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The expansion of 1/(1 - G(x)*x), where G(x) is the ordinary generating function of the Carlitz-Riordan q-Catalan numbers for q = 2 (A015083).
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%I #12 Oct 19 2024 22:15:50

%S 1,1,2,6,28,228,3592,113880,7267952,929696784,237968445472,

%T 121835841547872,124758916812038592,255505766282965942848,

%U 1046551115668335283290240,8573345713494489300568753536,140465691975467799273799959144192,4602779760325164559879331800453222656,301647773810532495378626041621616755442176,39537576990478498231890121766124629197694682624

%N The expansion of 1/(1 - G(x)*x), where G(x) is the ordinary generating function of the Carlitz-Riordan q-Catalan numbers for q = 2 (A015083).

%F O.g.f.: Continued fraction expansion 1/(1 - x/(1 - 2^0*x/(1 - 2^1*x/(1 - 2^2*x/(1 - 2^3*x/(...)))))).

%F a(n) = Sum_{k=0..n-1} a(n - k - 1)*A376923(n - 1, k)*(-1)^k, with a(0) = 1.

%F The Hankel determinant of a(0)..a(2*n) is 2^A016061(n). The Hankel determinant of a(1)..a(2*n+1) is 2^A002412(n).

%o (PARI)

%o A376923(n, k) = if(n < 0, return(0),return(if(k == 0, return(2^n), T(n-1, k) + 2^(n-1)*T(n-2, k-1) )))

%o a(n) = if(n==0, 1, sum(k=0, n-1, a(n-k-1)*A376923(n-1, k)*(-1)^k))

%Y Cf. A002412, A015083, A016061, A376923.

%K nonn

%O 0,3

%A _Thomas Scheuerle_, Oct 19 2024