OFFSET
0,2
COMMENTS
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)
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
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.
FORMULA
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)
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
Row sums of A225076. - Roger L. Bagula, Apr 27 2013
KEYWORD
nonn,easy,changed
AUTHOR
EXTENSIONS
More terms from Michael Somos
STATUS
approved