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A010747 Shifts 6 places left under inverse binomial transform.

%I A010747 #33 Mar 08 2023 06:27:09
%S A010747 1,2,4,8,16,32,1,1,1,1,1,1,-62,315,-1002,2505,-5377,10373,-18544,
%T A010747 32086,-60468,154687,-567986,2422043,-10225382,40740231,-152497274,
%U A010747 539809668,-1822828757,5928049329,-18782176673,58918636670,-187382010256,623524250516
%N A010747 Shifts 6 places left under inverse binomial transform.
%H A010747 Alois P. Heinz, <a href="/A010747/b010747.txt">Table of n, a(n) for n = 0..802</a>
%H A010747 M. Bernstein and N. J. A. Sloane, <a href="https://arxiv.org/abs/math/0205301">Some canonical sequences of integers</a>, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210; arXiv:math/0205301 [math.CO], 2002.
%H A010747 M. Bernstein and N. J. A. Sloane, <a href="/A003633/a003633_1.pdf">Some canonical sequences of integers</a>, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
%H A010747 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A010747 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
%F A010747 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
%p A010747 a:= proc(n) option remember; (m-> `if`(m<0, 2^n,
%p A010747       add(a(m-j)*binomial(m, j)*(-1)^j, j=0..m)))(n-6)
%p A010747     end:
%p A010747 seq(a(n), n=0..35);  # _Alois P. Heinz_, Feb 02 2022
%t A010747 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];
%t A010747 Table[a[n], {n, 0, 35}] (* _Jean-François Alcover_, Mar 08 2023, after _Alois P. Heinz_ *)
%Y A010747 Cf. A010739, A010741, A010743, A010745.
%K A010747 sign,eigen
%O A010747 0,2
%A A010747 _N. J. A. Sloane_, _Jonas Wallgren_