A010791 -id:A010791 - cp's OEIS frontend

A079484 a(n) = (2n-1)!! * (2n+1)!!, where the double factorial is A006882.

1, 3, 45, 1575, 99225, 9823275, 1404728325, 273922023375, 69850115960625, 22561587455281875, 9002073394657468125, 4348001449619557104375, 2500100833531245335015625, 1687568062633590601135546875, 1321365793042101440689133203125

Offset: 0

Benoit Cloitre, Jan 17 2003

nonn

a(n) is the determinant of M(2n+1) where M(k) is the k X k matrix with m(i,j)=j if i+j=k m(i,j)=i otherwise. - Adapted to offset 0, Rainer Rosenthal, Jun 19 2024
In the following two comments on the calculation of the terms using permanents, offset 1 is assumed. In the corresponding PARI code, this is implemented with offset 0. - Hugo Pfoertner, Jun 23 2024
(-1)^n*a(n)/2^(2n-1) is the permanent of the (m X m) matrix {1/(x_i-y_j), 1<=i<=m, 1<=j<=m}, where x_1,x_2,...,x_m are the zeros of x^m-1 and y_1,y_2,...,y_m the zeros of y^m+1 and m=2n-1.
In 1881, R. F. Scott posed a conjecture that the absolute value of permanent of square matrix with elements a(i,j)= (x_i - y_j)^(-1), where x_1,...,x_n are roots of x^n=1, while y_1,...,y_n are roots of y^n=-1, equals a((n-1)/2)/2^n, if n>=1 is odd, and 0, if n>=2 is even. After a century (in 1979), the conjecture was proved by H. Minc. - Vladimir Shevelev, Dec 01 2013
a(n) is the number of permutations in S_{2n+1} in which all cycles have odd length. - José H. Nieto S., Jan 09 2012
Number of 3-bundled increasing bilabeled trees with 2n labels. - Markus Kuba, Nov 18 2014
a(n) is the number of rooted, binary, leaf-labeled topologies with 2n+2 leaves that have n+1 cherry nodes. - Noah A Rosenberg, Feb 12 2019

			G.f. = 1 + 3*x + 45*x^2 + 1575*x^3 + 99225*x^4 + 9823275*x^5 + ...
M(5) =
  [1, 2, 3, 1, 5]
  [1, 2, 2, 4, 5]
  [1, 3, 3, 4, 5]
  [4, 2, 3, 4, 5]
  [1, 2, 3, 4, 5].
Integral_{x=0..oo} x^3*BesselK(1, sqrt(x)) = 1575*Pi. - _Olivier Gérard_, May 20 2009

Miklós Bóna, A walk through combinatorics, World Scientific, 2006.

Alois P. Heinz, Table of n, a(n) for n = 0..224
Cyril Banderier, Markus Kuba, and Michael Wallner, Analytic Combinatorics of Composition schemes and phase transitions with mixed Poisson distributions, arXiv:2103.03751 [math.PR], 2021.
Guo-Niu Han and Christian Krattenthaler, Rectangular Scott-type permanents, arXiv:math/0003072 [math.RA], 2000.
Markus Kuba and Alois Panholzer, Combinatorial families of multilabelled increasing trees and hook-length formulas, arXiv:1411.4587 [math.CO], 17 Nov 2014.
MathOverflow, Geometric / physical / probabilistic interpretations of Riemann zeta(n>1)?, answer by Tom Copeland posted in Aug 2021.
Henryk Minc, On a conjecture of R. F. Scott (1881), Linear Algebra Appl., Vol. 28 (1979), pp. 141-153.
Theodoros Theodoulidis, On the Closed-Form Expression of Carson’s Integral, Period. Polytech. Elec. Eng. Comp. Sci., Vol. 59, No. 1 (2015), pp. 26-29.
Eric Weisstein's World of Mathematics, Struve function.

Cf. A001818, A000165.
Bisection of A000246, A053195, |A013069|, |A046126|. Cf. A000909.
Cf. A001044, A010791, |A129464|, A114779, are also values of similar moments.
Equals the row sums of A162005.
Cf. A316728.
Diagonal elements of A306364 in even-numbered rows.

I:=[1, 3]; [n le 2 select I[n] else (4*n^2-8*n+3)*Self(n-1): n in [1..20]]; // Vincenzo Librandi, Nov 18 2014

a:= n-> (d-> d(2*n-1)*d(2*n+1))(doublefactorial):
seq(a(n), n=0..15);  # Alois P. Heinz, Jan 30 2013
# second Maple program:
A079484 := n-> LinearAlgebra[Determinant](Matrix(2*n+1, (i, j)-> `if`(i+j=2*n+1, j, i))): seq(A079484(n), n=0..14); # Rainer Rosenthal, Jun 18 2024

a[n_] := (2n - 1)!!*(2n + 1)!!; Table[a[n], {n, 0, 13}] (* Jean-François Alcover, Jan 30 2013 *)

/* Formula using the zeta function and a log integral:*/
L(n)= intnum(t=0, 1, log(1-1/t)^n);
Zetai(n)= -I*I^n*(2*Pi)^(n-1)/(n-1)*L(1-n);
a(m)={my(n=m+1);round(real(-I*2^(2*n-1)*Zetai(1/2-n)*L(-1/2+n)/(Zetai(-1/2+n)*L(1/2-n))))};
/* Gerry Martens, Mar 07 2011, adapted to offset 0 by Hugo Pfoertner, Jun 19 2024 */

{a(n) = if( n<0, -1 / self()(-1-n), (2*n + 1)! * (2*n)! / (n! * 2^n)^2 )}; /* Michael Somos, May 04 2017 */

{a(n) = if( n<0, -1 / self()(-1-n), my(m = 2*n + 1); m! * polcoeff( x / sqrt( 1 - x^2 + x * O(x^m) ), m))}; /* Michael Somos, May 04 2017 */

\\ using the Pochhammer symbol
a(n) = {my(P(x,k)=gamma(x+k)/gamma(x)); 4^n*round(P(1/2,n)*P(3/2,n))} \\ Hugo Pfoertner, Jun 20 2024

\\ Scott's (1881) method
a(n) = {my(m=2*n+1, X = polroots(x^m-1), Y = polroots(x^m+1), M = matrix(m, m, i, j, 1/(X[i]-Y[j]))); (-1)^n * round(2^m * real(matpermanent(M)))}; \\ Hugo Pfoertner, Jun 23 2024

D-finite with recurrence a(n) = (4*n^2 - 1) * a(n-1) for all n in Z.
a(n) = A001147(n)*A001147(n+1).
E.g.f.: 1/(1-x^2)^(3/2) (with interpolated zeros). - Paul Barry, May 26 2003
a(n) = (2n+1)! * C(2n, n) / 2^(2n). - Ralf Stephan, Mar 22 2004.
Alternatingly signed values have e.g.f. sqrt(1+x^2).
a(n) is the value of the n-th moment of (1/Pi)*BesselK(1, sqrt(x)) on the positive part of the real line. - Olivier Gérard, May 20 2009
a(n) = -2^(2*n-1)*exp(i*n*Pi)*gamma(1/2+n)/gamma(3/2-n). - Gerry Martens, Mar 07 2011
E.g.f. (odd powers) tan(arcsin(x)) = Sum_{n>=0} (2n-1)!!*(2n+1)!!*x^(2*n+1)/(2*n+1)!. - Vladimir Kruchinin, Apr 22 2011
G.f.: 1 + x*(G(0) - 1)/(x-1) where G(k) = 1 - ((2*k+2)^2-1)/(1-x/(x - 1/G(k+1))); ( continued fraction ). - Sergei N. Gladkovskii, Jan 15 2013
a(n) = (2^(2*n+3)*Gamma(n+3/2)*Gamma(n+5/2))/Pi. - Jean-François Alcover, Jul 20 2015
Limit_{n->oo} 4^n*(n!)^2/a(n) = Pi/2. - Daniel Suteu, Feb 05 2017
From Michael Somos, May 04 2017: (Start)
a(n) = (2*n + 1) * A001818(n).
E.g.f.: Sum_{n>=0} a(n) * x^(2*n+1) / (2*n+1)! = x / sqrt(1 - x^2) = tan(arcsin(x)).
Given e.g.f. A(x) = y, then x * y' = y + y^3.
a(n) = -1 / a(-1-n) for all n in Z.
0 = +a(n)*(+288*a(n+2) -60*a(n+3) +a(n+4)) +a(n+1)*(-36*a(n+2) -4*a(n+3)) +a(n+2)*(+3*a(n+2)) for all n in Z. (End)
a(n) = Sum_{k=0..2n} (k+1) * A316728(n,k). - Alois P. Heinz, Jul 12 2018
From Amiram Eldar, Mar 18 2022: (Start)
Sum_{n>=0} 1/a(n) = 1 + L_1(1)*Pi/2, where L is the modified Struve function.
Sum_{n>=0} (-1)^n/a(n) = 1 - H_1(1)*Pi/2, where H is the Struve function. (End)

Simpler description from Daniel Flath (deflath(AT)yahoo.com), Mar 05 2004

A302909 Determinant of n X n matrix whose main diagonal consists of the first n 5-gonal numbers and all other elements are 1's.

Original entry on oeis.org

1, 4, 44, 924, 31416, 1570800, 108385200, 9863053200, 1144114171200, 164752440652800, 28831677114240000, 6025820516876160000, 1482351847151535360000, 423952628285339112960000, 139480414705876568163840000, 52305155514703713061440000000

Offset: 1

Muniru A Asiru, Apr 15 2018

nonn

			The 7 X 7 matrix (as below) has determinant 108385200.
  1  1  1  1  1  1  1
  1  5  1  1  1  1  1
  1  1 12  1  1  1  1
  1  1  1 22  1  1  1
  1  1  1  1 35  1  1
  1  1  1  1  1 51  1
  1  1  1  1  1  1 70

Muniru A Asiru, Table of n, a(n) for n = 1..100

Cf. A000326, A115067.

Cf. Determinant of n X n matrix whose main diagonal consists of the first n k-gonal numbers and all other elements are 1's: A000142 (k=2), A067550 (k=3), A010791 (k=4, with offset 1), this sequence (k=5), A302910 (k=6), A302911 (k=7), A302912 (k=8), A302913 (k=9), A302914 (k=10).

d:=(i,j)->`if`(i<>j,1,i*(3*i-1)/2):
seq(LinearAlgebra[Determinant](Matrix(n,d)),n=1..17);

Table[FullSimplify[Gamma[n] * Gamma[n + 5/3] * 3^(n + 1) / (5 * Gamma[2/3] * 2^n)], {n, 1, 15}] (* Vaclav Kotesovec, Apr 16 2018 *)
Module[{nn=20,pn5},pn5=PolygonalNumber[5,Range[nn]];Table[Det[DiagonalMatrix[Take[pn5,n]]/.(0->1)],{n,nn}]] (* Harvey P. Dale, Feb 07 2025 *)

a(n) = matdet(matrix(n, n, i, j, if (i!=j, 1, i*(3*i-1)/2))); \\ Michel Marcus, Apr 16 2018

first(n) = my(res = vector(n)); res[1] = 1; for(i = 1, n - 1, res[i + 1] = res[i] * i*(3*i + 5)/2); res \\ David A. Corneth, Apr 16 2018

From Vaclav Kotesovec, Apr 16 2018: (Start)
a(n) = Gamma(n) * Gamma(n + 5/3) * 3^(n + 1) / (5 * Gamma(2/3) * 2^n).
a(n) ~ Gamma(1/3) * 3^(n + 3/2) * n^(2*n + 2/3) / (5 * 2^n * exp(2*n)).
(End)
a(n + 1) = A115067(n + 1) * a(n) = a(n) * n*(3*n + 5)/2. - David A. Corneth, Apr 16 2018

A302910 Determinant of n X n matrix whose main diagonal consists of the first n 6-gonal numbers and all other elements are 1's.

Original entry on oeis.org

1, 5, 70, 1890, 83160, 5405400, 486486000, 57891834000, 8799558768000, 1663116607152000, 382516819644960000, 105192125402364000000, 34082248630365936000000, 12849007733647957872000000, 5576469356403213716448000000, 2760352331419590789641760000000

Offset: 1

Muniru A Asiru, Apr 15 2018

nonn

			The matrix begins:
  1  1  1  1  1  1  1 ...
  1  6  1  1  1  1  1 ...
  1  1 15  1  1  1  1 ...
  1  1  1 28  1  1  1 ...
  1  1  1  1 45  1  1 ...
  1  1  1  1  1 66  1 ...
  1  1  1  1  1  1 91 ...

Cf. A000384 (hexagonal numbers).
Cf. Determinant of n X n matrix whose main diagonal consists of the first n k-gonal numbers and all other elements are 1's: A000142 (k=2), A067550 (k=3), A010791 (k=4, with offset 1), A302909 (k=5), this sequence (k=6), A302911 (k=7), A302912 (k=8), A302913 (k=9), A302914 (k=10).
Odd bisection of column k=1 of A097591.

d:=(i,j)->`if`(i<>j,1,i*(2*i-1)):
seq(LinearAlgebra[Determinant](Matrix(n,d)),n=1..20);

nmax = 20; Table[Det[Table[If[i == j, i*(2*i - 1), 1], {i, 1, k}, {j, 1, k}]], {k, 1, nmax}] (* Vaclav Kotesovec, Apr 16 2018 *)
Table[(n + 1/2) * (2*n - 1)! / (3 * 2^(n - 2)), {n, 1, 20}] (* Vaclav Kotesovec, Apr 16 2018 *)
Table[Det[DiagonalMatrix[PolygonalNumber[6,Range[n]]]/.(0->1)],{n,20}] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Mar 23 2020 *)

a(n) = matdet(matrix(n, n, i, j, if (i!=j, 1, i*(2*i-1)))); \\ Michel Marcus, Apr 16 2018

a(n) = (n + 1/2) * (2*n-1)! / (3 * 2^(n-2)). - Vaclav Kotesovec, Apr 16 2018

A010792 a(n) = n!*(n+3)! / 3!.

Original entry on oeis.org

1, 4, 40, 720, 20160, 806400, 43545600, 3048192000, 268240896000, 28970016768000, 3766102179840000, 579979735695360000, 104396352425164800000, 21714441304434278400000, 5168037030455358259200000, 1395369998222946729984000000, 424192479459775805915136000000

Offset: 0

N. J. A. Sloane

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for sequences related to factorial numbers.

Cf. A010791, A010793, A010794, A010795, A261879.

[Factorial(n)*Factorial(n+3)/6: n in [0..20]]; // Vincenzo Librandi, Feb 25 2017

Maple
```
f := n->n!*(n+3)!/3!;
```

Table[n! (n + 3)! / 3!, {n, 0, 20}] (* Vincenzo Librandi, Feb 25 2017 *)

a(n) = n!^2*binomial(n+3,3) \\ Charles R Greathouse IV, Oct 23 2023

From Amiram Eldar, Sep 27 2022: (Start)
Sum_{n>=0} 1/a(n) = 6*BesselI(3,2) = 6*A261879.
Sum_{n>=0} (-1)^n/a(n) = 6*BesselJ(3,2). (End)
a(n) = 1/([x^n] hypergeom([], [4], x)). - Peter Luschny, Sep 13 2024

More terms from Vincenzo Librandi, Feb 25 2017

A010793 a(n) = n!*(n+4)! / 4!.

Original entry on oeis.org

1, 5, 60, 1260, 40320, 1814400, 108864000, 8382528000, 804722688000, 94152554496000, 13181357629440000, 2174924008857600000, 417585409700659200000, 92286375543845683200000, 23256166637049112166400000, 6628007491558996967424000000

Offset: 0

N. J. A. Sloane

nonn

Delbert L. Johnson, Table of n, a(n) for n = 0..251
Index entries for sequences related to factorial numbers.

Cf. A010791, A010792, A010794, A010795.

Maple
```
f := n->n!*(n+4)!/4!;
```

Table[n! * (n + 4)!/24, {n, 0, 20}] (* Amiram Eldar, Sep 27 2022 *)

a(n) = n!*(n+4)! / 4!; \\ Michel Marcus, Mar 30 2023

From Amiram Eldar, Sep 27 2022: (Start)
Sum_{n>=0} 1/a(n) = 24*BesselI(4,2).
Sum_{n>=0} (-1)^n/a(n) = 24*BesselJ(4,2). (End)
a(n) = 1/([x^n] hypergeom([], [5], x)). - Peter Luschny, Sep 13 2024

A010794 a(n) = n!*(n+5)!/5!.

Original entry on oeis.org

1, 6, 84, 2016, 72576, 3628800, 239500800, 20118067200, 2092278988800, 263627152588800, 39544072888320000, 6959756828344320000, 1419790392982241280000, 332230951957844459520000, 88373433220786626232320000, 26512029966235987869696000000

Offset: 0

N. J. A. Sloane

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for sequences related to factorial numbers.

Cf. A010791, A010792, A010793, A010795.

[Factorial(n)*Factorial(n+5) / Factorial(5): n in [0..20]]; // Vincenzo Librandi, Oct 04 2011

Maple
```
f := n->n!*(n+5)!/5!;
```

With[{c=5!},Table[(n!(n+5)!)/c,{n,0,20}]] (* Harvey P. Dale, Oct 03 2011 *)

a(n)=n!^2*binomial(n+5,5) \\ Charles R Greathouse IV, Oct 23 2023

From Amiram Eldar, Sep 27 2022: (Start)
Sum_{n>=0} 1/a(n) = 120*BesselI(5,2).
Sum_{n>=0} (-1)^n/a(n) = 120*BesselJ(5,2). (End)
a(n) = 1/([x^n] hypergeom([], [6], x)). - Peter Luschny, Sep 13 2024

More terms from Harvey P. Dale, Oct 03 2011

A010795 a(n) = n!*(n+6)! / 6!.

Original entry on oeis.org

1, 7, 112, 3024, 120960, 6652800, 479001600, 43589145600, 4881984307200, 659067881472000, 105450861035520000, 19719311013642240000, 4259371178946723840000, 1052064681199840788480000, 294578110735955420774400000, 92792104881825957543936000000

Offset: 0

N. J. A. Sloane

nonn

Delbert L. Johnson, Table of n, a(n) for n = 0..250
Index entries for sequences related to factorial numbers.

Cf. A010791, A010792, A010793, A010794.

Maple
```
f := n->n!*(n+6)!/6!;
```

Table[n! * (n + 6)!/6!, {n, 0, 15}] (* Amiram Eldar, Sep 27 2022 *)

a(n)=n!^2*binomial(n+6,6) \\ Charles R Greathouse IV, Oct 23 2023

From Amiram Eldar, Sep 27 2022: (Start)
Sum_{n>=0} 1/a(n) = 720*BesselI(6,2).
Sum_{n>=0} (-1)^n/a(n) = 720*BesselJ(6,2). (End)
a(n) = 1/([x^n] hypergeom([], [7], x)).- Peter Luschny, Sep 13 2024

A175430 a(n) = (n-1)! * (n+1)!.

Original entry on oeis.org

2, 6, 48, 720, 17280, 604800, 29030400, 1828915200, 146313216000, 14485008384000, 1738201006080000, 248562743869440000, 41758540970065920000, 8142915489162854400000, 1824013069572479385600000, 465123332740982243328000000, 133955519829402886078464000000

Offset: 1

Jaroslav Krizek, May 10 2010

Also the number of automorphisms in the (n+1)-alternating group graph for n > 1. - Eric W. Weisstein, Jul 30 2018

			a(5) = (5-1)! * (5+1)! = 4! * 6! = 24 * 720 = 17280.
a(5) = ((1*2) * (2*3) * (3*4) * (4*5) * (5*6)) / 5 = 17280.

Amiram Eldar, Table of n, a(n) for n = 1..253
Eric Weisstein's World of Mathematics, Alternating Group Graph.
Eric Weisstein's World of Mathematics, Graph Automorphism.

Cf. A020725, A129464, A179442, A229020.

#[[1]] #[[3]] & /@ Partition[Range[0, 20]!, 3, 1] (* Harvey P. Dale, Jun 20 2017 *)
Table[(n - 1)! (n + 1)!, {n, 20}] (* or *) Table[Gamma[n] Gamma[n + 2], {n, 20}] (* Eric W. Weisstein, Jul 30 2018 *)

a(n) = (n-1)! * (n+1)!; \\ Michel Marcus, Feb 03 2016

a(n) = (Product_{k=1..n} k*A020725(k)) / n;
a(n) = (Product_{k=1..n} k*(k+1)) / n.
a(n) = (-1) * A129464(n) = A179442(n) * n. [Jaroslav Krizek, Jul 22 2010]
a(n) = 2*A010791(n-1). - Michel Marcus, Feb 03 2016
Sum_{n>=1} 1/a(n) = A229020 = BesselI(2,2). - Amiram Eldar, Oct 07 2020

A302911 Determinant of n X n matrix whose main diagonal consists of the first n 7-gonal numbers and all other elements are 1's.

Original entry on oeis.org

1, 6, 102, 3366, 181764, 14541120, 1614064320, 237267455040, 44606281547520, 10437869882119680, 2974792916404108800, 1014404384493801100800, 407790562566508042521600, 190845983281125763900108800, 102865984988526786742158643200

Offset: 1

Muniru A Asiru, Apr 15 2018

nonn

			The matrix begins:
  1   1   1   1   1   1   1 ...
  1   7   1   1   1   1   1 ...
  1   1  18   1   1   1   1 ...
  1   1   1  34   1   1   1 ...
  1   1   1   1  55   1   1 ...
  1   1   1   1   1  81   1 ...
  1   1   1   1   1   1 112 ...

Cf. A000566 (heptagonal numbers).

Cf. Determinant of n X n matrix whose main diagonal consists of the first n k-gonal numbers and all other elements are 1's: A000142 (k=2), A067550 (k=3), A010791 (k=4, with offset 1), A302909 (k=5), A302910 (k=6), this sequence (k=7), A302912 (k=8), A302913 (k=9), A302914 (k=10).

d:=(i,j)->`if`(i<>j,1,i*(5*i-3)/2):
seq(LinearAlgebra[Determinant](Matrix(n,d)),n=1..20);

nmax = 20; Table[Det[Table[If[i == j, i*(5*i - 3)/2, 1], {i, 1, k}, {j, 1, k}]], {k, 1, nmax}] (* Vaclav Kotesovec, Apr 16 2018 *)
Table[FullSimplify[5^(n + 1) * Gamma[n] * Gamma[n + 7/5] / (7 * Gamma[2/5] * 2^n)], {n, 1, 15}] (* Vaclav Kotesovec, Apr 16 2018 *)

a(n) = matdet(matrix(n, n, i, j, if (i!=j, 1, i*(5*i-3)/2))); \\ Michel Marcus, Apr 16 2018

From Vaclav Kotesovec, Apr 16 2018: (Start)
a(n) = 5^(n + 1) * Gamma(n) * Gamma(n + 7/5) / (7 * Gamma(2/5) * 2^n).
a(n) ~ Pi * 5^(n+1) * n^(2*n + 2/5) / (7 * Gamma(2/5) * 2^(n-1) * exp(2*n)).
a(n+1) = a(n) * n*(5*n + 7)/2.
(End)

A302912 Determinant of n X n matrix whose main diagonal consists of the first n 8-gonal numbers and all other elements are 1's.

Original entry on oeis.org

1, 7, 140, 5460, 349440, 33196800, 4381977600, 766846080000, 171773521920000, 47924812615680000, 16294436289331200000, 6631835569757798400000, 3183281073483743232000000, 1779454120077412466688000000, 1145968453329853628547072000000

Offset: 1

Muniru A Asiru, Apr 15 2018

nonn

			The matrix begins:
  1   1   1   1   1   1   1 ...
  1   8   1   1   1   1   1 ...
  1   1  21   1   1   1   1 ...
  1   1   1  40   1   1   1 ...
  1   1   1   1  65   1   1 ...
  1   1   1   1   1  96   1 ...
  1   1   1   1   1   1 133 ...

Cf. A000567 (octagonal numbers).

Cf. Determinant of n X n matrix whose main diagonal consists of the first n k-gonal numbers and all other elements are 1's: A000142 (k=2), A067550 (k=3), A010791 (k=4, with offset 1), A302909 (k=5), A302910 (k=6), A302911 (k=7), this sequence (k=8), A302913 (k=9), A302914 (k=10).

d:=(i,j)->`if`(i<>j,1,i*(3*i-2)):
seq(LinearAlgebra[Determinant](Matrix(n,d)),n=1..16);

nmax = 20; Table[Det[Table[If[i == j, i*(3*i - 2), 1], {i, 1, k}, {j, 1, k}]], {k, 1, nmax}] (* Vaclav Kotesovec, Apr 16 2018 *)
Table[FullSimplify[3^(n+1) * Gamma[n] * Gamma[n + 4/3] / (4*Gamma[1/3])], {n, 1, 15}] (* Vaclav Kotesovec, Apr 16 2018 *)RecurrenceTable[{a[n+1] == a[n] * n * (3*n + 4), a[1] == 1}, a, {n, 1, 20}] (* Vaclav Kotesovec, Apr 16 2018 *)

a(n) = matdet(matrix(n, n, i, j, if (i!=j, 1, i*(3*i-2)))); \\ Michel Marcus, Apr 16 2018

From Vaclav Kotesovec, Apr 16 2018: (Start)
a(n) = 3^(n+1) * Gamma(n) * Gamma(n + 4/3) / (4*Gamma(1/3)).
a(n) ~ Pi * 3^(n+1) * n^(2*n + 1/3) / (2 * Gamma(1/3) * exp(2*n)).
a(n+1) = a(n) * n*(3*n + 4).
(End)

cp's OEIS Frontend

A079484 a(n) = (2n-1)!! * (2n+1)!!, where the double factorial is A006882.

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A302909 Determinant of n X n matrix whose main diagonal consists of the first n 5-gonal numbers and all other elements are 1's.

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A302910 Determinant of n X n matrix whose main diagonal consists of the first n 6-gonal numbers and all other elements are 1's.

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A010792 a(n) = n!*(n+3)! / 3!.

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A010793 a(n) = n!*(n+4)! / 4!.

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A010794 a(n) = n!*(n+5)!/5!.

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