A001044 -id:A001044 - cp's OEIS frontend

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

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

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, A084939, A084940, A084941, A084943, A084944, A085356, A220605.

a := n->n!/2^n*product(7*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[9, #] &, 16, 0] (* Robert G. Wilson v, Dec 26 2016 *)

a(n)=n!/2^n*prod(i=1,n,7*i-5) \\ Charles R Greathouse IV, Dec 13 2016

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

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

A134375 a(n) = (n!)^4.

Original entry on oeis.org

1, 1, 16, 1296, 331776, 207360000, 268738560000, 645241282560000, 2642908293365760000, 17340121312772751360000, 173401213127727513600000000, 2538767161403058526617600000000, 52643875858853821607942553600000000, 1503561738404723998944447273369600000000

Offset: 0

Artur Jasinski, Oct 22 2007

nonn

a(n) is also the determinant of the symmetric n X n matrix M defined by M(i,j) = sigma_4(gcd(i,j)) for 1 <= i,j <= n, and n>0, where sigma_4 is A001159. - Enrique Pérez Herrero, Aug 13 2011

Alois P. Heinz, Table of n, a(n) for n = 0..100

Cf. A000142, A001044, A000442, A036740, A010050, A009445, A134366, A134367, A134368, A134369, A134371, A134372, A134373, A134374.

Row n=4 of A225816.

a:= n-> (n!)^4:
seq(a(n), n=0..20);  # Alois P. Heinz, Aug 15 2013

Mathematica
```
Table[((n)!)^(4), {n, 0, 10}]
```

a(n) = det(S(i+4,j), 1 <= i,j <= n), where S(n,k) are Stirling numbers of the second kind. - Mircea Merca, Apr 04 2013

A298851 a(n) = [x^n] Product_{k=1..n} 1/(1-k^2*x).

Original entry on oeis.org

1, 1, 21, 1408, 196053, 46587905, 16875270660, 8657594647800, 5974284925007685, 5336898188553325075, 5992171630749371157181, 8260051854943114812198756, 13714895317396748230146099660, 26998129079190909699998105620908, 62173633286588800021263427046090792

Offset: 0

Seiichi Manyama, Feb 01 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..200

Cf. A001044, A036969, A269945.
Cf. A007820, A299035, A299036.
Cf. A383853.

b:= proc(k, n) option remember; `if`(k=0, 1,
      add(b(k-1, j)*j^2, j=1..n))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..14);  # Alois P. Heinz, Feb 19 2022

Table[SeriesCoefficient[Product[1/(1 - k^2*x), {k, 1, n}], {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Feb 02 2018 *)
Join[{1}, Table[2*Sum[(-1)^(n-k) * Binomial[2*n, n-k] * k^(4*n), {k, 0, n}]/(2*n)!, {n, 1, 20}]] (* Vaclav Kotesovec, May 15 2025 *)

a(n):=if n<1 then 1 else 2*sum((n-k)^(4*n)/((2*n-k)!*k!*(-1)^k),k,0,n);
makelist(a(n), n, 0, 20); /* Tani Akinari, Mar 09 2021 */

From Vaclav Kotesovec, Feb 02 2018, updated May 12 2025: (Start)
a(n) ~ c * d^n * n^(2*n - 1/2), where d = 1.774513671664430848697327843228386312953174421074432567764556466987... and c = 0.617929515483613293691991371141292259390065108300160936187723552669...
In closed form, a(n) ~ n^(2*n - 1/2) * r^(4*n + 1) / (sqrt(Pi*(2 - r^2)) * (r^2 - 1)^n * exp(2*n)), where r = 1.04438203376083348498401390634474776086902815721... is the root of the equation (1-r)/(1+r) = -exp(-4/r). (End)
a(n) = 2*(Sum_{k=0..n} (n-k)^(4*n)/((2*n-k)!*k!*(-1)^k)) for n>0. - Tani Akinari, Mar 09 2021
a(n) = A036969(2n,n) = A269945(2n,n). - Alois P. Heinz, Feb 19 2022
From Seiichi Manyama, May 12 2025: (Start)
a(n) = Sum_{k=0..2*n} (-n)^k * binomial(4*n,k) * Stirling2(4*n-k,2*n).
a(n) = Sum_{k=0..2*n} (-1)^k * Stirling2(k+n,n) * Stirling2(3*n-k,n). (End)

A162990 Triangle of polynomial coefficients related to 3F2([1,n+1,n+1],[n+2,n+2],z).

Original entry on oeis.org

4, 36, 9, 576, 144, 64, 14400, 3600, 1600, 900, 518400, 129600, 57600, 32400, 20736, 25401600, 6350400, 2822400, 1587600, 1016064, 705600, 1625702400, 406425600, 180633600, 101606400, 65028096, 45158400, 33177600, 131681894400

Offset: 1

Johannes W. Meijer, Jul 21 2009

The hypergeometric function 3F2([1,n+1,n+1],[n+2,n+2],z) = (n+1)^2*Li2(z)/z^(n+1) - MN(z;n)/(n!^2*z^n) for n >= 1, with Li2(z) the dilogarithm. The polynomial coefficients of MN(z;n) lead to the triangle given above.
We observe that 3F2([1,1,1],[2,2],z) = Li2(z)/z and that 3F2([1,0,0],[1,1],z) = 1.
The generating function for the EG1[3,n] coefficients of the EG1 matrix, see A162005, is GFEG1(z;m=2) = 1/(1-z)*(3*zeta(3)/2-2*z*log(2)* 3F2([1,1,1],[2,2],z) + sum((2^(1-2*n)* factorial(2*n-1)*z^(n+1)*3F2([1,n+1,n+1],[n+2,n+2],z))/(factorial(n+1)^2), n=1..infinity)).
The zeros of the MN(z;n) polynomials for larger values of n get ever closer to the unit circle and resemble the full moon, hence we propose to call the MN(z;n) the moon polynomials.

			The first few rows of the triangle are:
  [4]
  [36, 9]
  [576, 144, 64]
  [14400, 3600, 1600, 900]
The first few MN(z;n) polynomials are:
  MN(z;n=1) = 4
  MN(z;n=2) = 36 + 9*z
  MN(z;n=3) = 576 + 144*z + 64*z^2
  MN(z;n=4) = 14400 + 3600*z + 1600*z^2 + 900*z^3

Lewin, L., Polylogarithms and Associated Functions. New York, North-Holland, 1981.

Paolo Xausa, Table of n, a(n) for n = 1..5050 (rows 1..100 of the triangle, flattened).
Eric Weisstein's World of Mathematics, Dilogarithm.

A162995 is a scaled version of this triangle.
A001819(n)*(n+1)^2 equals the row sums for n>=1.
A162991 and A162992 equal the first and second right hand columns.
A001048, A052747, A052759, A052778, A052794 are related to the square root of the first five right hand columns.
A001044, A162993 and A162994 equal the first, second and third left hand columns.
A000142, A001710, A002301, A133799, A129923, A001715 are related to the square root of the first six left hand columns.
A027451(n+1) equals the denominators of M(z, n)/(n!)^2.
A129202(n)/A129203(n) = (n+1)^2*Li2(z=1)/(Pi^2) = (n+1)^2/6.
Cf. A002378 and A035287.

a := proc(n, m): ((n+1)!/m)^2 end: seq(seq(a(n, m), m=1..n), n=1..7); # Johannes W. Meijer, revised Nov 29 2012

Table[((n+1)!/m)^2, {n, 10}, {m, n}] (* Paolo Xausa, Mar 30 2024 *)

a(n,m) = ((n+1)!/m)^2 for n >= 1 and 1 <= m <= n.

A134367 a(n) = (n!)^(n-2).

Original entry on oeis.org

1, 1, 1, 6, 576, 1728000, 268738560000, 3252016064102400000, 4296582355504620109824000000, 828592942960967278432052230225920000000, 30067980714167580599742311330438184960000000000000000

Offset: 0

Artur Jasinski, Oct 22 2007

nonn

Cf. A000142, A001044, A000442, A134367, A134368, A134369, A134370, A134371, A134374, A134375.

Mathematica
```
Table[(n!)^(n - 2), {n, 0, 10}]
```

a(n) ~ exp(1/12 + 2*n - n^2) * n^(n^2 - 3*n/2 - 1) * (2*Pi)^(n/2 - 1). - Vaclav Kotesovec, Oct 26 2017

A134374 a(n) = ((2n+1)!)^2.

Original entry on oeis.org

1, 36, 14400, 25401600, 131681894400, 1593350922240000, 38775788043632640000, 1710012252724199424000000, 126513546505547170185216000000, 14797530453474819213543604224000000

Offset: 0

Artur Jasinski, Oct 22 2007

Cf. A000142, A001044, A000442, A036740, A010050, A009445, A134366, A134367, A134368, A134369, A134371, A134372, A134373, A134375, A334378.

A134374:=n->((2*n+1)!)^2; seq(A134374(n), n=0..10); # Wesley Ivan Hurt, May 02 2014

Mathematica
```
Table[((2n+1)!)^(2), {n, 0, 10}]
```

a(n) = ((2*n+1)!)^2; \\ Michel Marcus, Nov 16 2020

a(n) = A009445(n)^2 = A001044(2n+1). - Wesley Ivan Hurt, May 02 2014
From Amiram Eldar, Nov 16 2020: (Start)
Sum_{n>=0} 1/a(n) = A334378.
Sum_{n>=0} (-1)^n/a(n) = Im(BesselJ(0, 2*exp(3*Pi*i/4))). (End)

A020549 a(n) = (n!)^2 + 1.

Original entry on oeis.org

2, 2, 5, 37, 577, 14401, 518401, 25401601, 1625702401, 131681894401, 13168189440001, 1593350922240001, 229442532802560001, 38775788043632640001, 7600054456551997440001, 1710012252724199424000001, 437763136697395052544000001

Offset: 0

N. J. A. Sloane, Simon Plouffe

nonn

Used to prove there are infinitely many primes of the form 4k+1 (see A282706). - N. J. A. Sloane, Feb 26 2017

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 147.
F. Iacobescu, Smarandache Partition Type and Other Sequences, Bull. Pure Appl. Sciences, Vol. 16E, No. 2 (1997), pp. 237-240.
H. Ibstedt, A Few Smarandache Sequences, Smarandache Notions Journal, Vol. 8, No. 1-2-3, 1997, 170-183.
M. Le, On the Interesting Smarandache Product Sequences, Smarandache Notions Journal, Vol. 9, No. 1-2, 1998, 133-134.
M. Le, The Primes in Smarandache Power Product Sequences, Smarandache Notions Journal, Vol. 9, No. 1-2, 1998, 96-97.

G. C. Greubel, Table of n, a(n) for n = 0..250
M. Fleuren, Smarandache Square Products.
F. Smarandache, Sequences of Numbers Involved in Unsolved Problems.
Apoloniusz Tyszka, On sets X, subset of N, whose finiteness implies that we know an algorithm which for every n, element of N, decides the inequality max (X) < n, (2019).
Apoloniusz Tyszka, On ZFC-formulae phi(x) for which we know a non-negative integer n such that max({x, element of N, phi(x)}) <= n if the set {x, element of N, phi(x)} is finite, 2019.
Eric Weisstein's World of Mathematics, Factorial
Eric Weisstein's World of Mathematics, Smarandache Sequences

Cf. A001044.

For smallest prime factor see A282706.

with(combinat):seq(fibonacci(3,n!), n=0..16); # Zerinvary Lajos, Apr 21 2008
[seq(n!^2+1,n=0..20)]; # N. J. A. Sloane, Feb 26 2017

Table[(n!)^2 + 1, {n, 0, 20}] (* Vladimir Joseph Stephan Orlovsky, Apr 08 2011 *)

a(n)=n!^2 + 1 \\ Charles R Greathouse IV, Nov 30 2016

A064618 Stirling transform of (n!)^2.

Original entry on oeis.org

1, 1, 5, 49, 821, 21121, 775205, 38516689, 2490976661, 203419086241, 20474978755205, 2490729330118129, 360263844701062901, 61114158974786823361, 12017074366801186956005, 2711409826920884006692369, 695820350706240448128979541, 201526362605605903609254528481

Offset: 0

Karol A. Penson, Sep 26 2001

From Thomas Wieder, Oct 21 2004: (Start)
"Also the number of hierarchies with labeled elements and labeled levels where the levels are permuted. Let l_x denote level x, e.g. l_2 is level 2. Let 1 denote an element and 2 a second element and so on. Then l_1:123 means elements 1,2 and 3 are on level 1.
"Let | indicate separation between levels. Then l_1:1|l_2:346|l_3:5 denotes a hierarchy of n=6 unlabeled elements with element 1 on level 1, elements 3,4 and 6 on level 2 and element 5 on level 3.
"E.g. for n=3 one has a(3) = 49 possible hierarchies:
"l_1:123,
"l_1:12|l_2:3, l_1:13|l_2:2, l_1:23|l_2:1,
"l_2:12|l_1:3, l_2:13|l_1:2, l_2:23|l_1:1,
"l_1:1|l_2:23, l_1:2|l_2:13, l_1:3|l_2:12,
"l_2:1|l_1:23, l_2:2|l_1:13, l_2:3|l_1:12,
"l_1:1|l_2:2|l_3:3 and further five permutations of the elements with levels fixed,
"l_3:1|l_1:2|l_2:3 and further five permutations of the elements with levels fixed,. etc., up to
"l_3:1|l_2:2|l_1:3 and further five permutations of the elements with levels fixed. this gives 1 + 6 +6 + 6*6 = 49 = a(3) possible hierarchies.
"See A001339 for the number of hierarchies with unlabeled elements and labeled levels."
(End)
Conjecture: for fixed k = 1,2,..., the sequence a(n) (mod k) is eventually periodic with the exact period dividing phi(k), where phi(k) is the Euler totient function A000010. For example, modulo 10 the sequence becomes (1, 1, 5, 9, 1, 1, 5, 9, ...), with an apparent period 1, 1, 5, 9 of length 4 = phi(10) beginning at a(0). - Peter Bala, Jan 15 2018

Alois P. Heinz, Table of n, a(n) for n = 0..253

Cf. A001044.
Cf. A001339.
Cf. A242282, A242283, A019774, A320502.

a:= n-> add(Stirling2(n, k)*(k!^2), k=0..n):
seq(a(n), n=0..20);  # Alois P. Heinz, Apr 21 2012

Table[Sum[(k!)^2*StirlingS2[n, k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, May 10 2014 *)

/* By Vladeta Jovovic's formula: */
{a(n) = my(X=x+x*O(x^n)); n!*polcoeff( sum(m=0,n, m!*(exp(X)-1)^m), n)} /* Paul D. Hanna, Feb 15 2012 */

a(n) = Sum_{k=0..n} Stirling2(n, k)*(k!)^2.
E.g.f: hypergeom([1, 1], [], exp(x)-1). - Vladeta Jovovic, Sep 14 2003
O.g.f.: Sum_{n>=0} n!^2 * Product_{k=1..n} x/(1 - k*x). - Paul D. Hanna, Nov 25 2012
a(n) ~ exp(1/2) * (n!)^2. - Vaclav Kotesovec, May 10 2014

A134368 a(n) = ((2n)!)^(n+1).

Original entry on oeis.org

1, 4, 13824, 268738560000, 106562062388507443200000, 2283380023591730815784976384000000000000, 5785737804304645733190746102656048717392091545600000000000000

Offset: 0

Artur Jasinski, Oct 22 2007

nonn

Cf. A000142, A001044, A000442, A036740, A134367, A134368, A134369, A134370, A134371, A134374, A134375.

Mathematica
```
Table[((2n)!)^(n + 1), {n, 0, 10}]
```

a(n) ~ 2^((n+1)*(2*n+1)) * exp(1/24 - 2*n*(n+1)) * n^((n+1)*(4*n+1)/2) * Pi^((n+1)/2). - Vaclav Kotesovec, Oct 26 2017

cp's OEIS Frontend

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

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A134375 a(n) = (n!)^4.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A298851 a(n) = [x^n] Product_{k=1..n} 1/(1-k^2*x).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Maxima

Formula

A162990 Triangle of polynomial coefficients related to 3F2([1,n+1,n+1],[n+2,n+2],z).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A134367 a(n) = (n!)^(n-2).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A134374 a(n) = ((2n+1)!)^2.

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A020549 a(n) = (n!)^2 + 1.

Views

Author

Keywords