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A007472 Shifts 2 places left when binomial transform is applied twice with a(0) = a(1) = 1.

%I A007472 M2812 #57 Apr 25 2025 16:52:37
%S A007472 1,1,1,3,9,29,105,431,1969,9785,52145,296155,1787385,11428949,
%T A007472 77124569,546987143,4062341601,31502219889,254500383457,2137863653811,
%U A007472 18639586581097,168387382189709,1573599537048265,15189509662516063,151243491212611217,1551565158004180137
%N A007472 Shifts 2 places left when binomial transform is applied twice with a(0) = a(1) = 1.
%D A007472 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A007472 Alois P. Heinz, <a href="/A007472/b007472.txt">Table of n, a(n) for n = 0..576</a>
%H A007472 M. Bernstein and N. J. A. Sloane, <a href="https://arxiv.org/abs/math/0205301">Some canonical sequences of integers</a>, arXiv:math/0205301 [math.CO], 2002; Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
%H A007472 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 A007472 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A007472 G.f. A(x) satisfies: A(x) = 1 + x + x^2 * A(x/(1 - 2*x)) / (1 - 2*x). - _Ilya Gutkovskiy_, Jan 30 2022
%F A007472 E.g.f.: (BesselK(0, 1) + BesselK(1, 1)) * BesselI(0, exp(x)) + (BesselI(1, 1) - BesselI(0, 1)) * BesselK(0, exp(x)). - _Ven Popov_, Apr 25 2025
%p A007472 bintr:= proc(p) local b;
%p A007472           b:= proc(n) option remember; add(p(k)*binomial(n,k), k=0..n) end
%p A007472         end:
%p A007472 b:= (bintr@@2)(a):
%p A007472 a:= n-> `if`(n<2, 1, b(n-2)):
%p A007472 seq(a(n), n=0..30);  # _Alois P. Heinz_, Oct 18 2012
%t A007472 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_ *)
%t A007472 (* another program *)
%t A007472 B[x_] := (BesselK[0, 1] + BesselK[1, 1])*BesselI[0, Exp[x]] + (BesselI[1, 1] - BesselI[0, 1])*BesselK[0, Exp[x]];
%t A007472 a[n_] := SeriesCoefficient[FullSimplify[Series[B[x], {x, 0, n}]],n] n!
%t A007472 Table[a[n], {n, 0, 30}] (* _Ven Popov_, Apr 25 2025 *)
%Y A007472 Cf. A351028, A351143.
%Y A007472 Row sums of triangle A383235.
%K A007472 nonn,nice,eigen
%O A007472 0,4
%A A007472 _N. J. A. Sloane_