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A038044 Shifts left under transform T where Ta is a DCONV a.

%I A038044 #27 May 20 2021 10:55:16
%S A038044 1,1,2,4,9,18,40,80,168,340,698,1396,2844,5688,11456,22948,46072,
%T A038044 92144,184696,369392,739536,1479232,2959860,5919720,11842696,23685473,
%U A038044 47376634,94753940,189519576,379039152,758102900,1516205800
%N A038044 Shifts left under transform T where Ta is a DCONV a.
%H A038044 Reinhard Zumkeller, <a href="/A038044/b038044.txt">Table of n, a(n) for n = 1..1000</a>
%H A038044 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A038044 From _Benoit Cloitre_, Aug 29 2004: (Start)
%F A038044 a(n+1) = Sum_{d|n} a(d)*a(n/d), a(1) = 1.
%F A038044 a(prime(k)+1) = 2*a(prime(k));
%F A038044 a(n) is asymptotic to c*2^n where c=0.353030198... (End)
%F A038044 G.f.: A(x) = Sum_{n>=1} a(n)*x^n = x * (1 + Sum_{i>=1} Sum_{j>=1} a(i)*a(j)*x^(i*j)). - _Ilya Gutkovskiy_, May 01 2019 [modified by _Ilya Gutkovskiy_, May 09 2019]
%F A038044 a(n+1) = Sum_{k=1..n} a(gcd(n,k))*a(n/gcd(n,k))/phi(n/gcd(n,k)) where phi = A000010. - _Richard L. Ollerton_, May 19 2021
%p A038044 with(numtheory); EIGENbyDIRCONV := proc(upto_n) local n,a,j,i,s,m; a := [1]; for i from 1 to upto_n do s := 0; m := convert(divisors(i),set); n := nops(m); for j from 1 to n do s := s+(a[m[j]]*a[m[(n-j)+1]]); od; a := [op(a),s]; od; RETURN(a); end;
%t A038044 dc[b_, c_] := Module[{p}, p[n_] := p[n] = Sum[b[d]*c[n/d], {d, If[n<0, {}, Divisors[n]]}]; p]; A[n_, k_] := Module[{f, b, t}, b[1] = dc[f, f]; For[t = 2, t <= k, t++, b[t] = dc[b[t-1], b[t-1]]]; f = Function[m, If[m == 1, 1, b[k][m-1]]]; f[n]]; a[n_] := A[n, 1]; Array[a, 40] (* _Jean-François Alcover_, Mar 20 2017, after A144324 *)
%o A038044 (Haskell)
%o A038044 import Data.Function (on)
%o A038044 a038044 n = a038044_list !! (n-1)
%o A038044 a038044_list = 1 : f 1 [1] where
%o A038044    f x ys = y : f (x + 1) (y:ys) where
%o A038044      y = sum $ zipWith ((*) `on` a038044) divs $ reverse divs
%o A038044          where divs = a027750_row x
%o A038044 -- _Reinhard Zumkeller_, Jan 21 2014
%Y A038044 Positions of odd terms are given by A003095. Other self-convolved sequences: A000108, A007460 - A007464, A025192, A061922, A062177.
%Y A038044 Column k=1 of A144324 and A144823. - _Alois P. Heinz_, Nov 04 2012
%Y A038044 Cf. A038040.
%Y A038044 Cf. A000010.
%K A038044 nonn,eigen
%O A038044 1,3
%A A038044 _Christian G. Bower_