, 5) = (A000142(n)/A000079(n))*A008544(n) = (n!/2^n)*Product_{i=0..n-1} (3*i+2) = (n!/2^n)*3^n*Pochhammer(2/3, n) = (n!/2^n)*3^n*GAMMA(n+2/3)/GAMMA(2/3).
a(n) ~ Gamma(1/3) * 3^(n+1/2) * n^(2*n+2/3) / (2^n * exp(2*n)). - Vaclav Kotesovec, Jul 17 2015
D-finite with recurrence a(n+1) = ((n+1)*(3*n+2)/2)*a(n) = A000326(n+1)*a(n). - Muniru A Asiru, Apr 05 2016
E.g.f.: hypergeom([2/3, 1], [], (3/2)*x). - Robert Israel, Apr 05 2016

A084940 Heptagorials: n-th polygorial for k=7.

Original entry on oeis.org

1, 1, 7, 126, 4284, 235620, 19085220, 2137544640, 316356606720, 59791398670080, 14050978687468800, 4018579904616076800, 1374354327378698265600, 553864793933615401036800, 259762588354865623086259200, 140271797711627436466579968000, 86407427390362500863413260288000
Offset: 0

Views

Author

Daniel Dockery (peritus(AT)gmail.com), Jun 13 2003

Keywords

Links

Crossrefs

Cf. A006472, A001044, A000680, A084939, A084941, A084942, A084943, A084944, A085356, A246745.

Programs

  • Maple
    a := n->n!/2^n*mul(5*i+2,i=0..n-1); [seq(a(j),j=0..30)];
  • Mathematica
    polygorial[k_, n_] := FullSimplify[ n!/2^n (k -2)^n*Pochhammer[2/(k -2), n]]; Array[ polygorial[7, #] &, 16, 0] (* Robert G. Wilson v, Dec 26 2016 *)
Join[{1},FoldList[Times,PolygonalNumber[7,Range[20]]]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jul 29 2019 *)
  • PARI
    a(n)=n!/2^n*prod(i=1,n,5*i-3) \\ Charles R Greathouse IV, Dec 13 2016

Formula

a(n) = polygorial(n, 7) = (A000142(n)/A000079(n))*A047055(n) = (n!/2^n)*Product_{i=0..n-1}(5*i+2) = (n!/2^n)*5^n*Pochhammer(2/5, n) = (n!/2^n)*5^n*Gamma(n+2/5)*sin(2*Pi/5)*Gamma(3/5)/Pi.
D-finite with recurrence 2*a(n) = n*(5*n-3)*a(n-1). - R. J. Mathar, Mar 12 2019
a(n) ~ 5^n * n^(2*n + 2/5) * Pi /(Gamma(2/5) * 2^(n-1) * exp(2*n)). - Amiram Eldar, Aug 28 2025

A084944 Hendecagorials: n-th polygorial for k=11.

Original entry on oeis.org

1, 1, 11, 330, 19140, 1818300, 256380300, 50250538800, 13065140088000, 4350691649304000, 1805537034461160000, 913601739437346960000, 553642654099032257760000, 395854497680808064298400000, 329746796568113117560567200000, 316556924705388592858144512000000, 346946389477105897772526385152000000
Offset: 0

Views

Author

Daniel Dockery (peritus(AT)gmail.com), Jun 13 2003

Keywords

Links

Crossrefs

Cf. A006472, A001044, A000680, A084939, A084940, A084941, A084942, A084943, A085356.

Programs

  • Maple
    a := n->n!/2^n*product(9*i+2,i=0..n-1); [seq(a(j),j=0..30)];
  • Mathematica
    polygorial[k_, n_] := FullSimplify[ n!/2^n (k -2)^n*Pochhammer[2/(k - 2), n]]; Array[polygorial[11, #] &, 16, 0] (* Robert G. Wilson v, Dec 13 2016 *)

Formula

a(n) = polygorial(n, 11) = (A000142(n)/A000079(n))*A084949(n) = (n!/2^n)*Product_{i=0..n-1} (9*i+2) = (n!/2^n)*9^n*Pochhammer(2/9, n) = (n!/2^n)*9^n*Gamma(n+2/9)/Gamma(2/9).
D-finite with recurrence 2*a(n) = n*(9*n-7)*a(n-1). - R. J. Mathar, Mar 12 2019
a(n) ~ 9^n * n^(2*n + 2/9) * Pi /(Gamma(2/9) * 2^(n-1) * exp(2*n)). - Amiram Eldar, Aug 28 2025

A084941 Octagorials: n-th polygorial for k=8.

Original entry on oeis.org

1, 1, 8, 168, 6720, 436800, 41932800, 5577062400, 981562982400, 220851671040000, 61838467891200000, 21086917550899200000, 8603462360766873600000, 4138265395528866201600000, 2317428621496165072896000000, 1494741460865026472017920000000, 1100129715196659483405189120000000
Offset: 0

Views

Author

Daniel Dockery (peritus(AT)gmail.com), Jun 13 2003

Keywords

Links

Crossrefs

Cf. A006472, A001044, A000680, A073005, A084939, A084940, A084942, A084943, A084944, A085356.

Programs

  • Maple
    a := n->n!/2^n*product(6*i+2,i=0..n-1); [seq(a(j),j=0..30)];
  • Mathematica
    polygorial[k_, n_] := FullSimplify[ n!/2^n (k -2)^n*Pochhammer[2/(k -2), n]]; Array[polygorial[8, #] &, 16, 0] (* Robert G. Wilson v, Dec 26 2016 *)
  • PARI
    a(n) = n! / 2^n * prod(i=0, n-1, 6*i+2) \\ Felix Fröhlich, Dec 13 2016

Formula

a(n) = polygorial(n, 8) = (A000142(n)/A000079(n))*A047657(n) = (n!/2^n)*Product_{i=0..n-1} (6*i+2) = (n!/2^n)*6^n*Pochhammer(1/3, n) = (n!/2)*3^n*sqrt(3)*Gamma(n+1/3)*Gamma(2/3)/Pi.
D-finite with recurrence a(n) = n*(3*n-2)*a(n-1). - R. J. Mathar, Mar 12 2019
a(n) ~ 2 * 3^n * n^(2*n + 1/3) * Pi /(Gamma(1/3) * exp(2*n)). - Amiram Eldar, Aug 28 2025

A084942 Enneagorials: n-th polygorial for k=9.

Original entry on oeis.org

1, 1, 9, 216, 9936, 745200, 82717200, 12738448800, 2598643555200, 678245967907200, 220429939569840000, 87290256069656640000, 41375581377017247360000, 23128949989752641274240000, 15056946443328969469530240000, 11292709832496727102147680000000, 9666559616617198399438414080000000
Offset: 0

Views

Author

Daniel Dockery (peritus(AT)gmail.com), Jun 13 2003

Keywords

Links

Crossrefs

Cf. A006472, A001044, A000680, A084939, A084940, A084941, A084943, A084944, A085356, A220605.

Programs

  • Maple
    a := n->n!/2^n*product(7*i+2,i=0..n-1); [seq(a(j),j=0..30)];
  • Mathematica
    polygorial[k_, n_] := FullSimplify[ n!/2^n (k -2)^n*Pochhammer[2/(k -2), n]]; Array[polygorial[9, #] &, 16, 0] (* Robert G. Wilson v, Dec 26 2016 *)
  • PARI
    a(n)=n!/2^n*prod(i=1,n,7*i-5) \\ Charles R Greathouse IV, Dec 13 2016

Formula

a(n) = polygorial(n, 9) = (A000142(n)/A000079(n))*A084947(n) = (n!/2^n)*Product_{i=0..n-1} (7*i+2) = (n!/2^n)*7^n*Pochhammer(2/7, n) = (n!/2^n)*7^n*Gamma(n+2/7)/Gamma(2/7).
D-finite with recurrence 2*a(n) = n*(7*n-5)*a(n-1). - R. J. Mathar, Mar 12 2019
a(n) ~ 7^n * n^(2*n + 2/7) * Pi /(Gamma(2/7) * 2^(n-1) * exp(2*n)). - Amiram Eldar, Aug 28 2025

A084943 Decagorials: n-th polygorial for k=10.

Original entry on oeis.org

1, 1, 10, 270, 14040, 1193400, 150368400, 26314470000, 6104957040000, 1813172240880000, 670873729125600000, 302564051835645600000, 163384587991248624000000, 104075982550425373488000000, 77224379052415627128096000000, 66026844089815361194522080000000, 64442199831659792525853550080000000
Offset: 0

Views

Author

Daniel Dockery (peritus(AT)gmail.com), Jun 13 2003

Keywords

Links

Crossrefs

Cf. A006472, A001044, A000680, A068466, A084939, A084940, A084941, A084942, A084944, A085356.

Programs

  • Maple
    a := n->n!/2^n*product(8*i+2,i=0..n-1); [seq(a(j),j=0..30)];
  • Mathematica
    polygorial[k_, n_] := FullSimplify[ n!/2^n (k -2)^n*Pochhammer[2/(k -2), n]]; Array[polygorial[10, #] &, 14, 0] (* Robert G. Wilson v, Dec 26 2016 *)
  • PARI
    a(n)=n!/2^n*prod(i=1,n,8*i-6) \\ Charles R Greathouse IV, Dec 13 2016

Formula

a(n) = polygorial(n, 10) = (A000142(n)/A000079(n))*A084948(n) = (n!/2^n)*Product_{i=0..n-1} (8*i+2) = (n!/2^n)*8^n*Pochhammer(1/4, n) = (n!/2)*4^n*Gamma(n+1/4)*sqrt(2)*Gamma(3/4)/Pi.
a(n) = Product_{k=1..n} k*(4k-3). - Daniel Suteu, Nov 01 2017
D-finite with recurrence a(n) -n*(4*n-3)*a(n-1)=0. - R. J. Mathar, May 02 2022
a(n) ~ 2^(2*n+1) * n^(2*n + 1/4) * Pi /(Gamma(1/4) * exp(2*n)). - Amiram Eldar, Aug 28 2025

A133401 Diagonal of polygorial array T(n,k) = n-th polygorial for k = n, for n > 2.

Original entry on oeis.org

18, 576, 46200, 7484400, 2137544640, 981562982400, 678245967907200, 670873729125600000, 913601739437346960000, 1660189302321994373529600, 3923769742187622047360640000, 11805614186177306251101945600000, 44403795869109177300313209696000000
Offset: 3

Views

Author

Jonathan Vos Post, Nov 25 2007

Keywords

Comments

Array T(n,k) = k-th polygorial(n,k) begins:
k | polygorial(n,k)
3 | 1 1 3 18 180 2700 56700 1587600 57153600
4 | 1 1 4 36 576 14400 518400 25401600 1625702400
5 | 1 1 5 60 1320 46200 2356200 164934000 15173928000
6 | 1 1 6 90 2520 113400 7484400 681080400 81729648000
7 | 1 1 7 126 4284 235620 19085220 2137544640 316356606720
8 | 1 1 8 168 6720 436800 41932800 5577062400 981562982400
9 | 1 1 9 216 9936 745200 82717200 12738448800 2598643555200
10 | 1 1 10 270 14040 1193400 150368400 26314470000 6104957040000

Examples

			a(3) = polygorial(3,3) = A006472(3) = product of the first 3 triangular numbers = 1*3*6 = 18.
a(4) = polygorial(4,4) = A001044(4) = product of the first 4 squares = 1*4*9*16 = 576.
a(5) = polygorial(5,5) = A084939(5) = product of the first 5 pentagonal numbers = 1*5*12*22*35 = 46200.

Links

Crossrefs

Cf. A006472, A001044, A000680, A084939, A084940, A084941, A084942, A084943, A084944, A085356.

Programs

  • Maple
    A133401 := proc(n) return mul((n/2-1)*m^2-(n/2-2)*m,m=1..n): end: seq(A133401(n),n=3..15); # Nathaniel Johnston, May 05 2011
  • Mathematica
    Table[Product[m*(4 - n + m*(n-2))/2, {m, 1, n}],{n, 3, 20}] (* Vaclav Kotesovec, Feb 20 2015 *)
Table[FullSimplify[(n-2)^n * Gamma[n+1] * Gamma[n+2/(n-2)] / (2^n*Gamma[2/(n-2)])],{n,3,15}] (* Vaclav Kotesovec, Feb 20 2015 *)
polygorial[k_, n_] := FullSimplify[ n!/2^n (k -2)^n*Pochhammer[2/(k - 2), n]]; Array[ polygorial[#, #] &, 13, 3] (* Robert G. Wilson v, Dec 13 2016 *)

Formula

a(n) ~ Pi * n^(3*n-1) / (2^(n-2) * exp(2*n+2)). - Vaclav Kotesovec, Feb 20 2015

Extensions

Edited by Nathaniel Johnston, May 05 2011
Showing 1-7 of 7 results.

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