A092605 -id:A092605 - cp's OEIS frontend

A334578 Double subfactorials: a(n) = (-1)^floor(n/2) * n!! * Sum_{i=0..floor(n/2)} (-1)^i/(n-2*i)!!.

1, 1, 1, 2, 5, 11, 29, 76, 233, 685, 2329, 7534, 27949, 97943, 391285, 1469144, 6260561, 24975449, 112690097, 474533530, 2253801941, 9965204131, 49583642701, 229199695012, 1190007424825, 5729992375301, 30940193045449, 154709794133126, 866325405272573

Offset: 0

Ryan Brooks, May 06 2020

nonn

			a(5) = (5*3*1)*(1/(1) - 1/(3*1) + 1/(5*3*1)) = 15-5+1 = 11.

Alois P. Heinz, Table of n, a(n) for n = 0..807

Even bisection gives A000354.

Cf. A000166, A006882.

a:= proc(n) option remember; `if`(n<2, [0$2, 1$2][n+3],
      (n-1)*a(n-2)+(n-2)*a(n-4))
    end:
seq(a(n), n=0..32);  # Alois P. Heinz, May 06 2020

RecurrenceTable[{a[0] == 1, a[1] == 1, a[n] == n a[n-2] + (-1)^Floor[n/2]}, a, {n, 0, 32}] (* Jean-François Alcover, Nov 27 2020 *)

a(n) = n*a(n-2) + (-1)^floor(n/2).
a(2n) = A000354(n).
From Ryan Brooks, Oct 25 2020: (Start)
a(2n)/A006882(2n) ~ 1/sqrt(e) = A092605.
a(2n+1)/A006882(2n+1) ~ sqrt(Pi/(2*e))*erfi(1/sqrt(2)) = A306858. (End)

A209050 Pyramid angle = arccos(1/sqrt(e)) in radians.

Original entry on oeis.org

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

Offset: 0

Robert G. Wilson v, Mar 04 2012

			= 0.91910665729358842204136171489674270575563073402035114...
= 52°.660932385299557203750879574284228798519442622806897...
= 52°39'.655943117973432225052774457053727911166557368413...
= 52°39'39.3565870784059335031664674232236746699934421048..."

Cf. A001113, A019774, A092605.

RealDigits[ ArcCos[ 1/Sqrt[ E]], 10, 111][[1]]

acos(exp(-1/2)) \\ Charles R Greathouse IV, Mar 05 2012

arccos(1/sqrt(e)) = arccos(1/sqrt(A001113)) = arccos(1/A019774) = arccos(A092605).

A217249 Decimal expansion of Pi^2/sqrt(e).

Original entry on oeis.org

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

Offset: 1

Bruno Berselli, Mar 20 2013

			5.9862176684954389749324507474236892501651010658993869833349491850614...

Cf. A002388, A092605, A222391.

Mathematica
```
RealDigits[Pi^2/Sqrt[E], 10, 90][[1]]
```

Equals A002388*A092605.

A352526 a(n) = Product_{k=0..n} Nimsum (2*k + 2), with Nimsum (2 + 2) = 0 replaced by 1.

Original entry on oeis.org

2, 2, 12, 48, 480, 3840, 53760, 645120, 11612160, 185794560, 4087480320, 81749606400, 2125489766400, 51011754393600, 1530352631808000, 42849873690624000, 1456895705481216000, 46620662575398912000, 1771585177865158656000, 63777066403145711616000, 2678636788932119887872000

Offset: 0

Peter McNair, Mar 19 2022

nonn

Nimsum 2*k + 2 = A004443(2*k).
Sum_{n>0} 1/a(n) = 1/sqrt(e) = A092605.
Sum_{n>0} 1/a(2*n-1) = sinh(1/2) = A334367.
Sum_{n>0} 1/a(2*n) = cosh(1/2) - 2*sinh(1/2).
a(n)/2^n = abs(A265376(n+1)) = Product_{k=0..n} Nimsum k + 1, with Nimsum 1 + 1 = 0 replaced by 1, n > 0.

Cf. A004443, A092605, A265376, A334367.

a[n_] := Product[If[k == 1, 1, BitXor[2*k, 2]], {k, 0, n}]; Array[a, 21, 0] (* Amiram Eldar, Mar 19 2022 *)

PARI
```
a(n) = 2*prod(k=2,n,bitxor(2*k, 2))
```

a(n) = 2*Product_{k=2..n} A004443(2*k).
a(n) = 2^(n-1)*(n+1)!/floor((n+1)/2), n > 0.
a(n) = 2^(n-1)*(1+(-1)^n)*((n-1)!+n!)-((-1)^n-1)*(2*n)!!/2, n > 0.
a(n) = 2*a(n-1)*(n+(-1)^n), n > 1, with a(1) = 2.

A369881 Decimal expansion of 1/(2*sqrt(e)).

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Feb 04 2024

			0.30326532985631671180189976749559022672095906774359...

D. M. Batinetu-Giurgiu, Problem 764, The Pentagon, Vol. 75, No. 2 (2016), p. 22.
Toyesh Prakash Sharma, The Applications of the Stirling's approximation to find limits, Revista Electronica MateInfo.ro, December 2020, pp. 44-49. See Problem 13, p. 49.

Cf. A001113, A019774, A019778, A066318, A092605, A187832, A249455.

Mathematica
```
RealDigits[1/(2*Sqrt[E]), 10, 105][[1]]
```
PARI
```
exp(-1/2)/2
```

Equals A092605 / 2.
Equals exp(-(1 + A187832)).
Equals Sum_{n>=1} (-1)^(n+1)/A066318(n).
Equals lim_{n->oo} sqrt(n)*(((n+1)!)^(1/(2*(n+1))) - (n!)^(1/(2*n))) (Batinetu-Giurgiu, 2016).

cp's OEIS Frontend

A334578 Double subfactorials: a(n) = (-1)^floor(n/2) * n!! * Sum_{i=0..floor(n/2)} (-1)^i/(n-2*i)!!.

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A209050 Pyramid angle = arccos(1/sqrt(e)) in radians.

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A217249 Decimal expansion of Pi^2/sqrt(e).

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A352526 a(n) = Product_{k=0..n} Nimsum (2*k + 2), with Nimsum (2 + 2) = 0 replaced by 1.

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A369881 Decimal expansion of 1/(2*sqrt(e)).

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