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 along with related series B(x) = A(x*B(x)) and C(x) = A(x/C(x)) satisfy the following formulas.
(1.a) [x^n] A(x)^n = [x^n] -1/A(x)^(n+1) for n > 1 with A(0) = 1, A'(0) = 1.
(1.b) [x^n] A(x)^n = [x^n] x*B'(x)/B(x) for n >= 1, where B(x) = A(x*B(x)).
(1.c) [x^n] 1/A(x)^(n+1) = [x^n] (x/C(x))' for n >= 0, where C(x) = A(x/C(x)).
(2.a) x*B'(x)/B(x) = 1 - x - (x/C(x))'.
(2.b) x*log(B(x))' = 1 - x - (1 - x*log(C(x))') / C(x).
(2.c) B(x) = exp( Integral (1-x - (x/C(x))')/x dx ).
(3.a) A(x) = B(x/A(x)).
(3.b) A(x) = C(x*A(x)).
(4.a) A(x) = x / Series_Reversion( x*B(x) ).
(4.b) A(x) = (1/x) * Series_Reversion( x/C(x) ).
(5.a) B(x) = A(x*B(x)) = C(x*B(x)^2).
(5.b) C(x) = A(x/C(x)) = B(x/C(x)^2).
(6.a) B(x) = (1/x) * Series_Reversion( x/A(x) ).
(6.b) C(x) = x / Series_Reversion( x*A(x) ).
a(n) ~ c * n!^2 * 2^n * n, where c = 0.3641560029438432044438855398899392... - Vaclav Kotesovec, May 18 2025
EXAMPLE
G.f.: A(x) = 1 + x + 7*x^2 + 163*x^3 + 7295*x^4 + 497193*x^5 + 46629734*x^6 + 5701678075*x^7 + 878881320340*x^8 + 166768493380169*x^9 + ...
RELATED SERIES.
Define B(x) by B(x) = A(x*B(x)), then B(x) begins
B(x) = 1 + x + 8*x^2 + 185*x^3 + 8088*x^4 + 541564*x^5 + 50149195*x^6 + 6074480142*x^7 + 929590478337*x^8 + ...
also, A(x) = B(x/A(x)).
Define C(x) by C(x) = A(x/C(x)), then C(x) begins
C(x) = 1 + x + 6*x^2 + 144*x^3 + 6610*x^4 + 457962*x^5 + 43451593*x^6 + 5359455081*x^7 + 831748005435*x^8 + ...
also, A(x) = C(x*A(x)).
RELATED TABLES.
In the tables below, the main diagonals agree (up to sign) after the initial terms.
The table of coefficients of x^k in A(x)^n begins
n = 0: [1, 0, 0, 0, 0, 0, 0, ...];
n = 1: [1, 1, 7, 163, 7295, 497193, 46629734, ...];
n = 2: [1, 2, 15, 340, 14965, 1011258, 94382553, ...];
n = 3: [1, 3, 24, 532, 23031, 1542831, 143287531, ...];
n = 4: [1, 4, 34, 740, 31515, 2092576, 193374688, ...];
n = 5: [1, 5, 45, 965, 40440, 2661186, 244675025, ...];
n = 6: [1, 6, 57, 1208, 49830, 3249384, 297220560, ...];
...
in which a diagonal gives the coefficients in B(x):
[1, 2/2, 24/3, 740/4, 40440/5, 3249384/6, ...] = [1, 1, 8, 185, 8088, 541564, ...].
The table of coefficients of x^k in 1/A(x)^(n+1) begins
n = 0: [1, -1, -6, -150, -6940, -480930, -45534811, ...];
n = 1: [1, -2, -11, -288, -13544, -946180, -90001982, ...];
n = 2: [1, -3, -15, -415, -19830, -1396308, -133427949, ...];
n = 3: [1, -4, -18, -532, -25815, -1831848, -175838332, ...];
n = 4: [1, -5, -20, -640, -31515, -2253311, -217257965, ...];
n = 5: [1, -6, -21, -740, -36945, -2661186, -257710925, ...];
n = 6: [1, -7, -21, -833, -42119, -3055941, -297220560, ...];
...
in which the main diagonal gives the coefficients in 1/C(x):
[1, -2/2, -15/3, -532/4, -31515/5, -2661186/6, -297220560/7, ...] = [1, -1, -5, -133, -6303, -443531, -42460080, ...].
where 1/C(x) = 1 - x - 5*x^2 - 133*x^3 - 6303*x^4 - 443531*x^5 - 42460080*x^6 - 5266805650*x^7 + ...
PROG
(PARI) {a(n) = my(A=[1, 1], m); for(i=1, n, A=concat(A, 0); m = #A-1;
A[#A] = polcoef(Ser(A)^m, m) - polcoef(-1/Ser(A)^(m+1), m) ); H=A; A[n+1]}
for(n=0, 20, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, May 17 2025
STATUS
approved