A049075 Eigensequence of a power series transformation.
1, 1, 2, 4, 8, 18, 43, 102, 247, 617, 1564, 4003, 10355, 27051, 71225, 188743, 503111, 1348301, 3630294, 9815159, 26637436, 72540432, 198162708, 542875096, 1491126550, 4105602719, 11329408543, 31328137525, 86795258650, 240898943969, 669730499207, 1864855943748
Offset: 1
Examples
x + x^2 + 2*x^3 + 4*x^4 + 8*x^5 + 18*x^6 + 43*x^7 + 102*x^8 + 247*x^9 + 617*x^10 + ...
Links
- Alois P. Heinz, Table of n, a(n) for n = 1..650
Programs
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Maple
with(numtheory): etr:= proc(p) local b; b:= proc(n) option remember; if n=0 then 1 else (add(d*p(d), d=divisors(n)) +add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n-1))/n fi end end: b:= etr(n-> a(n) -`if`(modp(n,4)<>0, 0,a(n/2))): a:= n-> b(n-1): seq(a(n), n=1..40); # Alois P. Heinz, Sep 06 2008
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Mathematica
s[ n_, k_ ] := s[ n, k ]=a[ n+1-k ]+If[ n<2k, 0, -s[ n-k, k ](-1)^k ]; a[ 1 ]=1; a[ n_ ] := a[ n ]=Sum[ a[ i ]s[ n-1, i ]i, {i, 1, n-1} ]/(n-1); Table[ a[ i ], {i, 1, 30} ] -
PARI
{a(n) = local(A=x); if( n<1, 0, for( k=1, n-1, A *= (1 + (-x)^k + x*O(x^n))^((-1)^k * polcoeff(A, k))); polcoeff(A, n))}
Formula
G.f.: A(x) = x exp(A(x) - A(-x^2)/2 + A(x^3)/3 - A(-x^4)/4 + ...). Also A(x) = Sum_{n >= 1} a(n)*x^n = x * Product_{n >= 1} (1+(-x)^n)^((-1)^n*a(n)).
G.f.: x prod_{n>0} (1-x^(4n))^a(2n)/(1-x^n)^a(n).
a(n) ~ c * d^n / n^(3/2), where d = 2.92045137601697174071599643..., c = 0.4299447159290328896620383... . - Vaclav Kotesovec, Aug 25 2014
Comments