A000996 Shifts 3 places left under binomial transform.

1, 0, 0, 1, 1, 1, 2, 6, 17, 44, 112, 304, 918, 3040, 10623, 38161, 140074, 528594, 2068751, 8436893, 35813251, 157448068, 713084042, 3315414747, 15805117878, 77273097114, 387692392570, 1996280632656, 10542604575130, 57034787751655, 315649657181821

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• eigen

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

• Alois P. Heinz, Table of n, a(n) for n = 0..300
• M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210; arXiv:math/0205301 [math.CO], 2002.
• M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
• N. J. A. Sloane, Transforms
• S. Tauber, On generalizations of the exponential function, Amer. Math. Monthly, 67 (1960), 763-767.

Crossrefs

Column k=3 of A143983.

Programs

Maple

a:= proc(n) option remember; local k; if n<=2 then [1,0,0][n+1] else 1+ add(binomial(n-3,k) *a(k), k=3..n-3) fi end: seq(a(n), n=0..29); # Alois P. Heinz, Sep 05 2008

Mathematica

a[n_] := a[n] = If[n <= 2 , {1, 0, 0}[[n+1]], 1+Sum [Binomial[n-3, k]*a[k], {k, 3, n-3}]]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 24 2014, after Alois P. Heinz *)

Formula

G.f. A(x) satisfies: A(x) = 1 + x^3 * A(x/(1 - x)) / (1 - x). - Ilya Gutkovskiy, Aug 09 2020

Extensions

More terms from Alois P. Heinz, Sep 05 2008