A188456 - OEIS

%I #18 Dec 09 2017 11:15:23

%S 1,1,4,44,1216,80640,12460032,4393091072,3479212916736,

%T 6113821454237696,23602899265140031488,198562423940692641316864,

%U 3615246879908004653107773440,141631725381846630255125115961344

%N G.f.: 1 = Sum_{n>=0} a(n)*x^n*(1 - 2^n*x)^(n+1).

%C G.f. satisfies a variant of an identity of the Catalan numbers (A000108):

%C 1 = Sum_{n>=0} A000108(n)*x^n*(1 - x)^(n+1).

%F 0 = Sum_{k=0..[(n+1)/2]} (-1)^k*C(n-k+1,k)*2^(k*(n-k))*a(n-k) for n > 0.

%e 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 + ...

%t 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}];

%t Table[a[n], {n, 0, 13}] (* _Jean-François Alcover_, Dec 09 2017 *)

%o (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)}

%Y Cf. A188455, A188457, A188458.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Mar 31 2011