A002673 Numerators of central difference coefficients M_{3}^(2n+1). (Formerly M4894 N2097)

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

Offset

1,3

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

Table of n, a(n) for n=1..23.

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.

Sequence in context: A201121 A116153 A116209 * A081301 A159527 A139277

Adjacent sequences: A002670 A002671 A002672 * A002674 A002675 A002676

Keyword

nonn,frac

Author

N. J. A. Sloane

Status

approved