A002672 Denominators of central difference coefficients M_{3}^(2n+1).

1, 8, 1920, 193536, 154828800, 1167851520, 892705701888000, 1428329123020800, 768472460034048000, 4058540589291090739200, 196433364521688791777280000, 5957759187690780937420800000, 30447485794244997427545243648000000, 341011840895543971188506728857600000

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• frac

Comments

From Peter Bala, Oct 03 2019: (Start)

Denominators in the expansion of (2*sinh(x/2))^3 = x^3 + (1/8)*x^5 + (13/1920)*x^7 + (41/193536)*x^9 + ....

Let f(x) be a polynomial in x. The expansion of (2*sinh(x/2))^3 leads to a formula for the third central differences: f(x+3/2) - 3*f(x+1/2) + 3*f(x-1/2) - f(x-3/2) = (2*sinh(D/2))^3(f(x)) = D^3(f(x)) + (1/8)*D^5(f(x)) + (13/1920)* D^7(f(x)) + ..., where D denotes the differential operator d/dx. (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

• 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 I.
• E. W. Weisstein, Central Difference. From MathWorld--A Wolfram Web Resource.

Crossrefs

Cf. A002673 (for numerators). Cf. A002671, A002674, A002675, A002676, A002677.

Formula

a(n) = denominator(3! * m(3, 2 * n + 1) / (2 * n + 1)!) where m(k, k) = 1; m(k, q) = 0 for k = 0, k > q, or k + q odd; m(1, q) = 1/2^(q-1) for odd q; m(2, q) = 1 for even q; m(k, q+2) = m(k-2, q) + (k/2)^2 * m(k, q) otherwise. [From Salzer] - Sean A. Irvine, Dec 20 2016