A084940 -id:A084940 - 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...

A279663 a(n) = (5/6)^nGamma(n+3/5)Gamma(n+1)*Gamma(n+2)/Gamma(3/5).

Original entry on oeis.org

1, 1, 8, 208, 12480, 1435200, 281299200, 86640153600, 39507910041600, 25482601976832000, 22424689739612160000, 26147188236387778560000, 39429959860472770068480000, 75350653293363463600865280000, 179334554838205043370059366400000, 523656900127558726640573349888000000

Offset: 0

Ilya Gutkovskiy, Dec 16 2016

nonn

Heptagonal pyramidal factorial numbers.

Eric Weisstein's World of Mathematics, Heptagonal Pyramidal Number
Index to sequences related to pyramidal numbers
Index entries for sequences related to factorial numbers

Cf. A002413.
Cf. A084940 (heptagonal factorial numbers).
Cf. A087047 (tetrahedral factorial numbers), A135438 (square pyramidal factorial numbers), A167484 (pentagonal pyramidal factorial numbers), A279662 (hexagonal pyramidal factorial numbers).

[Round((5/6)^n*Gamma(n+3/5)*Gamma(n+1)*Gamma(n+2)/Gamma(3/5)): n in [0..20]]; // Vincenzo Librandi Dec 17 2016

FullSimplify[Table[(5/6)^n Gamma[n + 3/5] Gamma[n + 1] Gamma[n + 2]/Gamma[3/5], {n, 0, 15}]]

a(n) = Product_{k=1..n} k*(k + 1)*(5*k - 2)/6, a(0)=1.
a(n) = Product_{k=1..n} A002413(k), a(0)=1.
a(n) ~ (2*Pi)^(3/2)*(5/6)^n*n^(3*n+21/10)/(Gamma(3/5)*exp(3*n)).

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).

A279663 a(n) = (5/6)^nGamma(n+3/5)Gamma(n+1)*Gamma(n+2)/Gamma(3/5).