A002671 a(n) = 4^n*(2*n+1)!. (Formerly M5174 N2246)

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

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.

Index to divisibility sequences.

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

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.

Sequence in context: A269338 A003738 A006685 * A328126 A225220 A262010

Adjacent sequences: A002668 A002669 A002670 * A002672 A002673 A002674

Keyword

nonn,easy

Author

N. J. A. Sloane and Simon Plouffe

Extensions

More terms from Michael Somos

Status

approved