A132455 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.

1, 1, 2, 5, 11, 18, 30, 45, 80, 147, 330, 882, 1935, 3298, 4676, 7613, 18409, 53724, 141094, 293407, 473604, 639418, 1138534, 3086050, 9159520, 23937239, 54502764, 106105954, 167076383, 237933308, 491988696, 1749801789, 6074959380

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

Cf. A095892 (variant); A000108.

Sequence in context: A264724 A298375 A260037 * A132459 A308517 A101057

Adjacent sequences: A132452 A132453 A132454 * A132456 A132457 A132458

Keyword

nonn

Author

Paul D. Hanna, Aug 21 2007

Status

approved