OFFSET
0,3
COMMENTS
Compare to: x = Sum_{n>=1} x^n * F(-x)^n if F(x) = 1/(1-x).
a(n) ~ c * 2^(n*(n+1)/2) where c = 0.148227873045508... - Paul D. Hanna, Oct 01 2023
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..100
EXAMPLE
G.f.: A(x) = 1 + x + 3*x^2 + 20*x^3 + 275*x^4 + 7715*x^5 + 442829*x^6 + 52013252*x^7 + 12463418970*x^8 + 6068733122286*x^9 + ...
where
x = x*A(-x) + x^2*A(-2*x)^2 + x^3*A(-3*x)^3 + x^4*A(-4*x)^4 + x^5*A(-5*x)^5 + x^6*A(-6*x)^6 + x^7*A(-7*x)^7 + x^8*A(-8*x)^8 + x^9*A(-9*x)^9 + ...
The array of coefficients in A(-n*x)^n begins:
n=1: [1, -1, 3, -20, 275, -7715, 442829, -52013252, ...];
n=2: [1, -4, 28, -368, 9584, -515200, 57800832, -13436089856, ...];
n=3: [1, -9, 108, -2133, 79461, -6133806, 1007579331, ...];
n=4: [1, -16, 288, -7680, 366336, -36085760, 7705305088, ...];
n=5: [1, -25, 625, -21250, 1225000, -144393750, 37527343750, ...];
n=6: [1, -36, 1188, -49680, 3343680, -453076416, 137425947840, ...];
n=7: [1, -49, 2058, -103243, 7932904, -1202725727, 413455645435, ...];
n=8: [1, -64, 3328, -196608, 16982016, -2826174464, 1077462171648, ...]; ...
in which the antidiagonal sums yield [1,0,0,0,0,0,0,0,...].
PROG
(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A]=(-1)^(#A)*Vec(sum(m=1, #A, subst(Ser(A), x, -m*x)^m*x^m))[#A] ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
(PARI) /* Quick print of terms 0..30 */
{A=[1]; for(i=1, 30, A=concat(A, 0);
A[#A]=(-1)^(#A)*Vec(sum(n=1, #A, subst(Ser(A), x, -n*x)^n*x^n))[#A] ); A}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jan 05 2016
STATUS
approved