A377132 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).

1, 1, 2, 6, 28, 228, 3592, 113880, 7267952, 929696784, 237968445472, 121835841547872, 124758916812038592, 255505766282965942848, 1046551115668335283290240, 8573345713494489300568753536, 140465691975467799273799959144192, 4602779760325164559879331800453222656, 301647773810532495378626041621616755442176, 39537576990478498231890121766124629197694682624

Offset

0,3

Links

Table of n, a(n) for n=0..19.

Formula

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/(...)))))).

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

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).

Prog

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

Crossrefs

Cf. A002412, A015083, A016061, A376923.

Sequence in context: A370926 A119966 A256599 * A355208 A002047 A126340

Adjacent sequences: A377129 A377130 A377131 * A377133 A377134 A377135

Keyword

nonn

Author

Thomas Scheuerle, Oct 19 2024

Status

approved