cp's OEIS Frontend

A357786 a(n) = coefficient of x^n, n >= 1, in A(x) such that: A(x)^2 = A( x^2/(1 - 4*x - 8*x^2) ) * sqrt(1 - 4*x - 8*x^2).

%I A357786 #13 Dec 04 2022 08:34:24
%S A357786 1,1,5,20,98,483,2499,13182,71030,388484,2152982,12061840,68212585,
%T A357786 388886050,2232764700,12898728750,74923372563,437303591874,
%U A357786 2563373794884,15083551143318,89060360731377,527477003037984,3132774700791126,18652891302520806,111314950683514698
%N A357786 a(n) = coefficient of x^n, n >= 1, in A(x) such that: A(x)^2 = A( x^2/(1 - 4*x - 8*x^2) ) * sqrt(1 - 4*x - 8*x^2).
%C A357786 Self convolution equals A357548.
%C A357786 Radius of convergence is r = (sqrt(57) - 5)/16, where r = r^2/(1 - 4*r - 8*r^2), with A(r) = sqrt(r).
%C A357786 Related identities:
%C A357786 (1) F(x)^2 = F( x^2/(1 - 4*x + 6*x^2) ) when F(x) = x/(1-2*x).
%C A357786 (2) C(x)^2 = C( x^2/(1 - 4*x + 4*x^2) ) when C(x) = (1-2*x - sqrt(1-4*x))/(2*x) is a g.f. of the Catalan numbers (A000108).
%F A357786 G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies:
%F A357786 (1) A(x) = -A( -x/(1 - 4*x) ) * sqrt(1 - 4*x).
%F A357786 (2) A(x)^2 = A( x^2/(1 - 4*x - 8*x^2) ) * sqrt(1 - 4*x - 8*x^2).
%F A357786 (3) A( x/(1 + 2*x) )^2 = A( x^2/(1 - 12*x^2) ) * sqrt(1 - 12*x^2) / (1 + 2*x).
%F A357786 (4) A( x/(1 + 2*x + 8*x^2) )^2 = A( x^2/(1 + 2^2*x^2 + 8^2*x^4) ) * sqrt(1 + 2^2*x^2 + 8^2*x^4) / (1 + 2*x + 8*x^2).
%e A357786 G.f.: A(x) = x + x^2 + 5*x^3 + 20*x^4 + 98*x^5 + 483*x^6 + 2499*x^7 + 13182*x^8 + 71030*x^9 + 388484*x^10 + 2152982*x^11 + ...
%e A357786 such that
%e A357786 A(x)^2 = A( x^2/(1 - 4*x - 8*x^2) ) * sqrt(1 - 4*x - 8*x^2)
%e A357786 where
%e A357786 A(x)^2 = x^2 + 2*x^3 + 11*x^4 + 50*x^5 + 261*x^6 + 1362*x^7 + 7344*x^8 + 40112*x^9 + 222338*x^10 + ... + A357548(n)*x^(n+1) + ...
%o A357786 (PARI) {a(n) = my(A=x); for(i=1, #binary(n+1),
%o A357786 A = sqrt( subst(A, x, x^2/(1 - 4*x - 8*x^2 +x*O(x^n)) )*sqrt(1 - 4*x - 8*x^2 +x*O(x^n)) )
%o A357786 ); polcoeff(H=A, n)}
%o A357786 for(n=1, 40, print1(a(n), ", "))
%Y A357786 Cf. A357548, A357785.
%K A357786 nonn
%O A357786 1,3
%A A357786 _Paul D. Hanna_, Dec 03 2022