OFFSET
0,4
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..500
FORMULA
G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) a(n) = [x^n] ( A(x)^(n-1) - 2*A(x)^n + A(x)^(n+1) ) for n > 1, with a(0) = a(1) = 1.
(2) A(x) = 1 + x + (B(x) + x*B'(x)) * (1 - 1/B(x))^2, where B(x) = A(x*B(x)) = (1/x)*Series_Reversion(x/A(x)).
EXAMPLE
G.f.: A(x) = 1 + x + x^2 + 4*x^3 + 21*x^4 + 138*x^5 + 1049*x^6 + 8878*x^7 + 81802*x^8 + 808584*x^9 + 8487493*x^10 + ...
RELATED SERIES.
The related series B(x) = A(x*B(x)) begins
B(x) = 1 + x + 2*x^2 + 8*x^3 + 46*x^4 + 324*x^5 + 2617*x^6 + 23397*x^7 + 226647*x^8 + 2346175*x^9 + 25707007*x^10 + ...
where B(x) = (1/x)*Series_Reversion(x/A(x))
also, A(x) = B(x/A(x)).
RELATED TABLE.
Given B(x) as defined above, the coefficients in (x*B(x))' forms the main diagonal in the table below:
[1, 2, 6, 32, 230, 1944, 18319, 187176, 2039823, ...].
The table of coefficients of x^k in A(x)^n begins:
n = 1: [1, 1, 1, 4, 21, 138, 1049, 8878, 81802, ...];
n = 2: [1, 2, 3, 10, 51, 326, 2432, 20298, 185003, ...];
n = 3: [1, 3, 6, 19, 93, 579, 4237, 34860, 314193, ...];
n = 4: [1, 4, 10, 32, 151, 916, 6574, 53300, 474911, ...];
n = 5: [1, 5, 15, 50, 230, 1361, 9580, 76520, 673840, ...];
n = 6: [1, 6, 21, 74, 336, 1944, 13425, 105624, 919041, ...];
n = 7: [1, 7, 28, 105, 476, 2702, 18319, 141961, 1220233, ...];
n = 8: [1, 8, 36, 144, 658, 3680, 24520, 187176, 1589127, ...];
n = 9: [1, 9, 45, 192, 891, 4932, 32343, 243270, 2039823, ...];
...
which obeys the following recurrence
a(n) = [x^n] ( A(x)^(n-1) - 2*A(x)^n + A(x)^(n+1) ) for n > 1
as illustrated by
a(2) = 1 - 2*3 + 6 = 1;
a(3) = 10 - 2*19 + 32 = 4;
a(4) = 93 - 2*151 + 230 = 21;
a(5) = 916 - 2*1361 + 1944 = 138;
a(6) = 9580 - 2*13425 + 18319 = 1049;
a(7) = 105624 - 2*141961 + 187176 = 8878;
a(8) = 1220233 - 2*1589127 + 2039823 = 81802;
...
PROG
(PARI) {a(n) = my(A=[1, 1], m); for(i=2, n, A = concat(A, 0); m = #A-1;
A[#A] = polcoef( Ser(A)^(m-1) - 2*Ser(A)^m + Ser(A)^(m+1), m) ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, May 23 2025
STATUS
approved