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A178319 E.g.f.: [ Sum_{n>=0} 3^(n(n+1)/2) * x^n/n! ]^(1/3). 0
1, 1, 7, 199, 17713, 4572529, 3426693463, 7575807034711, 49908659904426337, 983868034228748840161, 58130023275752925902247847, 10299771730830080877230000021479, 5474153833417147528343683843805979793 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Table of n, a(n) for n=0..12.

FORMULA

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

General conjecture: [ x^n/n! ] E(qx,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

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.

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

Sequence in context: A154936 A206473 A300616 * A202943 A057204 A124988

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

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Last modified October 29 21:18 EDT 2020. Contains 338074 sequences. (Running on oeis4.)