A084941 -id:A084941 - cp's OEIS frontend

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

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

Offset: 0

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

Daniel Dockery, Polygorials, Special "Factorials" of Polygonal Numbers, preprint, 2003.

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

a := n->n!/2^n*product(8*i+2,i=0..n-1); [seq(a(j),j=0..30)];

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 *)

a(n)=n!/2^n*prod(i=1,n,8*i-6) \\ Charles R Greathouse IV, Dec 13 2016

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

A085356 a(n) = polygorial(n,3)/polygorial(3,n), n >= 3.

Original entry on oeis.org

1, 5, 45, 630, 12600, 340200, 11907000, 523908000, 28291032000, 1838917080000, 141596615160000, 12743695364400000, 1325344317897600000, 157715973829814400000, 21291656467024944000000

Offset: 0

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

G. C. Greubel, Table of n, a(n) for n = 0..220
Daniel Dockery, Polygorials, Special "Factorials" of Polygonal Numbers, preprint, 2003.

Cf. A067550, A084939, A084940, A084941, A084942, A084943, A084944.

a := n->(n+1)!^2*(n+2)*(n+3)*(n+4)/2^n/24; [seq(a(j),j=0..15)];
seq(mul(binomial(k,2)-binomial(k,1), k =5..n), n=4..18 ); # Zerinvary Lajos, Aug 07 2007

polygorial[k_, n_] := FullSimplify[ n!/2^n (k -2)^n*Pochhammer[2/(k -2), n]]; Array[ polygorial[3, #]/polygorial[#, 3] &, 17, 3] (* Robert G. Wilson v, Dec 13 2016 *)

a(n) = polygorial(n+3, 3)/polygorial(3, n+3) = (n+1)!^2*(n+2)*(n+3)*(n+4)/(2^n*24) = A067550(n+2)/2.
a(n) ~ (1/12)*Pi*n^(2*n+6)/(2^n*exp(2*n)). - Ilya Gutkovskiy, Dec 17 2016
D-finite with recurrence 2*a(n) = (n+4)*(n+1)*a(n-1). - R. J. Mathar, Mar 12 2019

A210277 a(n) = (3*n)!/3^n.

Original entry on oeis.org

1, 2, 80, 13440, 5913600, 5381376000, 8782405632000, 23361198981120000, 94566133475573760000, 553211880832106496000000, 4492080472356704747520000000, 49017582114356362204938240000000, 699971072593008852286518067200000000

Offset: 0

Mohammad K. Azarian, Mar 20 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..100
D. Bevan, D. Levin, P. Nugent, J. Pantone and L. Pudwell, Pattern avoidance in forests of binary shrubs, arXiv preprint arXiv:1510:08036 [math.CO], 2015-2016.

Cf. A210278, A000680, A067630, A084939, A084940, A084941, A084942, A084943, A084944, A087127, A001147, A132101.

[Factorial(3*n)/3^n: n in [0..15]]; // Vincenzo Librandi, Feb 15 2013

Table[(3 n)!/3^n, {n, 0, 15}] (* Vincenzo Librandi, Feb 15 2013 *)

E.g.f.: 1/(1-x^3/3).
a(n) = Product_{i=1..n} (2*binomial(3i,3)). - James Mahoney, Apr 04 2012
From Amiram Eldar, Jan 18 2021: (Start)
Sum_{n>=0} 1/a(n) = exp(3^(1/3))/3 + (2/3)*exp(-3^(1/3)/2)*cos(3^(5/6)/2).
Sum_{n>=0} (-1)^n/a(n) = exp(-3^(1/3))/3 + (2/3)*exp(3^(1/3)/2)*cos(3^(5/6)/2). (End)

A210278 (5n)!/5^n.

Original entry on oeis.org

1, 24, 145152, 10461394944, 3892643213082624, 4963587213865915514880, 16976183027980227752723742720, 132264293969742655099733137120296960, 2088743125114618199924764850166056689336320, 61246577083125859615725138685776750112964471685120

Offset: 0

Mohammad K. Azarian, Mar 20 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..90

Cf. A210277, A000680, A067630, A084939, A084940, A084941, A084942, A084943, A084944, A087127, A001147, A132101.

[Factorial(5*n)/5^n: n in [0..10]]; // Vincenzo Librandi, Feb 15 2013

Table[(5 n)!/5^n, {n, 0, 10}] (* Vincenzo Librandi, Feb 15 2013 *)
With[{nn=100},Take[CoefficientList[Series[1/(1-x^5/5),{x,0,nn}],x] Range[0,nn]!,{1,-1,5}]] (* Harvey P. Dale, May 27 2025 *)

E.g.f.: 1/(1-x^5/5).

A210279 (6n)!/6^n.

Original entry on oeis.org

1, 120, 13305600, 29640619008000, 478741050720092160000, 34111736086958726676480000000, 7973107998754741458076119859200000000, 5019026197962676820927435579005599744000000000

Offset: 0

Mohammad K. Azarian, Apr 12 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..80

Cf. A210278, A210277, A000680, A067630, A084939, A084940, A084941, A084942, A084943, A084944, A087127, A001147, A132101

[Factorial(6*n)/6^n: n in [0..10]]; // Vincenzo Librandi, Feb 15 2013

Table[(6 n)!/6^n, {n, 0, 11}] (* Vincenzo Librandi, Feb 15 2013 *)
With[{nn=50},Take[CoefficientList[Series[1/(1-x^6/6),{x,0,nn}],x] Range[0,nn-2]!,{1,-1,6}]] (* Harvey P. Dale, Sep 25 2023 *)

E.g.f.: 1/(1-x^6/6).

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

Jonathan Vos Post, Nov 25 2007

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

			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.

Nathaniel Johnston, Table of n, a(n) for n = 3..100
Daniel Dockery, Polygorials, Special "Factorials" of Polygonal Numbers, preprint, 2003.

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

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

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 *)

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

Edited by Nathaniel Johnston, May 05 2011

A210280 (7n)!/7^n.

Original entry on oeis.org

1, 720, 1779148800, 148953184174080000, 126983900296423931904000000, 614812159599342234168301977600000000, 11942354952042770431904585727413846016000000000

Offset: 0

Mohammad K. Azarian, Apr 12 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..60

Cf. A210279, A210278, A210277, A000680, A067630, A084939, A084940, A084941, A084942, A084943, A084944, A087127, A001147, A132101

[Factorial(7*n)/7^n: n in [0..10]]; // Vincenzo Librandi, Feb 15 2013

Table[(7 n)!/7^n, {n, 0, 10}] (* Vincenzo Librandi, Feb 15 2013 *)

E.g.f.: 1/(1-x^7/7).

A210281 (8n)!/8^n.

Original entry on oeis.org

1, 5040, 326918592000, 1211813284635233280000, 64240926985765022013480960000000, 24899758399899222849902687670779904000000000, 47355329866546908076714664639943599847875543040000000000

Offset: 0

Mohammad K. Azarian, Apr 12 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..50

Cf. A210280, A210279, A210278, A210277, A000680, A067630, A084939, A084940, A084941, A084942, A084943, A084944, A087127, A001147, A132101

[Factorial(8*n)/8^n: n in [0..10]]; // Vincenzo Librandi, Feb 15 2013

Table[(8 n)!/8^n, {n, 0, 10}] (* Vincenzo Librandi, Feb 15 2013 *)

E.g.f.: 1/(1-x^8/8).

A276482 a(n) = 5^nGamma(n+1/5)Gamma(n+1)/Gamma(1/5).

Original entry on oeis.org

1, 1, 12, 396, 25344, 2661120, 415134720, 90084234240, 25944259461120, 9573431741153280, 4403778600930508800, 2470519795122015436800, 1660189302321994373529600, 1316530116741341538208972800, 1216473827868999581305090867200, 1295544626680484554089921773568000

Offset: 0

Ilya Gutkovskiy, Sep 05 2016

12-gonal (or dodecagonal) factorial numbers, also polygorial(n, 12).

More generally, the m-gonal factorial numbers (or polygorial(n, m)) is 2^(-n)*(m - 2)^n*Gamma(n+2/(m-2))*Gamma(n+1)/Gamma(2/(m-2)), m>2.

Robert Israel, Table of n, a(n) for n = 0..220
Daniel Dockery, Polygorials, Special "Factorials" of Polygonal Numbers, preprint, 2003.
Index entries for sequences related to factorial numbers

Cf. A000142, A008548, A051624.

Cf. similar sequences of m-gonal factorial numbers (or polygorial(n, m)): A006472 (m=3), A001044 (m=4), A084939 (m=5), A000680 (m=6), A084940 (m=7), A084941 (m=8), A084942 (m=9), A084943 (m=10), A084944 (m=11).

seq(mul(k*(5*k-4),k=1..n), n=0..20); # Robert Israel, Sep 18 2016

FullSimplify[Table[5^n Gamma[n + 1/5] (Gamma[n + 1]/Gamma[1/5]), {n, 0, 15}]]
polygorial[k_, n_] := FullSimplify[ n!/2^n (k -2)^n*Pochhammer[2/(k -2),n]]; Array[polygorial[12, #] &, 16, 0] (* Robert G. Wilson v, Dec 13 2016 *)

a(n) = prod(k=1, n, k*(5*k - 4)); \\ Michel Marcus, Sep 06 2016

a(n) = Product_{k=1..n} k*(5*k - 4), a(0)=1.
a(n) = Product_{k=1..n} A051624(k), a(0)=1.
a(n) = A000142(n)*A008548(n).
a(n) ~ 2*Pi*5^n*n^(2*n+1/5)/(Gamma(1/5)*exp(2*n)).
Sum_{n>=0} 1/a(n) = BesselI(-4/5,2/sqrt(5))*Gamma(1/5)/5^(2/5) = Hypergeometric0F1(1/5, 1/5) = 2.085898421130914...

cp's OEIS Frontend

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

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A085356 a(n) = polygorial(n,3)/polygorial(3,n), n >= 3.

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A210277 a(n) = (3*n)!/3^n.

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A210278 (5n)!/5^n.

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A210279 (6n)!/6^n.

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

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A210280 (7n)!/7^n.

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A210281 (8n)!/8^n.

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A276482 a(n) = 5^nGamma(n+1/5)Gamma(n+1)/Gamma(1/5).