A175430 -id:A175430 - cp's OEIS frontend

A010791 a(n) = n!*(n+2)!/2.

1, 3, 24, 360, 8640, 302400, 14515200, 914457600, 73156608000, 7242504192000, 869100503040000, 124281371934720000, 20879270485032960000, 4071457744581427200000, 912006534786239692800000, 232561666370491121664000000, 66977759914701443039232000000

Offset: 0

N. J. A. Sloane

Also determinant of n X n matrix with m(i,j) = i^2 if i=j, otherwise 1. - Robert G. Wilson v, Jan 28 2002
Partial products of positive values of A005563. - Jonathan Vos Post, Oct 21 2008
This sequence has been shown to contain infinitely many squares. From the Hong and Liu abstract: Recently Cilleruelo proved that the product Product_{k=1..n} (k^2 + 1) is a square only for n = 3 which confirms a conjecture of Amdeberhan, Medina and Moll. In this paper, we show that the sequence Product_{k=2..n} (k^2 - 1) contains infinitely many squares. Furthermore, we determine all squares in this sequence. We also give a formula for the p-adic valuation of the terms in this sequence. - Jonathan Vos Post, Oct 21 2008
Equals (-1)^n * (1, 1, 3, 24, 360, ...) dot (1, -4, 9, -16, 25, ...). E.g., a(4) = (1, 1, 3, 24, 360) dot (1, -4, 9, -16, 25) = 1 - 4 + 27 - 384 + 9000 = 8640. - Gary W. Adamson, Apr 21 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Javier Cilleruelo, Squares in (1^2+1)...(n^2+1), Journal of Number Theory 128:8 (2008), pp. 2488-2491.
Shaofang Hong and Xingjiang Liu, Squares in (2^2-1)...(n^2-1) and p-adic valuation, arXiv:0810.3366 [math.NT], 2008-2009.
Index entries for sequences related to factorial numbers.

Cf. A005563, A010792, A010793, A010794, A010795, A175430, A229020.

[Factorial(n)* Factorial(n+2) / 2: n in [0..20]]; // Vincenzo Librandi, Jun 11 2013

Maple
```
f := n->n!*(n+2)!/2;
```

Table[n!(n+2)!/2,{n,0,20}] (* Vladimir Joseph Stephan Orlovsky, Apr 03 2011 *)

a(n) = n!*(n+2)!/2; \\ Michel Marcus, Feb 03 2016

From Amiram Eldar, Sep 27 2022: (Start)
a(n) = A175430(n+1)/2.
Sum_{n>=0} 1/a(n) = 2*BesselI(2,2) = 2*A229020.
Sum_{n>=0} (-1)^n/a(n) = 2*BesselJ(2,2). (End)
a(n) = 1/([x^n] hypergeom([], [3], x)). - Peter Luschny, Sep 13 2024

A179442 a(n) = ((n-1)! * (n+1)!) / n.

Original entry on oeis.org

2, 3, 16, 180, 3456, 100800, 4147200, 228614400, 16257024000, 1448500838400, 158018273280000, 20713561989120000, 3212195459235840000, 581636820654489600000, 121600871304831959040000

Offset: 1

Jaroslav Krizek, Jul 14 2010

nonn

			a(5) = ((5-1)! * (5+1)!) / 5 = (4! * 6!) / 5 = (24 * 720) / 5 = 17280 / 5 = 3456.
a(5) = ((5 -1)!^2) * (5+1) = 24^2 * 6 = 3456.

Cf. A001044, A020725, A175430, A229020, A261879.

Table[(n - 1)!*(n + 1)!/n, {n, 1, 15}] (* Amiram Eldar, Jan 18 2021 *)

a(n) = (n-1)!^2*(n+1) \\ Charles R Greathouse IV, Oct 23 2023

a(n) = Product_{k=1..n} (k * A020725(k)) / (n^2) = Product_{k=1..n} (k * (k+1)) / (n^2).
a(n) = A175430(n) / n = A001044(n-1) * (n+1) = ((n -1)^2)! * (n+1).
G.f.: 1 + G(0), where G(k)= 1 + x*(k+1)/(1 - (k+2)/(k+2 + 1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 08 2013
From Amiram Eldar, Jan 18 2021: (Start)
Sum_{n>=1} 1/a(n) = BesselI(2,2) + BesselI(3,2) = A229020 + A261879.
Sum_{n>=1} (-1)^(n+1)/a(n) = BesselJ(2,2) - BesselJ(3,2). (End)

A179870 a(n) = ((n-1)! * (n+1)!) ^ 2.

Original entry on oeis.org

4, 36, 2304, 518400, 298598400, 365783040000, 842764124160000, 3344930808791040000, 21407557176262656000000, 209815467884550291456000000, 3021342737537524196966400000000, 61783437639904832303765913600000000

Offset: 1

Jaroslav Krizek, Jul 30 2010

Table[((n-1)!(n+1)!)^2,{n,15}] (* Harvey P. Dale, Jul 12 2020 *)

a(n)=(n-1)!^4*(n^4+2*n^3+n^2) \\ Charles R Greathouse IV, Jan 03 2013

a(n) = A175430(n) ^ 2 = ((Product_(k=1,2,...,n) k*A020725(k)) / n) ^ 2 = ((Product_(k=1,2,...,n) k*(k+1)) / n) ^ 2.

A370418 Triangle read by rows. T(n, k) = (n - k)! * (n + k)!.

Original entry on oeis.org

1, 1, 2, 4, 6, 24, 36, 48, 120, 720, 576, 720, 1440, 5040, 40320, 14400, 17280, 30240, 80640, 362880, 3628800, 518400, 604800, 967680, 2177280, 7257600, 39916800, 479001600, 25401600, 29030400, 43545600, 87091200, 239500800, 958003200, 6227020800, 87178291200

Offset: 0

Peter Luschny, Feb 27 2024

			Triangle starts:
[0]      1;
[1]      1,      2;
[2]      4,      6,     24;
[3]     36,     48,    120,     720;
[4]    576,    720,   1440,    5040,   40320;
[5]  14400,  17280,  30240,   80640,  362880,  3628800;
[6] 518400, 604800, 967680, 2177280, 7257600, 39916800, 479001600;

Cf. A010050 (main diagonal), A009445 (subdiagonal), A001044 (column 0), A175430 (column 1), A024420 (bisection is alternating sum).

Cf. A119502, A143084.

T := (n, k) -> (n - k)! * (n + k)!:
seq(seq(T(n, k), k = 0..n), n = 0..7);

Table[(n - k)!*(n + k)!, {n, 0, 7}, {k, 0, n}] // Flatten (* Michael De Vlieger, Mar 05 2024 *)

Sum_{k=0..n} (-1)^k*T(n, k) = n!^2 / 2 + (-1)^n * (2*n + 2)! / (2*n + 2)^2.

cp's OEIS Frontend

A010791 a(n) = n!*(n+2)!/2.

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A179442 a(n) = ((n-1)! * (n+1)!) / n.

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Mathematica

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A179870 a(n) = ((n-1)! * (n+1)!) ^ 2.

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Author

Keywords

Programs

Mathematica

PARI

Formula

A370418 Triangle read by rows. T(n, k) = (n - k)! * (n + k)!.

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Maple

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