0,3

Definition is a variation of the following property of the Catalan numbers: A000108(n) = Sum_{k=0..[n/2]} ( [x^n] (x*A(x^2))^(n-2k+1) )^2.

a(n) = Sum_{k=0..[n/2]} ( [x^n] (x/A(x^2))^(n-2k+1) )^2 for n>0 with a(0)=1.

a(5) = 18 = 1^2 + (-4)^2 + (-1)^2;

a(6) = 30 = 1^2 + (-5)^2 + 0^2 + (-2)^2;

a(7) = 45 = 1^2 + (-6)^2 + 2^2 + (-2)^2;

a(8) = 80 = 1^2 + (-7)^2 + 5^2 + (-1)^2 + (-2)^2;

a(9) =147 = 1^2 + (-8)^2 + 9^2 + 0^2 + 1^2;

indicated by the initial coefficients of x^m*A(x^2)^-m:

A(x).........: 1,1,2,5,11,18,30,45,80,147,...

x^1*A(x^2)^-1: 1,_-1,__-1,___-2,___-2,.......

x^2*A(x^2)^-2: __1,_-2,___-1,___-2,_____1,...

x^3*A(x^2)^-3: ____1,__-3,____0,___-1,.......

x^4*A(x^2)^-4: ______1,___-4,____2,_____0,...

x^5*A(x^2)^-5: _________1,___-5,____5,.......

x^6*A(x^2)^-6: ____________1,___-6,_____9,...

x^7*A(x^2)^-7: _______________1,___-7,.......

x^8*A(x^2)^-8: __________________1,____-8,...

x^9*A(x^2)^-9: _____________________1,.......

x^10*A(x^2)^-10: _______________________1,...

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

Cf. A095892 (variant); A000108.

Sequence in context: A141778 A117877 A025200 this_sequence A132459 A101057 A045362

Adjacent sequences: A132452 A132453 A132454 this_sequence A132456 A132457 A132458

nonn

Paul D. Hanna (pauldhanna(AT)juno.com), Aug 21 2007