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

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

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]

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

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.

Cf. A000010.

Sequence in context: A219755 A289846 A193201 * A189911 A026732 A171003

Adjacent sequences: A038041 A038042 A038043 * A038045 A038046 A038047

Keyword

nonn,eigen

Author

Christian G. Bower

Status

approved