|
| |
|
|
A359721
|
|
a(n) = coefficient of x^n in the power series A(x) such that: x = Sum_{n=-oo..+oo} (-1)^n * x^n * (1 + x^n)^n * A(x)^n.
|
|
5
|
|
|
|
1, 1, 3, 10, 37, 127, 460, 1710, 6461, 24851, 96921, 382358, 1522997, 6116518, 24740564, 100698617, 412126133, 1694982357, 7001729420, 29037602898, 120856092153, 504647152650, 2113469775619, 8875358529059, 37364827472930, 157668052571948, 666735804080597, 2825054673048981
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,3
|
|
|
COMMENTS
|
Related identity: 0 = Sum_{n=-oo..+oo} x^n * (y - x^n)^n, which holds formally for all y.
|
|
|
LINKS
|
|
|
|
FORMULA
|
G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following.
(1) x = Sum_{n=-oo..+oo} (-1)^n * x^n * (1 + x^n)^n * A(x)^n.
(2) x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / ((1 + x^n)^n * A(x)^n).
(3) a(n) = Sum_{k=0..floor(2*n/3)} A359720(n,k), for n >= 0.
a(n) ~ c * d^n / n^(3/2), where d = 4.470597712126170109... and c = 1.18164918660560739... - Vaclav Kotesovec, Mar 14 2023
|
|
|
EXAMPLE
|
G.f.: A(x) = 1 + x + 3*x^2 + 10*x^3 + 37*x^4 + 127*x^5 + 460*x^6 + 1710*x^7 + 6461*x^8 + 24851*x^9 + 96921*x^10 + 382358*x^11 + 1522997*x^12 + ...
SPECIFIC VALUES.
A(2/9) = 2.24070435506724977359903344036738515875266644317987374...
A(x) = 2 at x = 0.21791735938682393028374635435485389216073583164032813...
A(1/5) = 1.63325728843716074555468074513852677972333543319428229...
A(1/6) = 1.36828627213340815002770404510072582545059876619425902...
|
|
|
PROG
|
(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff(x - sum(n=-#A-1, #A+1, (-1)^n * x^n * (1 + x^n +x*O(x^#A) )^n * Ser(A)^n ), #A-1) ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
