A132455 - OEIS

%I #5 Jun 14 2012 01:18:04

%S 1,1,2,5,11,18,30,45,80,147,330,882,1935,3298,4676,7613,18409,53724,

%T 141094,293407,473604,639418,1138534,3086050,9159520,23937239,

%U 54502764,106105954,167076383,237933308,491988696,1749801789,6074959380

%N a(n) = sum of the squares of the coefficients of x^n in x^(n-2*k)/A(x^2)^(n-2*k+1), as k varies from 0 to floor(n/2), with a(0)=1, where A(x) is the g.f. of this sequence.

%C 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.

%H Paul D. Hanna, <a href="/A132455/b132455.txt">Table of n, a(n) for n = 0..100</a>

%F 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.

%e a(n) is the sum of squares of coefficients in x^(n-2*k)/A(x^2)^(n-2*k+1):

%e a(2) =  2 = 1^2 + (-1)^2;

%e a(3) =  5 = 1^2 + (-2)^2;

%e a(4) = 11 = 1^2 + (-3)^2 + (-1)^2;

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

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

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

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

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

%e as can be seen from the initial coefficients in x^n/A(x^2)^n:

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

%e x^1/A(x^2)^1: 1,_-1,__-1,___-2,___-2, ......

%e x^2/A(x^2)^2: __1,_-2,___-1,___-2,_____1,...

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

%e x^4/A(x^2)^4: ______1,___-4,____2,_____0,...

%e x^5/A(x^2)^5: _________1,___-5,____5, ......

%e x^6/A(x^2)^6: ____________1,___-6,_____9,...

%e x^7/A(x^2)^7: _______________1,___-7, ......

%e x^8/A(x^2)^8: __________________1,____-8,...

%e x^9/A(x^2)^9: _____________________1, ......

%e x^10/A(x^2)^10: _______________________1,...

%o (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))}

%Y Cf. A095892 (variant); A000108.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Aug 21 2007