OFFSET
0,3
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..300
FORMULA
G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
(1) [x^n] A(x)^(n+1) = [x^n] (1 + x*A(x)^(2*n+1))^(n+1) for n>=0.
(2) A(x) = Sum_{n>=0} b(n) * x^n/A(x)^n, where b(n) = [x^n] (1 + x*A(x)^(2*n+1))^(n+1) / (n+1).
a(n) ~ c * d^n * n! * n^alpha, where d = 3.93464558322824528799..., alpha = 1.635402029299..., c = 0.0308525091280143... - Vaclav Kotesovec, Feb 06 2023
EXAMPLE
G.f.: A(x) = 1 + x + 5*x^2 + 62*x^3 + 1214*x^4 + 31269*x^5 + 973485*x^6 + 34993597*x^7 + 1412846469*x^8 + 62926155294*x^9 + ...
RELATED SERIES.
G.f. A(x) = B(x/A(x)) where B(x) = B(x*A(x)) begins:
B(x) = 1 + x + 6*x^2 + 78*x^3 + 1543*x^4 + 39810*x^5 + 1239252*x^6 + 44537587*x^7 + 1798314384*x^8 + ... + b(n)*x^n + ...
such that b(n) = [x^n] (1 + x*A(x)^(2*n+1))^(n+1) / (n+1),
as well as b(n) = [x^n] A(x)^(n+1) / (n+1),
so that b(n) begin:
[1/1, 2/2, 18/3, 312/4, 7715/5, 238860/6, 8674764/7, 356300696/8, ...].
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in A(x)^(n+1) begins:
n=0: [1, 1, 5, 62, 1214, 31269, 973485, 34993597, ...];
n=1: [1, 2, 11, 134, 2577, 65586, 2025492, 72397390, ...];
n=2: [1, 3, 18, 217, 4104, 103212, 3161648, 112357788, ...];
n=3: [1, 4, 26, 312, 5811, 144428, 4387978, 155030276, ...];
n=4: [1, 5, 35, 420, 7715, 189536, 5710930, 200579975, ...];
n=5: [1, 6, 45, 542, 9834, 238860, 7137401, 249182232, ...];
n=6: [1, 7, 56, 679, 12187, 292747, 8674764, 301023241, ...];
n=7: [1, 8, 68, 832, 14794, 351568, 10330896, 356300696, ...]; ...
Compare to the table of coefficients in (1 + x*A(x)^(2*n+1))^(n+1):
n=0: [1, 1, 1, 5, 62, 1214, 31269, 973485, ...];
n=1: [1, 2, 7, 42, 479, 8750, 216258, 6562156, ...];
n=2: [1, 3, 18, 136, 1560, 26895, 633608, 18631701, ...];
n=3: [1, 4, 34, 312, 3767, 62888, 1412530, 40031684, ...];
n=4: [1, 5, 55, 595, 7715, 128041, 2763270, 75234930, ...];
n=5: [1, 6, 81, 1010, 14172, 238860, 5016947, 131313798, ...];
n=6: [1, 7, 112, 1582, 24059, 418166, 8674764, 219340759, ...];
n=7: [1, 8, 148, 2336, 38450, 696216, 14466592, 356300696, ...]; ...
to see that the main diagonals of the tables are the same.
PROG
(PARI) {a(n) = my(A=[1]); for(m=1, n, A=concat(A, 0); A[m+1] = (Vec((1+x*Ser(A)^(2*m+1))^(m+1))[m+1] - Vec(Ser(A)^(m+1))[m+1])/(m+1) ); A[n+1]}
for(n=0, 20, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 05 2023
STATUS
approved