A261594 G.f.: sqrt( Sum_{n>=0} x^n * Sum_{k=0..n} C(n,k) * 2^(k*(k-1)) ).

1, 1, 3, 37, 2149, 532611, 539508291, 2202251249193, 36044200375109487, 2361471528989758715269, 618991271516919971774301613, 649043297118583276751832395970903, 2722266074808493870871954767765560237289, 45671958833739479081570180837023756023304348531, 3064991675467024774224369897734145197065069681513495767

Offset: 0

Author

Paul D. Hanna, Aug 25 2015

Keywords

• nonn

Comments

a(k) = 1 (mod 3) iff k = 9*A005836(n) + [0,1,3,4] for n>=0, with a(k) = 0 (mod 3) elsewhere, where A005836 lists numbers n whose base 3 representation contains no 2 (conjecture).

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 37*x^3 + 2149*x^4 + 532611*x^5 + 539508291*x^6 +...
where
A(x)^2 = 1 + 2*x + 7*x^2 + 80*x^3 + 4381*x^4 + 1069742*x^5 + 1080096067*x^6 +...+ A135756(n)*x^n +...
such that
A135756(n) = Sum_{k=0..n} binomial(n,k) * 2^(k*(k-1)).
The residue of the terms modulo 3 begin:
[1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, ...]
where a(k) appears to be congruent to 1 (mod 3) at k = 9*A005836(n) + [0,1,3,4] for n>=0, and congruent to zeros elsewhere.
		

Crossrefs

Cf. A135756.

Programs

PARI

{a(n) = polcoeff( sum(m=0,n, x^m*sum(k=0, m, binomial(m, k)*2^(k*(k-1))) +x*O(x^n))^(1/2), n)}
for(n=0,20,print1(a(n),", "))