|
%I #12 Sep 27 2014 11:03:01
%S 1,3,69,5005,806148,239220375,116532061510,86173621173099,
%T 91417549409916684,133300597778263476112,258360728839130761571757,
%U 647880493609691058921741273,2055869510173976408422116133220,8103111707775918586405906798540650,39047811321420953231675462397758519802
%N G.f.: 1 = Sum_{n>=0} a(n)*x^n * [ Sum_{k=0..n+1} binomial(n+1, k)^3 * (-x)^k ]^3.
%C Compare to a g.f. of A006632(n) = 3*binomial(4*n+3,n)/(4*n+3):
%C 1 = Sum_{n>=0} A006632(n)*x^n * [ Sum_{k=0..n+1} binomial(n+1, k)*(-x)^k ]^3
%C where A006632 equals the self-convolution cube of A002293.
%C ...
%C a(3*n) == 2 (mod 3) iff A245658(n) == 2 (mod 3), where A245658 is the self-convolution cube root of this sequence (conjecture).
%H Paul D. Hanna, <a href="/A247031/b247031.txt">Table of n, a(n) for n = 0..200</a>
%e O.g.f.: A(x) = 1 + 3*x + 69*x^2 + 5005*x^3 + 806148*x^4 + 239220375*x^5 +...
%e such that the coefficients satisfy:
%e 1 = 1*(1-x)^3 + 3*x*(1-2^3*x+x^2)^3 + 69*x^2*(1-3^3*x+3^3*x^2-x^3)^3 + 5005*x^3*(1-4^3*x+6^3*x^2-4^3*x^3+x^4)^3 + 806148*x^4*(1-5^3*x+10^3*x^2-10^3*x^3+5^3*x^4-x^5)^3 +...
%e Note that the cube-root of the o.g.f., A(x)^(1/3), is an integer series:
%e A(x)^(1/3) = 1 + x + 22*x^2 + 1624*x^3 + 264962*x^4 + 79136637*x^5 + 38671111558*x^6 + 28642761340956*x^7 + 30413158977739302*x^8 +...+ A245658(n)*x^n +...
%e CONJECTURE: given
%e G(x,m,j) = Sum_{n>=0} a(n) * (m*x)^n * [ Sum_{k=0..n+1} C(n+1, k)^3 * (j*x)^k ]^3
%e then G(x,m,j)^(1/3) is an integer series in x whenever m, j, are integers.
%e OBSERVATIONS.
%e The terms of this sequence are congruent to 2 modulo 3 at positions:
%e [12, 36, 93, 108, 111, 114, 120, 174, 255, 279, ...].
%e The terms of A245658 are congruent to 2 modulo 3 at positions:
%e [4, 12, 31, 36, 37, 38, 40, 58, 85, 93, ...].
%o (PARI) {a(n)=if(n==0, 1, -polcoeff(sum(m=0, n-1, a(m)*x^m*sum(k=0, m+1, binomial(m+1, k)^3 * (-x)^k +x*O(x^n) )^3 ), n))}
%o for(n=0, 20, print1(a(n), ", "))
%Y Cf. A245658, A247030, A212370.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Sep 09 2014
|
