A118023 - OEIS

%I #16 Nov 04 2024 18:36:11

%S 1,1,3,19,243,6227,319251,32737427,6714170259,2754046149011,

%T 2259333156408723,3706972573115098515,12164337831474297132435,

%U 79833941280970262512121235,1047892334589811621056371520915

%N Column 0 of triangle A118022, where the matrix square of A118022 shifts each column up 1 row, dropping the main diagonal of powers of 2.

%C Numerators of the q-Catalan numbers for q = 1/2. - _John Keith_, Feb 19 2021

%F G.f.: 1 = Sum_{n>=0} a(n)*x^n*prod_{k=0, n} (1-2^k*x) with a(0)=1.

%F a(n) = 2^(n*(n-1)/2)*b(n) where b(0)=1 and b(n)=sum(i=0,n-1,b(i)*b(n-1-i)/2^i). - _Benoit Cloitre_, Oct 25 2006

%F G.f.: Sum_{n>=0} a(n)*x^n/2^(n*(n+1)/2) = 1/(1 - (x/2)/(1 - (x/2^2)/(1 - (x/2^3)/(1 - (x/2^4)/(1 - (x/2^5)/(1 - ...)))))), a continued fraction. - _Paul D. Hanna_, Sep 28 2012

%e 1 = (1-x) + 1*x*(1-x)*(1-2*x) + 3*x^2*(1-x)*(1-2*x)*(1-4*x) + 19*x^3*(1-x)*(1-2*x)*(1-4*x)*(1-8*x) + 243*x^4*(1-x)*(1-2*x)*(1-4*x)*(1-8*x)*(1-16*x) + ...

%o (PARI) {a(n)=if(n==0, 1, polcoeff(1-sum(k=0, n-1, a(k)*x^k*prod(j=0, k, 1-2^j*x+x*O(x^n))), n))}

%o {a(n)=local(CF=1+x*O(x^n)); for(k=1, n, CF=1/(1-x/2^(n-k+1)*CF)); 2^(n*(n+1)/2)*polcoeff(CF, n)} \\ _Paul D. Hanna_, Sep 28 2012

%Y Cf. A118022.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Apr 10 2006