OFFSET
0,3
COMMENTS
FORMULA
a(n) = Sum_{k=0..n\2} ( [x^(2*k)] A(x^2)^{2*(n-2*k+1)} )^2 for n>0 with a(0)=1.
EXAMPLE
G.f.: A(x) = 1 + x + 5*x^2 + 17*x^3 + 158*x^4 + 741*x^5 + 4062*x^6 +...
To illustrate the recurrence, list coefficients of A(x^2)^(2*n):
A^2: 1, 2, 11, 44, 375, 1968, 11475, ...;
A^4: 1, 4, 26, 132, 1047, 6404, 41008, ...;
A^6: 1, 6, 45, 272, 2148, 14562, ...;
A^8: 1, 8, 68, 472, 3826, 28048, ...;
A^10: 1, 10, 95, 740, 6245, ...;
A^12: 1, 12, 126, 1084, 9585, ...;
A^14: 1, 14, 161, 1512, ...;
A^16: 1, 16, 200, 2032, ...;
A^18: 1, 18, 243, ...;
A^20: 1, 20, 290, ...;
A^22: 1, 22, ...;
A^24: 1, 24, ...;
A^26: 1, ...;
A^28: 1, ...; ...
then sum the squares of the coefficients in each column:
a(0) = 1^2 = 1;
a(1) = 1^2 = 1;
a(2) = 2^2 + 1^2 = 5;
a(3) = 4^2 + 1^2 = 17;
a(4) = 11^2 + 6^2 + 1^2 = 158;
a(5) = 26^2 + 8^2 + 1^2 = 741;
a(6) = 44^2 + 45^2 + 10^2 + 1^2 = 4062;
a(7) = 132^2 + 68^2 + 12^2 + 1^2 = 22193;
a(8) = 375^2 + 272^2 + 95^2 + 14^2 + 1^2 = 223831;
a(9) = 1047^2 + 472^2 + 126^2 + 16^2 + 1^2 = 1335126.
PROG
(PARI) {a(n)=local(A=1+sum(j=1, n\2, a(j)*x^(2*j))+x*O(x^n)); if(n==0, 1, sum(k=0, n\2, polcoeff(A^(2*(n-2*k+1)), 2*k)^2))}
for(n=0, 40, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Apr 09 2012
STATUS
approved