A186376 - OEIS

%I #19 Sep 05 2018 12:23:11

%S 1,14,294,7292,198310,5717124,171485916,5290989816,166688596998,

%T 5335746337460,172951272017524,5662840724506056,186960502253087836,

%U 6215612039963043368,207865952390729881080,6987002286567227550192

%N a(n) is the sum of the squares of the coefficients of (x + 2*y + 3*z)^n.

%C a(n) equals the least sum of the squares of the coefficients in (1 + 2*x^k + 3*x^p)^n found at sufficiently large p for some fixed k>0.

%C Equivalently, a(n) equals the sum of the squares of the coefficients in any one of the following polynomials:

%C . (1 + 3*x^k + 2*x^p)^n, or

%C . (2 + x^k + 3*x^p)^n, or

%C . (3 + 2*x^k + x^p)^n, etc.,

%C for all p>(n+1)k and fixed k>0.

%C a(n) is the sum of the squares of the coefficients of (x + 2*y + 3*z)^n. - _Michael Somos_, Aug 25 2018

%F (1) a(n) = Sum_{k=0..n} C(n,k)^2 *Sum_{j=0..k} C(k,j)^2*4^(k-j)*9^j.

%F Let g.f. A(x) = Sum_{n>=0} a(n)*x^n/n!^2, then

%F (2) A(x) = B(x) * B(2^2*x) * B(3^2*x)

%F where B(x) = Sum_{n>=0} x^n/n!^2 = BesselI(0, 2*sqrt(x)).

%F Recurrence: n^2*(4*n - 7)*a(n) = 14*(16*n^3 - 44*n^2 + 34*n - 9)*a(n-1) - 196*(2*n - 3)^2*(4*n - 3)*a(n-2) + 144*(4*n - 9)*(4*n - 7)*(4*n - 3)*a(n-3). - _Vaclav Kotesovec_, Feb 12 2015

%F a(n) ~ 2^(2*n-1) * 3^(2*n+1) / (Pi*n). - _Vaclav Kotesovec_, Feb 12 2015

%e G.f.: A(x) = 1 + 14*x + 294*x^2/2!^2 + 7292*x^3/3!^2 +...

%e The g.f. may be expressed as:

%e [Sum_{n>=0}x^n/n!^2]*[Sum_{n>=0}(4x)^n/n!^2]*[Sum_{n>=0}(9x)^n/n!^2].

%t Table[Sum[Binomial[n,k]^2 * Sum[Binomial[k,j]^2 * 4^(k-j) * 9^j, {j,0,k}], {k,0,n}], {n,0,20}] (* _Vaclav Kotesovec_, Feb 11 2015 *)

%t a[ n_] := If[ n < 0, 0, Block[ {x, y, z}, Expand[(x + y + 2 z)^n] /. {t_Integer -> t^2, x -> 1, y -> 2, z -> 3}]]; (* _Michael Somos_, Aug 25 2018 *)

%o (PARI) {a(n)=local(V=Vec((1+2*x+3*x^(n+2))^n));V*V~}

%o (PARI) {a(n)=sum(k=0,n,binomial(n,k)^2*sum(j=0,k,binomial(k,j)^2*4^(k-j)*9^j))}

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

%Y Cf. A186375, A186377, A186378, A186391.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Feb 19 2011

%E Name changed to match the definition given by _Michael Somos_. - _Paul D. Hanna_, Sep 05 2018