OFFSET
0,2
COMMENTS
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.
Equivalently, a(n) equals the sum of the squares of the coefficients in any one of the following polynomials:
. (1 + 3*x^k + 2*x^p)^n, or
. (2 + x^k + 3*x^p)^n, or
. (3 + 2*x^k + x^p)^n, etc.,
for all p>(n+1)k and fixed k>0.
a(n) is the sum of the squares of the coefficients of (x + 2*y + 3*z)^n. - Michael Somos, Aug 25 2018
FORMULA
(1) a(n) = Sum_{k=0..n} C(n,k)^2 *Sum_{j=0..k} C(k,j)^2*4^(k-j)*9^j.
Let g.f. A(x) = Sum_{n>=0} a(n)*x^n/n!^2, then
(2) A(x) = B(x) * B(2^2*x) * B(3^2*x)
where B(x) = Sum_{n>=0} x^n/n!^2 = BesselI(0, 2*sqrt(x)).
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
a(n) ~ 2^(2*n-1) * 3^(2*n+1) / (Pi*n). - Vaclav Kotesovec, Feb 12 2015
EXAMPLE
G.f.: A(x) = 1 + 14*x + 294*x^2/2!^2 + 7292*x^3/3!^2 +...
The g.f. may be expressed as:
[Sum_{n>=0}x^n/n!^2]*[Sum_{n>=0}(4x)^n/n!^2]*[Sum_{n>=0}(9x)^n/n!^2].
MATHEMATICA
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 *)
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 *)
PROG
(PARI) {a(n)=local(V=Vec((1+2*x+3*x^(n+2))^n)); V*V~}
(PARI) {a(n)=sum(k=0, n, binomial(n, k)^2*sum(j=0, k, binomial(k, j)^2*4^(k-j)*9^j))}
(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)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 19 2011
EXTENSIONS
Name changed to match the definition given by Michael Somos. - Paul D. Hanna, Sep 05 2018
STATUS
approved