A385617 G.f. A(x) satisfies A(x) = 1/( 1 - x*(A(x) + A(2*x)) ).
1, 2, 10, 82, 1062, 22646, 846570, 58644858, 7808479582, 2038568219422, 1054007965984050, 1084591195956246130, 2226674324358059364150, 9131600163886719149539590, 74851744440590132840318820090, 1226745312860243142951267683147178, 40204124737879503807503331117931168974
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Keywords
Programs
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Mathematica
terms = 17; A[] = 1; Do[A[x] = 1/( 1 - x*(A[x] + A[2*x]) ) + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, Jul 05 2025 *)
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PARI
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=0, i-1, (2^j+1)*v[j+1]*v[i-j])); v;
Formula
a(0) = 1; a(n) = Sum_{k=0..n-1} (2^k+1) * a(k) * a(n-1-k).
a(n) ~ c * 2^(n*(n-1)/2), where c = 30.250837358072598377515060923766952434821313428993180484... - Vaclav Kotesovec, Jul 05 2025