A178319 E.g.f.: ( Sum_{n>=0} 3^(n*(n + 1)/2) * x^n/n! )^(1/3).

1, 1, 7, 199, 17713, 4572529, 3426693463, 7575807034711, 49908659904426337, 983868034228748840161, 58130023275752925902247847, 10299771730830080877230000021479, 5474153833417147528343683843805979793, 8727821227226586439546709016484604992020049

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..64

Richard Stanley, Proof of the general conjecture, MathOverflow, March 2021.

Formula

a(n) = 1 (mod 6) for n >= 0 (conjecture).

General conjecture: [x^n/n!] E(q*x, q)^(1/q) = 1 (mod q(q-1)) for n >= 0 and integer q > 1 where E(x, q) = Sum_{n>=0} q^(n*(n - 1)/2)*x^n/n!.

Example

E.g.f.: A(x) = 1 + x + 7*x^2/2! + 199*x^3/3! + 17713*x^4/4! +...
A(x)^3 = 1 + 3*x + 3^3*x^2/2! + 3^6*x^3/3! + 3^10*x^4/4! +...
Let E(x, q) = Sum_{n>=0} q^(n*(n - 1)/2)*x^n/n!, then the coefficients of (x^n/n!) in E(qx, q)^(1/q) begin:
  1;
  1;
  q^2 - q + 1;
  q^5 - 3*q^3 + 5*q^2 - 3*q + 1;
  q^9 - 4*q^6 + q^5 + 15*q^4 - 24*q^3 + 17*q^2 - 6*q + 1;
  q^14 - 5*q^10 + 5*q^9 - 10*q^8 + 30*q^7 - 95*q^5 + 149*q^4 - 110*q^3 + 45*q^2 - 10*q + 1; ...
Setting q = 3 generates this sequence.

Maple

a:= n-> n!*coeff(series(add(3^binomial(j+1, 2)
        *x^j/j!, j=0..n)^(1/3), x, n+1), x, n):
seq(a(n), n=0..14);  # Alois P. Heinz, Mar 15 2021

Prog

(PARI) {a(n)=n!*polcoeff(sum(m=0, n, 3^(m*(m+1)/2)*x^m/m!+x*O(x^n))^(1/3), n)}

Crossrefs

Cf. A178315 (sqrt case).

Column k=3 of A346061.

Sequence in context: A206473 A300616 A370084 * A202943 A355088 A057204

Adjacent sequences: A178316 A178317 A178318 * A178320 A178321 A178322

Keyword

nonn

Author

Paul D. Hanna, May 24 2010

Extensions

General conjecture restated by Paul D. Hanna, May 25 2010

Status

approved