A176827 G.f.: Sum_{n>=0} a(n)*x^n/3^(n(n-1)/2) = exp( Sum_{n>=1} A(x/3^n)^n*x^n/n ).

1, 1, 4, 43, 1288, 111874, 28513474, 21447796411, 47864888436352, 318138497623694374, 6313455845922307358476, 374710369870861989860068222, 66586966437639051221223067992430

Offset

0,3

Comments

More generally, coefficients a(n,q) that satisfy:

. A(x,q) = Sum_{n>=0} a(n,q)*x^n/q^(n(n-1)/2) such that

. A(x,q) = exp( Sum_{n>=1} A(x/q^n,q)^n * x^n/n )

begin:

a(0,q) = 1; a(1,q) = 1; a(2,q) = q + 1; a(3,q) = q^3 + q^2 + 2*q + 1;

a(4,q) = q^6 + q^5 + 5/2*q^4 + 3*q^3 + 5/2*q^2 + 3*q + 1;

a(5,q) = q^10 + q^9 + 5/2*q^8 + 5*q^7 + 11/2*q^6 + 4*q^5 + 15/2*q^4 + 6*q^3 + 9/2*q^2 + 4*q + 1; ...

Conjecture: {a(n,q), n>=0} are integers when q is a positive integer.

Conjecture: a(n,q) = 1 (mod q) for n>=0 when q is a positive integer. [From Paul D. Hanna, Apr 29 2010]

Links

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

Example

G.f.: A(x) = 1 + x + 4*x^2/3 + 43*x^3/3^3 + 1288*x^4/3^6 + 111874*x^5/3^10 +...
log(A(x)) = A(x/3)*x + A(x/9)^2*x^2/2 + A(x/27)^3*x^3/3 + A(x/81)^4*x^4/4 +...

Prog

(PARI) {a(n)=local(A=1+x); for(n=2, n, A=exp(sum(k=1, n, subst(A, x, x/3^k+x*O(x^n))^k*x^k/k))); 3^(n*(n-1)/2)*polcoeff(A, n)}

Crossrefs

Cf. A159592.

Sequence in context: A153255 A015084 A355081 * A220675 A222979 A284192

Adjacent sequences: A176824 A176825 A176826 * A176828 A176829 A176830

Keyword

nonn

Author

Paul D. Hanna, Apr 27 2010

Status

approved