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 = 0.5984002265754..., c = 0.08321697608093... - Vaclav Kotesovec, Feb 06 2023
EXAMPLE
G.f.: A(x) = 1 + x + 3*x^2 + 31*x^3 + 526*x^4 + 11907*x^5 + 328980*x^6 + 10580531*x^7 + 384937042*x^8 + 15549217485*x^9 + ...
RELATED SERIES.
G.f. A(x) = B(x/A(x)) where B(x) = B(x*A(x)) begins:
B(x) = 1 + x + 4*x^2 + 41*x^3 + 687*x^4 + 15433*x^5 + 424524*x^6 + 13620842*x^7 + 495005025*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)} begins:
[1/1, 2/2, 12/3, 164/4, 3435/5, 92598/6, 2971668/7, 108966736/8, ...].
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in A(x)^(n+1) begins:
n=0: [1, 1, 3, 31, 526, 11907, 328980, 10580531, ...];
n=1: [1, 2, 7, 68, 1123, 25052, 685891, 21923076, ...];
n=2: [1, 3, 12, 112, 1800, 39555, 1072896, 34076544, ...];
n=3: [1, 4, 18, 164, 2567, 55548, 1492336, 47093172, ...];
n=4: [1, 5, 25, 225, 3435, 73176, 1946745, 61028770, ...];
n=5: [1, 6, 33, 296, 4416, 92598, 2438866, 75942984, ...];
n=6: [1, 7, 42, 378, 5523, 113988, 2971668, 91899578, ...];
n=7: [1, 8, 52, 472, 6770, 137536, 3548364, 108966736, ...]; ...
Compare to the table of coefficients in (1 + x*A(x)^(2*n-1))^(n+1):
n=0: [1, 1, -1, -2, -26, -463, -10778, -303048, ...];
n=1: [1, 2, 3, 8, 69, 1120, 24937, 683012, ...];
n=2: [1, 3, 12, 55, 444, 6351, 132492, 3504654, ...];
n=3: [1, 4, 26, 164, 1411, 18560, 357624, 9024812, ...];
n=4: [1, 5, 45, 360, 3435, 43926, 785715, 18700710, ...];
n=5: [1, 6, 69, 668, 7134, 92598, 1570420, 35086104, ...];
n=6: [1, 7, 98, 1113, 13279, 179816, 2971668, 62645353, ...];
n=7: [1, 8, 132, 1720, 22794, 327032, 5403036, 108966736, ...]; ...
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