OFFSET
0,3
COMMENTS
Conjecture: for n >= 0, a(n) is odd iff n+1 is a power of 2.
Conjecture: for n >= 0, a(n) is not divisible by 3 iff 2*(n+1) is a sum of two powers of 3 (A055235).
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..400
FORMULA
a(n) ~ c * (n-1)!^2, where c = 8.09432041660756179176311954708910172717555... - Vaclav Kotesovec, Jul 03 2026
EXAMPLE
G.f.: A(x) = 1 + x + 4*x^2 + 27*x^3 + 304*x^4 + 5000*x^5 + 126144*x^6 + 4372221*x^7 + 213232960*x^8 + 13375072836*x^9 + ...
RELATED SEQUENCES.
1/A(x) = 1 - x - 3*x^2 - 20*x^3 - 245*x^4 - 4290*x^5 - 114422*x^6 - 4086800*x^7 + ... + (-1)^n*A237652(n)*x^n + ...
The table of coefficients of x^k in 1/A(x)^(n^2) begins
n = 1: [1, (-1), -3, -20, -245, -4290, -114422, ...];
n = 2: [1, -4, (-6), -48, -721, -13836, -399342, ...];
n = 3: [1, -9, 9, (-48), -1071, -22572, -732768, ...];
n = 4: [1, -16, 72, -160, (-1260), -26688, -1018704, ...];
n = 5: [1, -25, 225, -1000, 525, (-25680), -1230900, ...];
n = 6: [1, -36, 522, -4080, 16695, -61452, (-1360590), ...]; ...
Compare to the table of coefficients of x^k in 1/A(x)^(n^2-1):
n = 1: [1, (0), 0, 0, 0, 0, 0, ...];
n = 2: [1, -3, (-6), -43, -597, -11127, -313038, ...];
n = 3: [1, -8, 4, (-48), -1022, -21328, -677040, ...];
n = 4: [1, -15, 60, -125, (-1260), -26508, -986720, ...];
n = 5: [1, -24, 204, -848, 54, (-25680), -1211936, ...];
n = 6: [1, -35, 490, -3675, 14035, -52927, (-1360590), ...]; ...
to see that the diagonals in parenthesis are equal after initial terms; that is, [x^n] 1/A(x)^(n^2) = [x^n] 1/A(x)^(n^2-1) for n > 1.
PROG
(PARI) {a(n) = my(A=[1, 1]); for(i=2, n, A=concat(A, 0); A[#A] = (Vec(1/Ser(A)^((#A-1)^2)) - Vec(1/Ser(A)^((#A-1)^2-1)))[#A]); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jul 03 2026
STATUS
approved
