|
| |
|
|
A323676
|
|
Odd coefficients in Sum_{n>=0} x^n * (x^n + i)^n / (1 + i*x^(n+1))^(n+1), where i^2 = -1.
|
|
3
|
|
|
|
1, -1, -7, -9, 39, 533, 2701, 9567, 5259, 23687, -531597, -6683401, -27445177, -251078037, -1245962509, -5523256133, -24464598853, -121368642607, -6413965885, 3365940994511, 26590133272861, 203250747797947, 1619896949491523, 8079808176937707, 31536734523928147, 157633940775344629, 661826129231253629, 2551518281305096845, -1162181865120937657, -42134613433736484823, -346386816982406236291, -3086583429684434309889, -23969200997893207053885
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,3
|
|
|
COMMENTS
|
a(n) = A323675(n*(n+1)) for n >= 0.
|
|
|
LINKS
|
|
|
|
FORMULA
|
a(n) = [x^(n*(n+1))] Sum_{n>=0} x^n * (x^n + i)^n / (1 + i*x^(n+1))^(n+1).
a(n) = [x^(n*(n+1))] Sum_{n>=0} x^n * (x^n - i)^n / (1 - i*x^(n+1))^(n+1).
|
|
|
EXAMPLE
|
Given the g.f. of A323675, G(x) = Sum_{n>=0} x^n * (x^n + i)^n / (1 + i*x^(n+1))^(n+1), i.e.,
G(x) = 1/(1 + i*x) + x*(x + i)/(1 + i*x^2)^2 + x^2*(x^2 + i)^2/(1 + i*x^3)^3 + x^3*(x^3 + i)^3/(1 + i*x^4)^4 + x^4*(x^4 + i)^4/(1 + i*x^5)^5 + x^5*(x^5 + i)^5/(1 + i*x^6)^6 + x^6*(x^6 + i)^6/(1 + i*x^7)^7 + x^7*(x^7 + i)^7/(1 + i*x^8)^8 + ...
and writing G(x) as a power series in x starting as
G(x) = 1 - x^2 + 2*x^3 + 2*x^4 - 7*x^6 - 2*x^7 + 8*x^8 + 8*x^10 + 12*x^11 - 9*x^12 - 28*x^13 - 16*x^14 + 4*x^15 + 2*x^16 + 46*x^18 + 104*x^19 + 39*x^20 - 100*x^21 - 144*x^22 + 4*x^23 + 16*x^24 - 144*x^25 - 66*x^26 + 28*x^27 + 72*x^28 + 336*x^29 + 533*x^30 + 178*x^31 - 496*x^32 - 448*x^33 + 242*x^34 + 288*x^35 - 298*x^36 - 1032*x^37 - 1212*x^38 - 480*x^39 + 142*x^40 + 1008*x^41 + 2701*x^42 + ...
then the odd coefficients of x^n in G(x), occurring at n = k*(k+1) for k>=0, form this sequence.
|
|
|
PROG
|
(PARI) {A323675(n) = my(A=sum(m=0, n, x^m * (x^m + I +x*O(x^n))^m/(1 + I*x^(m+1) +x*O(x^n))^(m+1) )); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
sign
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
