OFFSET
0,3
COMMENTS
The definition is a variation of the following property of the Catalan numbers: A000108(n) = Sum_{k=0..[n/2]} ( [x^n] (x*C(x^2))^(n-2*k+1) )^2, where C(x) is the g.f. of the Catalan numbers.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..100
FORMULA
a(n) = Sum_{k=0..[n/2]} ( [x^n] (x/A(x^2))^(n-2*k+1) )^2 for n>0 with a(0)=1.
EXAMPLE
a(n) is the sum of squares of coefficients in x^(n-2*k)/A(x^2)^(n-2*k+1):
a(2) = 2 = 1^2 + (-1)^2;
a(3) = 5 = 1^2 + (-2)^2;
a(4) = 11 = 1^2 + (-3)^2 + (-1)^2;
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; ...
as can be seen from the initial coefficients in x^n/A(x^2)^n:
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,...
PROG
(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))}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Aug 21 2007
STATUS
approved