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 A038044 Shifts left under transform T where Ta is a DCONV a. 15
 1, 1, 2, 4, 9, 18, 40, 80, 168, 340, 698, 1396, 2844, 5688, 11456, 22948, 46072, 92144, 184696, 369392, 739536, 1479232, 2959860, 5919720, 11842696, 23685473, 47376634, 94753940, 189519576, 379039152, 758102900, 1516205800 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 LINKS Reinhard Zumkeller, Table of n, a(n) for n = 1..1000 N. J. A. Sloane, Transforms FORMULA From Benoit Cloitre, Aug 29 2004: (Start) a(n+1) = Sum_{d|n} a(d)*a(n/d), a(1) = 1. a(prime(k)+1) = 2*a(prime(k)); a(n) is asymptotic to c*2^n where c=0.353030198... (End) 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] MAPLE 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; MATHEMATICA 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 *) PROG (Haskell) import Data.Function (on) a038044 n = a038044_list !! (n-1) a038044_list = 1 : f 1 [1] where    f x ys = y : f (x + 1) (y:ys) where      y = sum \$ zipWith ((*) `on` a038044) divs \$ reverse divs          where divs = a027750_row x -- Reinhard Zumkeller, Jan 21 2014 CROSSREFS Positions of odd terms are given by A003095. Other self-convolved sequences: A000108, A007460 - A007464, A025192, A061922, A062177. Column k=1 of A144324 and A144823. - Alois P. Heinz, Nov 04 2012 Cf. A038040. Sequence in context: A219755 A289846 A193201 * A189911 A026732 A171003 Adjacent sequences:  A038041 A038042 A038043 * A038045 A038046 A038047 KEYWORD nonn,eigen AUTHOR STATUS approved

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Last modified April 13 19:34 EDT 2021. Contains 342941 sequences. (Running on oeis4.)