|
| |
|
|
A325155
|
|
G.f. A(x) satisfies: x = Sum_{n>=1} x^n * (1+x)^(n^2/2) / A(x)^(n/2).
|
|
1
|
|
|
|
1, 3, 5, 7, 11, 21, 49, 133, 408, 1376, 5020, 19564, 80741, 350551, 1593066, 7547792, 37163568, 189662934, 1001046684, 5453972462, 30622950955, 176942133603, 1050773432990, 6405898358012, 40048848677954, 256521565555908, 1681897617101795, 11278819380424173, 77301464920178158, 541084956406886214, 3865540113371340736, 28167799470180443028, 209238063076396838375, 1583562040116769584091, 12204247180832799551059, 95731651337427271893873
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,2
|
|
|
LINKS
|
|
|
|
FORMULA
|
G.f. A(x) satisfies the following identities.
(1) 1 + x = Sum_{n>=0} x^n * (1+x)^(n^2/2) / A(x)^(n/2).
(2) 1 + x = 1/(1 - q*x/(sqrt(A(x)) - q*(q^2-1)*x/(1 - q^5*x/(sqrt(A(x)) - q^3*(q^4-1)*x/(1 - q^9*x/(sqrt(A(x)) - q^5*(q^6-1)*x/(1 - q^13*x/(sqrt(A(x)) - q^7*(q^8-1)*x/(1 - ...))))))))), where q = sqrt(1+x), a continued fraction due to a partial elliptic theta function identity.
(3) 1 + x = Sum_{n>=0} x^n * (1+x)^(n/2) / A(x)^(n/2) * Product_{k=1..n} (sqrt(A(x)) - x*sqrt(1+x)^(4*k-3)) / (sqrt(A(x)) - x*sqrt(1+x)^(4*k-1)), due to a q-series identity.
(4) A(x) = (1+x)*G(x)^2 where G(x) is the g.f. of A318644.
|
|
|
EXAMPLE
|
G.f.: A(x) = 1 + 3*x + 5*x^2 + 7*x^3 + 11*x^4 + 21*x^5 + 49*x^6 + 133*x^7 + 408*x^8 + 1376*x^9 + 5020*x^10 + 19564*x^11 + 80741*x^12 + ...
such that
A(x) = 1 + x*(1+x)^(1/2)/A(x)^(1/2) + x^2*(1+x)^2/A(x) + x^3*(1+x)^(9/2)/A(x)^(3/2) + x^4*(1+x)^8/A(x)^2 + x^5*(1+x)^(25/2)/A(x)^(5/2) + x^6*(1+x)^18/A(x)^3 + x^7*(1+x)^(49/2)/A(x)^(7/2) + x^8*(1+x)^32/A(x)^4 + ...
Note that
sqrt(A(x))*sqrt(1+x) = 1 + x + x^2 + x^3 + 2*x^4 + 4*x^5 + 11*x^6 + 32*x^7 + 106*x^8 + 376*x^9 + 1433*x^10 + 5782*x^11 + ... + A318644(n)*x^n + ...
|
|
|
PROG
|
(PARI) {a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0); A[#A] = 2*polcoeff( sum(n=0, #A+1, x^n*(1+x +x*O(x^#A))^(n^2/2) / Ser(A)^(n/2) ), #A)); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
