A002673 Numerators of central difference coefficients M_{3}^(2n+1).

1, 1, 13, 41, 671, 73, 597871, 7913, 28009, 792451, 170549237, 19397633, 317733228541, 9860686403, 75397891, 170314355593, 2084647712458321, 29327731093, 168856464709124011, 3063310184201, 499338236699611, 535201577273701757, 23571643935246013553

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• frac

Comments

From Peter Bala, Oct 03 2019: (Start)

Numerators 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. A002671, A002672, A002674, A002675, A002676, A002677.