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

1,3

Comments

Euler transform of a(n) - if( n%4, 0, a(n/2)) is sequence itself with offset 0.

Links

Alois P. Heinz, Table of n, a(n) for n = 1..650

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

Cf. A045648, A000081, A004111.

Sequence in context: A027056 A024428 A330269 * A318797 A318850 A052327

Adjacent sequences: A049072 A049073 A049074 * A049076 A049077 A049078

Keyword

nonn,eigen

Author

Michael Somos, Aug 08 1999

Status

approved