A007346 -id:A007346 - cp's OEIS frontend

A002671 a(n) = 4^n(2n+1)!.

1, 24, 1920, 322560, 92897280, 40874803200, 25505877196800, 21424936845312000, 23310331287699456000, 31888533201572855808000, 53572735778642397757440000, 108431217215972213061058560000

Offset: 0

N. J. A. Sloane and Simon Plouffe

From Sanjar Abrarov, Mar 30 2019: (Start)
There is a formula for numerical integration (see MATLAB Central file ID# 71037):
Integral_{x=0..1} f(x) dx = 2*Sum_{m=1..M} Sum_{n>=0} 1/((2*M)^(2*n + 1)*(2*n + 1)!)*f^(2*n)(x)|_x = (m - 1/2)/M, where the notation f^(2*n)(x)|_x = (m - 1/2)/M is the (2*n)-th derivative of the function f(x) at the points x = (m - 1/2)/M.
When we choose M = 1, then the corresponding coefficients are generated as 2*1/(2^(2*n + 1)*(2*n + 1)!) = 1/(4^n*(2*n + 1)!).
Therefore, this sequence also occurs in the denominator of the numerical integration formula at M = 1. (End)
From Peter Bala, Oct 03 2019: (Start)
Denominators in the expansion of 2*sinh(x/2) = x + x^3/24 + x^5/1920 + x^7/322560 + ....
If f(x) is a polynomial in x then the central difference f(x+1/2) - f(x-1/2) = 2*sinh(D/2)(f(x)) = D(f(x)) + (1/24)*D^3(f(x)) + (1/1920)*D^5(f(x)) + ..., where D denotes the differential operator d/dx. Formulas for higher central differences in terms of powers of the operator D can be obtained from the expansion of powers of the function 2*sinh(x/2). For example, the expansion (2*sinh(x/2))^2 = x^2 + (1/12)*x^4 + (1/360)*x^6 + .. leads to the second central difference formula f(x+1) - 2*f(x) + f(x-1) = D^2(f(x)) + (1/12)*D^4(f(x)) + (1/360)* D^6(f(x)) + .... See A002674. (End)

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Delbert L. Johnson, Table of n, a(n) for n = 0..201
S. M. Abrarov and B. M. Quine, Array numerical integration by enhanced midpoint rule, MATLAB Central file ID #: 71037.
H. E. Salzer, Tables of coefficients for obtaining central differences from the derivatives, Journal of Mathematics and Physics (this journal is also called Studies in Applied Mathematics), 42 (1963), 162-165, plus several inserted tables. [Note that there is a mistake in the definition of this sequence on line 2 of page 164.]
H. E. Salzer, Annotated scanned copy of left side of Table I.
Eric Weisstein's World of Mathematics, Central Difference.
Index to divisibility sequences.

A bisection of A002866 and (apart from initial term) also a bisection of A007346.
Row sums of A225076. - Roger L. Bagula, Apr 27 2013
Cf. A002672, A002673, A002674, A002675, A002676, A002677.

a[n_] := 4^n*(2*n + 1)!; Array[a, 12, 0] (* Amiram Eldar, Apr 09 2022 *)

PARI
```
a(n)=4^n*(2*n+1)!
```

a(n) = 16^n * Pochhammer(1,n) * Pochhammer(3/2,n). - Roger L. Bagula, Apr 26 2013
From Amiram Eldar, Apr 09 2022: (Start)
Sum_{n>=0} 1/a(n) = 2*sinh(1/2).
Sum_{n>=0} (-1)^n/a(n) = 2*sin(1/2). (End)

More terms from Michael Somos

A274299 A bisection of A002326.

Original entry on oeis.org

2, 3, 10, 4, 18, 11, 18, 5, 12, 12, 14, 23, 8, 20, 58, 6, 66, 35, 20, 39, 82, 28, 12, 36, 30, 51, 106, 36, 44, 24, 20, 7, 130, 36, 138, 60, 42, 15, 20, 52, 162, 83, 18, 60, 178, 60, 40, 95, 12, 99, 84, 66, 210, 28, 18, 37, 226, 30, 92, 119, 162, 36, 50, 8, 36, 131, 22, 135, 20, 30, 94, 60, 48, 116

Offset: 0

N. J. A. Sloane, Jun 20 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Steve Butler, Persi Diaconis and R. L. Graham, The mathematics of the flip and horseshoe shuffles, arXiv:1412.8533 [math.CO], 2014.
Steve Butler, Persi Diaconis and R. L. Graham, The mathematics of the flip and horseshoe shuffles, The American Mathematical Monthly 123.6 (2016): 542-556.

Cf. A002326, A007346, A274298.

Table[MultiplicativeOrder[2, 4 n + 3], {n, 0, 73}] (* Stan Wagon, Jun 24 2016 *)

A274305 Order of shuffle group generated by in- and out-horseshoe shuffles of a deck of 2n cards.

Original entry on oeis.org

2, 12, 120, 32, 3628800, 95040, 87178291200, 80, 6402373705728000, 1216451004088320000, 1124000727777607680000, 310224200866619719680000, 403291461126605635584000000, 152444172305856930250752000000, 265252859812191058636308480000000, 192, 295232799039604140847618609643520000000

Offset: 1

N. J. A. Sloane, Jun 20 2016

nonn

Jean-François Alcover, Table of n, a(n) for n = 1..64
Steve Butler, Persi Diaconis and R. L. Graham, The mathematics of the flip and horseshoe shuffles, arXiv:1412.8533 [math.CO], 2014.
Steve Butler, Persi Diaconis and R. L. Graham, The mathematics of the flip and horseshoe shuffles, The American Mathematical Monthly 123.6 (2016): 542-556.

Cf. A007346, A002326.

f:=proc(n) local k,i,np;
if n=1 then 2
elif n=2 then 12
elif n=3 then 120
elif n=6 then 12!/7!
elif (n mod 2) = 1 then (2*n)!
else
np:=n; k:=1;
for i while (np mod 2) = 0 do
   np:=np/2; k:=k+1; od;
   if (n=2^(k-1)) then (k+1)*2^k else (2*n)!/2; fi;
fi;
end;
[seq(f(n),n=1..64)];

a[n_] := Which[n == 1, 2, n == 2, 12, n == 3, 120, n == 6, 12!/7!, OddQ[n], (2 n)!, True, np = n; k = 1; While[EvenQ[np], np = np/2; k++]; If[n == 2^(k - 1), (k + 1)*2^k, (2n)!/2]];
Array[a, 17] (* Jean-François Alcover, Nov 30 2017, from Maple *)

See Maple program.

cp's OEIS Frontend

A274303 A bisection of A007346.

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A002671 a(n) = 4^n(2n+1)!.

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A274299 A bisection of A002326.

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A274305 Order of shuffle group generated by in- and out-horseshoe shuffles of a deck of 2n cards.

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Mathematica

Formula

cp's OEIS Frontend

A274303 A bisection of A007346.

Views

Author

Keywords

Crossrefs

A002671 a(n) = 4^n*(2*n+1)!.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A274299 A bisection of A002326.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

A274305 Order of shuffle group generated by in- and out-horseshoe shuffles of a deck of 2n cards.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A002671 a(n) = 4^n(2n+1)!.