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%I M5174 N2246 #71 Mar 28 2023 10:49:06
%S 1,24,1920,322560,92897280,40874803200,25505877196800,
%T 21424936845312000,23310331287699456000,31888533201572855808000,
%U 53572735778642397757440000,108431217215972213061058560000
%N a(n) = 4^n*(2*n+1)!.
%C From _Sanjar Abrarov_, Mar 30 2019: (Start)
%C There is a formula for numerical integration (see MATLAB Central file ID# 71037):
%C 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.
%C 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)!).
%C Therefore, this sequence also occurs in the denominator of the numerical integration formula at M = 1. (End)
%C From _Peter Bala_, Oct 03 2019: (Start)
%C Denominators in the expansion of 2*sinh(x/2) = x + x^3/24 + x^5/1920 + x^7/322560 + ....
%C 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)
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Delbert L. Johnson, <a href="/A002671/b002671.txt">Table of n, a(n) for n = 0..201</a>
%H S. M. Abrarov and B. M. Quine, <a href="https://www.mathworks.com/matlabcentral/fileexchange/71037-array-numerical-integration-by-enhanced-midpoint-rule">Array numerical integration by enhanced midpoint rule</a>, MATLAB Central file ID #: 71037.
%H H. E. Salzer, <a href="https://doi.org/10.1002/sapm1963421162">Tables of coefficients for obtaining central differences from the derivatives</a>, 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 H. E. Salzer, <a href="/A002673/a002673.png">Annotated scanned copy of left side of Table I</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CentralDifference.html">Central Difference</a>.
%H <a href="/index/Di#divseq">Index to divisibility sequences</a>.
%F a(n) = 16^n * Pochhammer(1,n) * Pochhammer(3/2,n). - _Roger L. Bagula_, Apr 26 2013
%F From _Amiram Eldar_, Apr 09 2022: (Start)
%F Sum_{n>=0} 1/a(n) = 2*sinh(1/2).
%F Sum_{n>=0} (-1)^n/a(n) = 2*sin(1/2). (End)
%t a[n_] := 4^n*(2*n + 1)!; Array[a, 12, 0] (* _Amiram Eldar_, Apr 09 2022 *)
%o (PARI) a(n)=4^n*(2*n+1)!
%Y A bisection of A002866 and (apart from initial term) also a bisection of A007346.
%Y Row sums of A225076. - _Roger L. Bagula_, Apr 27 2013
%Y Cf. A002672, A002673, A002674, A002675, A002676, A002677.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_ and _Simon Plouffe_
%E More terms from _Michael Somos_
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