A195067 G.f. A(x) satisfies A(x) = Sum{n>=0} x^n * A(2*n*x).
1, 1, 3, 17, 191, 4261, 189123, 16723689, 2949213319, 1037964817357, 729449200732395, 1024041038817726353, 2872628913886690237679, 16105674069113302453209781, 180504701103754829110217971731, 4044484405239396750189431682523833
Offset: 0
Keywords
Examples
G.f.: A(x) = 1 + x + 3*x^2 + 17*x^3 + 191*x^4 + 4261*x^5 +... where: A(x) = 1 + x*A(2*x) + x^2*A(4*x) + x^3*A(6*x) + x^4*A(8*x) + x^5*A(10*x) +...
Programs
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Mathematica
a[0] = 1; a[n_] := a[n] = Sum[2^k * (n-k)^k * a[k], {k, 0, n-1}]; Table[a[n], {n, 0, 20}] (* Vaclav Kotesovec, Jul 03 2025 *) -
PARI
{a(n)=local(A=1+x+x*O(x^n)); for(k=1, n, A=1+sum(j=1, n, x^j*subst(A,x,2*j*x))); polcoeff(A, n)} -
PARI
{a(n)=if(n==0, 1, sum(k=0, n-1, 2^k*(n-k)^k*a(k)))}
Formula
a(n) = Sum_{k=0..n-1} 2^k*(n-k)^k * a(k) for n>0 with a(0)=1.
a(n) ~ c * (1 + sqrt(3))^n * 2^(n*(n-3)/2), where c = 0.9296543230172164460137009343716233391546324099495685771220234877636263909188... - Vaclav Kotesovec, Jul 03 2025