|
| |
|
|
A358965
|
|
a(n) = coefficient of x^n in A(x) such that: 1 = Sum_{n=-oo..+oo} x^n * (A(x) - x^(6*n+5))^(n-1).
|
|
6
|
|
|
|
1, 2, 7, 30, 143, 729, 3876, 21321, 120195, 690816, 4032807, 23846485, 142530516, 859719414, 5226571568, 31992109155, 197002217763, 1219554190530, 7585453430037, 47380560231549, 297081856642195, 1869191995298989, 11797744585161792, 74678247991840230, 473954364916279312
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,2
|
|
|
COMMENTS
|
Related identity: 0 = Sum_{n=-oo..+oo} x^n * (y - x^(6*n+5))^n, which holds formally for all y.
|
|
|
LINKS
|
|
|
|
FORMULA
|
G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
(1) 1 = Sum_{n=-oo..+oo} x^n * (A(x) - x^(6*n+5))^(n-1).
(2) x^5 = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(6*n^2) / (1 - x^(6*n-5)*A(x))^(n+1).
(3) A(x) = Sum_{n=-oo..+oo} x^(7*n+5)* (A(x) - x^(6*n+5))^(n-1).
(4) A(x) = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(6*n*(n-1)) / (1 - x^(6*n-5)*A(x))^(n+1).
(5) 0 = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(6*n*(n-1)) / (1 - x^(6*n-5)*A(x))^n.
|
|
|
EXAMPLE
|
G.f.: A(x) = 1 + 2*x + 7*x^2 + 30*x^3 + 143*x^4 + 729*x^5 + 3876*x^6 + 21321*x^7 + 120195*x^8 + 690816*x^9 + 4032807*x^10 + ...
where A = A(x) satisfies the doubly infinite sum
1 = ... + x^(-2)*(A - x^(-7))^(-3) + x^(-1)*(A - x^(-1))^(-2) + (A - x^5)^(-1) + x*(A - x^11)^0 + x^2*(A - x^17) + x^3*(A - x^23)^2 + x^4*(A - x^29)^3 + ... + x^n * (A - x^(6*n+5))^(n-1) + ...
also,
A(x) = ... + x^72/(1 - x^(-23)*A)^(-2) - x^36/(1 - x^(-17)*A)^(-1) + x^12 - 1/(1 - x^(-5)*A) + 1/(1 - x*A)^2 - x^12/(1 - x^7*A)^3 + x^36/(1 - x^13*A)^4 - x^72/(1 - x^19*A)^5 + ... + (-1)^(n+1)*x^(6*n*(n-1))/(1 - x^(6*n-5)*A)^(n+1) + ...
|
|
|
PROG
|
(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff( sum(n=-#A, #A, x^n * (Ser(A) - x^(6*n+5))^(n-1) ), #A-1) ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
