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A188456
G.f.: 1 = Sum_{n>=0} a(n)*x^n*(1 - 2^n*x)^(n+1).
2
1, 1, 4, 44, 1216, 80640, 12460032, 4393091072, 3479212916736, 6113821454237696, 23602899265140031488, 198562423940692641316864, 3615246879908004653107773440, 141631725381846630255125115961344
OFFSET
0,3
COMMENTS
G.f. satisfies a variant of an identity of the Catalan numbers (A000108):
1 = Sum_{n>=0} A000108(n)*x^n*(1 - x)^(n+1).
FORMULA
0 = Sum_{k=0..[(n+1)/2]} (-1)^k*C(n-k+1,k)*2^(k*(n-k))*a(n-k) for n > 0.
EXAMPLE
G.f.: 1 = (1-x) + x*(1-2*x)^2 + 4*x^2*(1-4*x)^3 + 44*x^3*(1-8*x)^4 + 1216*x^4*(1-16*x)^5 + 80640*x^5*(1-32*x)^6 + ...
MATHEMATICA
a[0] = 1; a[n_] := a[n] = SeriesCoefficient[1-Sum[a[k]*x^k*(1-2^k*x)^(k+1), {k, 0, n-1}], {x, 0, n}];
Table[a[n], {n, 0, 13}] (* Jean-François Alcover, Dec 09 2017 *)
PROG
(PARI) {a(n)=polcoeff(1-sum(k=0, n-1, a(k)*x^k*(1-2^k*x+x*O(x^n))^(k+1)), n)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Mar 31 2011
STATUS
approved