

A002671


a(n) = 4^n*(2*n+1)!.
(Formerly M5174 N2246)


16



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

0,2


COMMENTS

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)
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(x1/2) = 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(x1) = D^2(f(x)) + (1/12)*D^4(f(x)) + (1/360)* D^6(f(x)) + .... See A002674. (End)


REFERENCES

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).


LINKS



FORMULA

a(n) = 16^n * Pochhammer(1,n) * Pochhammer(3/2,n).  Roger L. Bagula, Apr 26 2013
Sum_{n>=0} 1/a(n) = 2*sinh(1/2).
Sum_{n>=0} (1)^n/a(n) = 2*sin(1/2). (End)


MATHEMATICA

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


PROG

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


CROSSREFS

A bisection of A002866 and (apart from initial term) also a bisection of A007346.


KEYWORD

nonn,easy


AUTHOR



EXTENSIONS



STATUS

approved



