A092042 -id:A092042 - cp's OEIS frontend

A047053 a(n) = 4^n * n!.

1, 4, 32, 384, 6144, 122880, 2949120, 82575360, 2642411520, 95126814720, 3805072588800, 167423193907200, 8036313307545600, 417888291992371200, 23401744351572787200, 1404104661094367232000, 89862698310039502848000

Offset: 0

Joe Keane (jgk(AT)jgk.org)

Original name was "Quadruple factorial numbers".
For n >= 1, a(n) is the order of the wreath product of the cyclic group C_4 and the symmetric group S_n. - Ahmed Fares (ahmedfares(AT)my-deja.com), May 07 2001
Number of n X n monomial matrices with entries 0, +/-1, +/-i.
a(n) is the product of the positive integers <= 4*n that are multiples of 4. - Peter Luschny, Jun 23 2011
Also, a(n) is the number of signed permutations of length 2*n that are equal to their reverse-complements. (See the Hardt and Troyka reference.) - Justin M. Troyka, Aug 13 2011.
Pi^n/a(n) is the volume of a 2*n-dimensional ball with radius 1/2. - Peter Luschny, Jul 24 2012
Equals the first right hand column of A167557, and also equals the first right hand column of A167569. - Johannes W. Meijer, Nov 12 2009
a(n) is the order of the group U_n(Z[i]) = {A in M_n(Z[i]): A*A^H = I_n}, the group of n X n unitary matrices over the Gaussian integers. Here A^H is the conjugate transpose of A. - Jianing Song, Mar 29 2021

			G.f. = 1 + 4*x + 32*x^2 + 384*x^3 + 6144*x^4 + 122880*x^5 + 2949120*x^6 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..100
R. B. Brent, Generalizing Tuenter's Binomial Sums, J. Int. Seq. 18 (2015), # 15.3.2.
CombOS - Combinatorial Object Server, Generate colored permutations
R. Coquereaux and J.-B. Zuber, Maps, immersions and permutations, arXiv preprint arXiv:1507.03163 [math.CO], 2015-2016. Also J. Knot Theory Ramifications 25, 1650047 (2016), DOI.
Sylvie Corteel and Lauren Williams, Tableaux Combinatorics for the Asymmetric Exclusion Process II, arXiv:0810.2916 [math.CO], 2008-2009.
A. Hardt and J. M. Troyka, Restricted symmetric signed permutations, Pure Mathematics and Applications, Vol. 23 (No. 3, 2012), pp. 179-217.
A. Hardt and J. M. Troyka, Slides (associated with the Hardt and Troyka reference above).
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 492.
Alexsandar Petojevic, The Function vM_m(s; a; z) and Some Well-Known Sequences, Journal of Integer Sequences, 5 (2002), Article 02.1.7.
M. D. Schmidt, Generalized j-Factorial Functions, Polynomials, and Applications, J. Int. Seq. 13 (2010), Article 10.6.7, p. 39.
Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the Hankel Transform, Journal of Integer Sequences, 9 (2006), Article 06.1.1.

Cf. A000142, A000165, A007696, A008545, A032031, A047058, A087299, A092042, A092616.

a(n)= A051142(n+1, 0) (first column of triangle).

[4^n*Factorial(n): n in [0..20]]; // Vincenzo Librandi, Jul 20 2011

A047053:= n -> mul(k, k = select(k-> k mod 4 = 0, [$1..4*n])): seq(A047053(n), n = 0.. 16); # Peter Luschny, Jun 23 2011

a[n_]:= With[{m=2n}, If[ m<0, 0, m!*SeriesCoefficient[1 +Sqrt[Pi]*x*Exp[x^2]*Erf[x], {x, 0, m}]]]; (* Michael Somos, Jan 03 2015 *)
Table[4^n n!,{n,0,20}] (* Harvey P. Dale, Sep 19 2021 *)

PARI
```
a(n)=4^n*n!;
```

a(n) = 4^n * n!.
E.g.f.: 1/(1 - 4*x).
Integral representation as the n-th moment of a positive function on a positive half-axis: a(n) = Integral_{x=0..oo} x^n*exp(-x/4)/4, n >= 0. This representation is unique. - Karol A. Penson, Jan 28 2002 [corrected by Jason Yuen, May 04 2025]
Sum_{k>=0} (-1)^k/(2*k + 1)^n = (-1)^n * n * (PolyGamma[n-1, 1/4] - PolyGamma[n-1, 3/4]) / a(n) for n > 0. - Joseph Biberstine (jrbibers(AT)indiana.edu), Jul 27 2006
a(n) = Sum_{k=0..n} C(n,k)*(2k)!*(2(n-k))!/(k!(n-k)!) = Sum_{k=0..n} C(n,k)*A001813(k)*A001813(n-k). - Paul Barry, May 04 2007
E.g.f.: With interpolated zeros, 1 + sqrt(Pi)*x*exp(x^2)*erf(x). - Paul Barry, Apr 10 2010
From Gary W. Adamson, Jul 19 2011: (Start)
a(n) = sum of top row terms of M^n, M = an infinite square production matrix as follows:
  2, 2, 0, 0, 0, 0, ...
  4, 4, 4, 0, 0, 0, ...
  6, 6, 6, 6, 0, 0, ...
  8, 8, 8, 8, 8, 0, ...
  ... (End)
G.f.: 1/(1 - 4*x/(1 - 4*x/(1 - 8*x/(1 - 8*x/(1 - 12*x/(1 - 12*x/(1 - 16*x/(1 - ... (continued fraction). - Philippe Deléham, Jan 08 2012
G.f.: 2/G(0), where G(k) = 1 + 1/(1 - 8*x*(k + 1)/(8*x*(k + 1) - 1 + 8*x*(k + 1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 30 2013
G.f.: 1/Q(0), where Q(k) = 1 - 4*x*(2*k + 1) - 16*x^2*(k + 1)^2/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Sep 28 2013
a(n) = A000142(n) * A000302(n). - Michel Marcus, Nov 28 2013
a(n) = A087299(2*n). - Michael Somos, Jan 03 2015
D-finite with recurrence: a(n) - 4*n*a(n-1) = 0. - R. J. Mathar, Jan 27 2020
From Amiram Eldar, Jun 25 2020: (Start)
Sum_{n>=0} 1/a(n) = e^(1/4) (A092042).
Sum_{n>=0} (-1)^n/a(n) = e^(-1/4) (A092616). (End)

Edited by Karol A. Penson, Jan 22 2002

A331501 Decimal expansion of exp(3/4).

Original entry on oeis.org

2, 1, 1, 7, 0, 0, 0, 0, 1, 6, 6, 1, 2, 6, 7, 4, 6, 6, 8, 5, 4, 5, 3, 6, 9, 8, 1, 9, 8, 3, 7, 0, 9, 5, 6, 1, 0, 1, 3, 4, 4, 9, 1, 5, 8, 4, 7, 0, 2, 4, 0, 3, 4, 2, 1, 7, 7, 9, 1, 3, 3, 0, 3, 0, 8, 1, 0, 9, 8, 4, 5, 3, 3, 3, 6, 4, 0, 1, 2, 8, 2, 0, 0, 0, 2, 7, 9, 1, 5, 6, 0, 2, 6, 6, 6, 1, 5, 7, 9, 8, 2, 1, 8, 8, 8

Offset: 1

Washington Bomfim, Feb 27 2020

Considering graph evolutions (see the Flajolet link) with 2n vertices initially isolated, the probability of the occurrence of an acyclic graph at the critical point n in the uniform model, will be denoted by P(n). In the case of the permutation model, the respective probability will be denoted by Pp(n).

Pp(n) / P(n) ~ exp(3/4) since Pp(n) = A302112(n) / A331505(2n) = A302112(n) / C(C(2n,2), n), and P(n) = A302112(n) * n! * 2^n / (2n)^(2n), Pp(n) / P(n) = (2n)^(2n) / (C(C(2n,2), n) * n! * 2^n), and lim_{n->oo} Pp(n) / P(n) = exp(3/4).

			2.1170000166126746685453698198370956101344915847024...

Philippe Flajolet, Donald E. Knuth, and Boris Pittel, The first cycles in an evolving graph, Discrete Mathematics, Vol. 75, No. 1-3 (1989), pp. 167-215.
Leonard Giugiuc and Dan Stefan Marinescu, Problem 4257, Crux Mathematicorum, Vol. 43, No. 6 (2017), pp. 263 and 265; Solution to Problem 4257, ibid., Vol. 44, No. 6 (2018), pp. 268-270.
Index entries for transcendental numbers

Cf. A000178, A001113, A019774, A092042, A302112, A331500, A331502, A331505.

Maple
```
evalf(exp(3/4), 134);
```

RealDigits[Exp[3/4], 10, 100][[1]] (* Amiram Eldar, Apr 12 2022 *)

exp(3/4) \\ Charles R Greathouse IV, Nov 21 2024

Equals lim_{n->oo} Pp(n) / P(n) = lim_{n->oo} (2*n)^(2*n) / (binomial(binomial(2n,2), n) * n! * 2^n).

Equals lim_{n->oo} sqrt(n)/A000178(n)^(1/(n*(n+1))) (Giugiuc and Marinescu, 2017). - Amiram Eldar, Apr 12 2022

A379411 a(n) = n + floor(ns/r) + floor(nt/r), where r = e^(1/4), s = e^(1/2), t = e^(3/4).

Original entry on oeis.org

3, 7, 10, 15, 19, 22, 26, 31, 34, 38, 43, 46, 50, 54, 58, 62, 66, 70, 74, 77, 81, 86, 89, 93, 98, 101, 105, 109, 113, 117, 121, 125, 129, 133, 136, 141, 145, 148, 153, 156, 160, 164, 168, 172, 176, 180, 184, 188, 191, 196, 200, 203, 208, 212, 215, 219, 223

Offset: 1

Clark Kimberling, Jan 18 2025

nonn

This sequence and A379412 and A379413 partition the positive integers; see A184812 for a proof. For each k in A000027, write "a" if k=A379411(n) for some n, "b" if k=A379412(n) for some n, and "c" if k=A379413(n) for some n. Concatenating these letters for k = 1,2,3,... spells the following infinite word:

cbacbcabcacbcbacbcabcacbcabcbcacbacbcabcbcacbacbcabccabcbacbcabccabcbacbcacbacbcabcbcacbacbcabccbacbacbcabccabcbacbcacbcabcbacbcacbcabcabccbacb...

Cf. A184812, A379412, A379413.

Cf. A092042, A019774, A331501.

r = E^(1/4); s = E^(1/2); t = E^(3/4);
Table[n + Floor[n*s/r] + Floor[n*t/r], {n, 1, 120}]  (* A379411 *)
Table[n + Floor[n*r/s] + Floor[n*t/s], {n, 1, 120}]  (* A379412 *)
Table[n + Floor[n*r/t] + Floor[n*s/t], {n, 1, 120}]  (* A379413 *)

a(n) = n + floor(n*r) + floor(n*r^2), where r = e^(1/4).

A379413 a(n) = n + floor(nr/t) + floor(ns/t), where r = e^(1/4), s = e^(1/2), t = e^(3/4).

Original entry on oeis.org

1, 4, 6, 9, 11, 13, 16, 18, 21, 23, 25, 28, 30, 32, 35, 37, 40, 42, 44, 47, 49, 52, 53, 56, 59, 61, 64, 65, 68, 71, 73, 75, 78, 80, 83, 85, 87, 90, 92, 95, 96, 99, 102, 104, 107, 108, 111, 114, 116, 118, 120, 123, 126, 128, 130, 132, 135, 138, 139, 142, 144

Offset: 1

Clark Kimberling, Jan 18 2025

nonn

This sequence and A379411 and A379412 partition the positive integers; see A378142 for a proof.

Cf. A378142, A379411, A379412.

Cf. A092042, A019774, A331501.

r = E^(1/4); s = E^(1/2); t = E^(3/4);
Table[n + Floor[n*s/r] + Floor[n*t/r], {n, 1, 120}]  (* A379411 *)
Table[n + Floor[n*r/s] + Floor[n*t/s], {n, 1, 120}]  (* A379412 *)
Table[n + Floor[n*r/t] + Floor[n*s/t], {n, 1, 120}]  (* A379413 *)

a(n) = n + floor(n/r) + floor(n*r^2), where r = e^(1/4).

A113550 a(n) = product of n successive numbers up to n, if n is odd a(n) = n(n-1).. = n!, if n is even a(n) = n(n+1)(n+2)... 'n' terms.

Original entry on oeis.org

1, 6, 6, 840, 120, 332640, 5040, 259459200, 362880, 335221286400, 39916800, 647647525324800, 6227020800, 1748648318376960000, 1307674368000, 6288139352883548160000, 355687428096000, 29051203810321992499200000, 121645100408832000, 167683548393178540705382400000

Offset: 1

Amarnath Murthy, Nov 03 2005

			a(3) = 3*2*1 = 6.
a(4) = 4*5*6*7 = 840.

G. C. Greubel, Table of n, a(n) for n = 1..350

Cf. A113549.

Cf. A019704, A073742, A092042, A092616, A333419, A367563.

n = 1; anfunc[n_] := (If [EvenQ[n], {an = n, Do[an = an*(n + i), {i, n - 1}]}, an = n! ]; an); Table[anfunc[n], {n, 1, 20}] (* Elizabeth A. Blickley (Elizabeth.Blickley(AT)gmail.com), Mar 10 2006 *)

a(2n-1) = (2n-1)!, a(2n) = (4n-1)!/(2n-1)!.
a(2n-1)*a(2n) = (4n-1)!.
Sum_{n>=1} 1/a(n) = sinh(1) + (sqrt(Pi)/2) * (exp(1/4) * erf(1/2) - exp(-1/4) * erfi(1/2)). - Amiram Eldar, Aug 15 2025

More terms from Elizabeth A. Blickley (Elizabeth.Blickley(AT)gmail.com), Mar 10 2006

A331502 Decimal expansion of exp(4/9).

Original entry on oeis.org

1, 5, 5, 9, 6, 2, 3, 4, 9, 7, 6, 0, 6, 7, 8, 0, 7, 1, 5, 5, 3, 3, 7, 0, 9, 2, 8, 6, 9, 7, 9, 4, 7, 1, 1, 8, 6, 3, 9, 4, 8, 2, 4, 0, 1, 1, 4, 2, 2, 1, 4, 2, 3, 5, 4, 3, 9, 0, 2, 7, 8, 4, 3, 1, 5, 4, 3, 5, 6, 3, 8, 5, 0, 1, 3, 3, 1, 0, 6, 3, 2, 6, 4, 2, 7, 5, 8, 1, 6, 1, 2, 4, 9, 2, 9, 9, 4, 0, 1, 5, 4, 2, 9, 1, 6, 9

Offset: 1

Washington Bomfim, Mar 04 2020

Considering graph evolutions (see the Flajolet link) with 3n vertices initially isolated, the probability of the occurrence of an acyclic graph at the point n, (n = 1/3 * 3n), in the uniform model, will be denoted by P13(n). In the case of the permutation model, the respective probability will be denoted by Pp13(n).
Pp13(n) / P13(n) ~ exp(4/9) since Pp13(n) = f(n) / C(N,n), where f(n) is the number of labeled forests with 3n nodes and n edges, and C(N,n), N = 3n *(3n-1)/2 (see the Lucatero link) is the number of labeled graphs with 3n nodes and n edges.
Because P13(n) = f(n)* n!* 2^n / (3n)^(2n), Pp13(n) / P13(n) = (3n)^(2n) / (C(N,n)* n! *2^n), and Lim_{n->oo} Pp13(n) / P13(n) = exp(4/9).

			1.55962349760678071553370928697947118639482401142214...

P. Flajolet, D. E. Knuth, and B. Pittel, The first cycles in an evolving graph, Discrete Mathematics, 75(1-3):167-215, 1989.
Carlos R. Lucatero, Combinatorial Enumeration of Graphs.
Index entries for transcendental numbers

Cf. A001113, A019774, A092042, A302112, A331500, A331501, A331505.

Maple
```
evalf(exp(4/9), 134);
```

RealDigits[Exp[4/9],10,120][[1]] (* Harvey P. Dale, Jun 05 2023 *)

PARI
```
exp(4/9)
```

Equals Lim_{n->oo} Pp13(n) / P13(n) = Lim_{n->oo} (3*n)^(2*n) / (binomial((3*n *(3*n-1)/2), n) * n! * 2^n).

A367549 Decimal expansion of 1 - DawsonF(1/2).

Original entry on oeis.org

5, 7, 5, 5, 6, 3, 6, 1, 6, 4, 9, 7, 9, 7, 7, 7, 0, 4, 0, 6, 5, 9, 5, 7, 6, 4, 7, 5, 1, 0, 3, 3, 0, 4, 2, 8, 9, 0, 3, 5, 7, 0, 5, 2, 2, 6, 4, 0, 3, 0, 7, 9, 6, 1, 8, 4, 8, 6, 6, 0, 3, 0, 3, 3, 6, 6, 7, 5, 4, 8, 4, 5, 2, 4, 0, 4, 0, 8, 0, 5, 2, 3, 8, 3, 2, 2, 8, 7, 9, 8, 7, 1, 5, 2, 1, 3, 8, 7, 7, 7, 8, 5, 7, 4, 0, 3, 8, 3, 0, 2

Offset: 0

Peter Luschny, Nov 23 2023

			0.57556361649797770406595764751033042890357052264030796184866030336675484524040...

Eric Weisstein's World of Mathematics, Dawson's Integral.

Cf. A019704, A092042, A367563.

1 - sqrt(Pi/4)*erfi(1/2)/exp(1/4): evalf(%, 109);

N[1 - DawsonF[1/2], 110] // RealDigits // First

Equals 1 - sqrt(Pi/4) * erfi(1/2) / exp(1/4) = 1 - A019704 * A367563 / A092042.
Let C denote the constant. Then:
2*C - 1 = Sum_{n>=0} (-1)^n / Pochhammer(n, n).
2*(C - 1) = Sum_{n>=1} (-1)^n*Gamma(n) / Gamma(2*n).
Equals Integral_{x=0..oo} exp(-x)*cos(sqrt(x)) dx. - Kritsada Moomuang, Jun 06 2025

cp's OEIS Frontend

A047053 a(n) = 4^n * n!.

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A331501 Decimal expansion of exp(3/4).

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A379411 a(n) = n + floor(n*s/r) + floor(n*t/r), where r = e^(1/4), s = e^(1/2), t = e^(3/4).

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A379413 a(n) = n + floor(n*r/t) + floor(n*s/t), where r = e^(1/4), s = e^(1/2), t = e^(3/4).

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A113550 a(n) = product of n successive numbers up to n, if n is odd a(n) = n*(n-1)*.. = n!, if n is even a(n) = n(n+1)(n+2)... 'n' terms.

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A331502 Decimal expansion of exp(4/9).

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Maple

Mathematica

PARI

Formula

A367549 Decimal expansion of 1 - DawsonF(1/2).

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Maple

A379411 a(n) = n + floor(ns/r) + floor(nt/r), where r = e^(1/4), s = e^(1/2), t = e^(3/4).

A379413 a(n) = n + floor(nr/t) + floor(ns/t), where r = e^(1/4), s = e^(1/2), t = e^(3/4).

A113550 a(n) = product of n successive numbers up to n, if n is odd a(n) = n(n-1).. = n!, if n is even a(n) = n(n+1)(n+2)... 'n' terms.