A352850 E.g.f. A(x) satisfies: [x^k] A(x)^n = 0 for k = n*(n-1)/2 + 1 through n*(n+1)/2, for n > 1.

1, 1, -1, 3, -18, 150, -1590, 20790, -321930, 5751270, -116263350, 2628687600, -65723427000, 1800165767400, -53590470733800, 1722471401851200, -59462981199121500, 2194469988757546500, -86216322855144982500, 3592621186732990912500, -158253385898375658891000

Offset

0,4

Links

Table of n, a(n) for n=0..20.

Example

E.g.f.: A(x) = 1 + x - x^2/2! + 3*x^3/3! - 18*x^4/4! + 150*x^5/5! - 1590*x^6/6! + 20790*x^7/7! - 321930*x^8/8! + 5751270*x^9/9! - 116263350*x^10/10! + ...
The series of reduced fractional coefficients begins:
A(x) = 1 + x - 1/2*x^2 + 1/2*x^3 - 3/4*x^4 + 5/4*x^5 - 53/24*x^6 + 33/8*x^7 - 511/64*x^8 + 3043/192*x^9 - 4101/128*x^10 + ...
Related table.
The table of coefficients of x^k/k! in A(x)^n begins:
n=1: [1, 1, -1, 3, -18, 150, -1590, 20790, -321930, 5751270, ...];
n=2: [1, 2, 0, 0, -6, 60, -660, 9240, -149100, 2729160, ...];
n=3: [1, 3, 3, -3, 0, 0, 0, 1260, -28350, 595350, -13012650, ...];
n=4: [1, 4, 8, 0, -12, 0, 120, 0, 0, 0, 0, 30353400, -1166734800, ...];
n=5: [1, 5, 15, 15, -30, -30, 150, 1050, -1050, -47250, 1445850, 0, 0, 0, 0, 0, -141443655853500, ...];
...
which illustrates that A(x)^n has n contiguous zeros for coefficients of x^k starting at k = n*(n-1)/2 + 1 through k = n*(n+1)/2, for n > 1.

Prog

(PARI) {a(n) = my(A=1+x, m=floor(sqrt(2*n)+1/2));
for(k=1, m, A = ( truncate(A^k) + x*O(x^(k*(k+1)/2)) )^(1/k));
n!*polcoeff(H=A, n)}
for(n=0, 21, print1(a(n), ", "))

Crossrefs

Sequence in context: A138420 A152409 A200320 * A347020 A371538 A207569

Adjacent sequences: A352847 A352848 A352849 * A352851 A352852 A352853

Keyword

sign

Author

Paul D. Hanna, Apr 05 2022

Status

approved