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

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• frac

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

• 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

Crossrefs

Numerators are A002675.

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

Programs

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

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)

Extensions

More terms from Sean A. Irvine, Dec 20 2016