OFFSET
0,3
COMMENTS
Compare to: [x^n] (x*F(x)^n)' / F(x)^(n*(n+1)) = 0 for n>1 holds when F(0) = 1.
Note that [x^n] G(x,k)^(k*(n+1)-1) / (x*G(x,k)^k)' = 0 is satisfied by an integer series G(x,k) when k is a fixed positive integer; the g.f. of this sequence explores the case when k varies with n.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..300
FORMULA
G.f. A(x) satisfies: [x^n] A(x)^(n^2) / (A(x) + n*x*A'(x)) = 0 for n>1.
a(n) ~ c * 2^n * n!^3 / n, where c = 0.2754104367997800017735528000516901713205624917... - Vaclav Kotesovec, Oct 20 2020
EXAMPLE
G.f.: A(x) = 1 + x + 3*x^2 + 107*x^3 + 11627*x^4 + 2513589*x^5 + 949355653*x^6 + 575357369483*x^7 + 525974349806337*x^8 + 691365121056215549*x^9 + ...
such that [x^n] A(x)^(n*(n+1)-1) / (x*A(x)^n)' = 0 for n>1.
ILLUSTRATION OF DEFINITION.
The table of coefficients in A(x)^(n*(n+1)-1) / (x*A(x)^n)' begins:
n=0: [1, -1, -2, -102, -11412, -2490030, -944283630, -573448825894, ...];
n=1: [1, -1, -4, -304, -45436, -12414490, -5655451828, -4009336016960, ...];
n=2: [1, 1, 0, -296, -56621, -17380683, -8487839136, -6303946190960, ...];
n=3: [1, 5, 22, 0, -43410, -17309652, -9440759988, -7462899694108, ...];
n=4: [1, 11, 86, 874, 0, -11796810, -8449485806, -7468455619310, ...];
n=5: [1, 19, 228, 3068, 88298, 0, -5377376960, -6278167743244, ...];
n=6: [1, 29, 496, 8136, 256299, 19641657, 0, -3822351028528, ...];
n=7: [1, 41, 950, 18924, 581824, 50072326, 8025251308, 0, ...]; ...
in which the main diagonal consists of all zeros after the initial terms, illustrating that [x^n] A(x)^(n*(n+1)-1) / (x*A(x)^n)' = 0 for n>1.
RELATED SERIES.
log(A(x)) = x + 5*x^2/2 + 313*x^3/3 + 46073*x^4/4 + 12508771*x^5/5 + 5680881713*x^6/6 + 4020812685695*x^7/7 + 4203174178089489*x^8/8 + 6217540835502410521*x^9/9 + ...
PROG
(PARI) {a(n) = my(A=[1, 1]); for(i=1, n, A = concat(A, 0); A[#A] = Vec( Ser(A)^(#A*(#A-1)-1)/(x*Ser(A)^(#A-1))' )[#A] ); A[n+1]}
for(n=0, 20, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Mar 31 2018
STATUS
approved