A095892 a(n) = sum of the squares of the coefficients of x^n in x^(n-2k)*A(x^2)^(n-2k), as k varies from 0 to floor(n/2), with a(0)=1.

1, 1, 1, 2, 5, 11, 26, 66, 173, 469, 1307, 3676, 10506, 30260, 87922, 256990, 756381, 2236409, 6648893, 19846902, 59507503, 179051839, 540778436, 1638019284, 4977317066, 15162490432, 46314401884, 141773634376, 434993794258

Offset

0,4

Comments

Defined using a variation of this property of the Catalan sequence: A000108(n) = sum of the square of the coefficients of x^n in x^(n-2k)*G000108(x^2)^(n-2k+1), as k varies from 0 to floor(n/2).

Links

Paul D. Hanna, Table of n, a(n) for n = 0..200

Formula

a(n) = Sum_{k=0..[n/2]} ( [x^n] (x*A(x^2))^(n-2k) )^2, where [x^n] f(x) denotes the coefficient of x^n in f(x).

Example

a(6) = 26 since 26 = 1^2 + 4^2 + 3^2 + 0^2;
a(7) = 66 since 66 = 1^2 + 5^2 + 6^2 + 2^2;
a(8) = 173 since 173 = 1^2 + 6^2 + 10^2 + 6^2 + 0^2;
indicated by this array of coefficients of x^m*A(x^2)^m:
A(x)........: 1,1,1,2,5,11,26,66,173,...
x^0*A(x^2)^0: 1,__0,__0,____0,_____0,...
x^1*A(x^2)^1: __1,__1,___1,____2,____...
x^2*A(x^2)^2: ____1,__2,____3,_____6,...
x^3*A(x^2)^3: ______1,___3,____6,____...
x^4*A(x^2)^4: ________1,____4,____10,...
x^5*A(x^2)^5: ___________1,____5,____...
x^6*A(x^2)^6: ______________1,_____6,...
x^7*A(x^2)^7: _________________1,____...
x^8*A(x^2)^8: _____________________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), n)^2))}

Crossrefs

Sequence in context: A300125 A307576 A079223 * A239311 A319760 A343872

Adjacent sequences: A095889 A095890 A095891 * A095893 A095894 A095895

Keyword

nonn

Author

Paul D. Hanna, Jun 22 2004

Status

approved