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%I M2812 #31 Jan 30 2022 15:59:40
%S 1,1,1,3,9,29,105,431,1969,9785,52145,296155,1787385,11428949,
%T 77124569,546987143,4062341601,31502219889,254500383457,2137863653811,
%U 18639586581097,168387382189709,1573599537048265,15189509662516063,151243491212611217,1551565158004180137
%N Shifts 2 places left when binomial transform is applied twice.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Alois P. Heinz, <a href="/A007472/b007472.txt">Table of n, a(n) for n = 0..250</a>
%H M. Bernstein and N. J. A. Sloane, <a href="http://arXiv.org/abs/math.CO/0205301">Some canonical sequences of integers</a>, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
%H 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 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F G.f. A(x) satisfies: A(x) = 1 + x + x^2 * A(x/(1 - 2*x)) / (1 - 2*x). - _Ilya Gutkovskiy_, Jan 30 2022
%p bintr:= proc(p) local b;
%p b:=proc(n) option remember; add (p(k)*binomial(n,k), k=0..n) end
%p end:
%p b:= (bintr@@2)(a):
%p a:= n-> `if`(n<2, 1, b(n-2)):
%p seq (a(n), n=0..30); # _Alois P. Heinz_, Oct 18 2012
%t bintr[p_] := Module[{b}, b[n_] := b[n] = Sum [p[k]*Binomial[n, k], {k, 0, n}]; b]; b = a // bintr // bintr; a[n_] := If[n<2, 1, b[n-2]]; Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, Jan 27 2014, after _Alois P. Heinz_ *)
%K nonn,nice,eigen
%O 0,4
%A _N. J. A. Sloane_
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