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

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

Offset: 0

Author

N. J. A. Sloane and Simon Plouffe

Keywords

• nonn
• easy

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.
• Index to divisibility sequences.

Crossrefs

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.

Programs

Mathematica

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

PARI

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

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)

Extensions

More terms from Michael Somos