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A049075
Eigensequence of a power series transformation.
13
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
1,3
COMMENTS
Euler transform of a(n) - if( n%4, 0, a(n/2)) is sequence itself with offset 0.
LINKS
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
EXAMPLE
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 + ...
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
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} ]
PROG
(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))}
CROSSREFS
KEYWORD
nonn,eigen
AUTHOR
Michael Somos, Aug 08 1999
STATUS
approved