A002677 Denominators of coefficients for central differences M_{3}'^(2*n+1). (Formerly M3676 N1499)

1, 4, 40, 12096, 604800, 760320, 217945728000, 697426329600, 16937496576000, 30964207376793600, 187333454629601280000, 111407096483020800000, 1814811575069725360128000000, 10162944820390462016716800000

Offset

1,2

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

Table of n, a(n) for n=1..14.

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.

H. E. Salzer, Annotated scanned copy of left side of Table III

Formula

From Peter Bala, Oct 03 2019: (Start)

a(n) are the denominators in the expansion of (1/2)*(d/dx)(2*sinh(sqrt(x)/2))^4 =

x + (1/4)*x^2 + (1/40)*x^3 + (17/12096)*x^4 + (31/604800)*x^5 + ...

The a(n) also appear as denominators in the difference formula: (1/2)*f(x+2) - f(x+1) + f(x-1) - (1/2)*f(x-2) = D^3(f(x)) + (1/4)*D^5(f(x)) + (1/40)*D^7(f(x)) + (17/12096)*D^9(f(x)) + ..., where D denotes the differential operator d/dx.

(End)

Maple

gf := (sinh(2*sqrt(x)) - 2*sinh(sqrt(x)))/sqrt(x): ser := series(gf, x, 20):
seq(denom(coeff(ser, x, n)), n=1..14); # Peter Luschny, Oct 05 2019

Crossrefs

Numerators are A002675.

Cf. A002671, A002672, A002673, A002674, A002676.

Sequence in context: A000841 A059918 A296101 * A119527 A074991 A294972

Adjacent sequences: A002674 A002675 A002676 * A002678 A002679 A002680

Keyword

nonn,frac

Author

N. J. A. Sloane

Extensions

More terms from Sean A. Irvine, Dec 20 2016

Status

approved