A010747 Shifts 6 places left under inverse binomial transform.

1, 2, 4, 8, 16, 32, 1, 1, 1, 1, 1, 1, -62, 315, -1002, 2505, -5377, 10373, -18544, 32086, -60468, 154687, -567986, 2422043, -10225382, 40740231, -152497274, 539809668, -1822828757, 5928049329, -18782176673, 58918636670, -187382010256, 623524250516

Offset: 0

Author

N. J. A. Sloane, Jonas Wallgren

Keywords

• sign
• eigen

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..802
• 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

Crossrefs

Cf. A010739, A010741, A010743, A010745.

Programs

Maple

a:= proc(n) option remember; (m-> `if`(m<0, 2^n,
      add(a(m-j)*binomial(m, j)*(-1)^j, j=0..m)))(n-6)
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, Feb 02 2022

Mathematica

a[n_] := a[n] = Function[m, If[m < 0, 2^n, Sum[a[m - j]* Binomial[m, j]*(-1)^j, {j, 0, m}]]][n - 6];
Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Mar 08 2023, after Alois P. Heinz *)

Formula

G.f. A(x) satisfies: A(x) = 1 + 2*x + 4*x^2 + 8*x^3 + 16*x^4 + 32*x^5 + x^6*A(x/(1 + x))/(1 + x). - Ilya Gutkovskiy, Feb 02 2022

a(n) = 2^n for n<6; otherwise a(n) = Sum_{j=0..n-6} a(n-6-j)*binomial(n-6,j)*(-1)^j. - Michel Marcus, Mar 08 2023