A186375 a(n) equals the sum of the squares of the expansion coefficients for (x + y + 2*z)^n.

1, 6, 54, 588, 7110, 91476, 1224636, 16849944, 236523078, 3371140740, 48630906324, 708412918824, 10403176168476, 153813188724552, 2287366047735480, 34185974267420208, 513159651195396678, 7732530110414488932

Offset: 0

Author

Paul D. Hanna, Feb 19 2011

Keywords

• nonn

Comments

Equivalently, a(n) equals the sum of the squares of the coefficients in any one of the following polynomials: (1 + 2*x^k + x^p)^n, (2 + x^k + x^p)^n, or (1 + x^k + 2*x^p)^n, for all p > n*k and fixed k > 0.

Rescaling the g.f. G(x) to T(u)=G(3*u/16) moves the singular point x=1/16 to u=1/3. Period function T(u) measures precession of the J-vector along an algebraic sphere curve with local cyclic C_3 symmetry. For precise definitions, pictures, a proof certificate, and more information, see A318245. - Bradley Klee, Aug 22 2018

Examples

			G.f.: A(x) = 1 + 6*x + 54*x^2/2!^2 + 588*x^3/3!^2 + 7110*x^4/4!^2 + ...
The g.f. may be expressed as:
A(x) = [Sum_{n>=0} x^n/n!^2]^2 *[Sum_{n>=0} (4x)^n/n!^2] where
[Sum_{n>=0} x^n/n!^2]^2 = 1 + 2*x + 6*x^2/2!^2 + 20*x^3/3!^2 + 70*x^4/4!^2 + ... + (2n)!/n!^2 *x^n/n!^2 + ...
a(4) =   256
       + 1024 + 1024
       + 576  + 2304 + 576
       + 64   + 576  + 576 + 64
       + 1    + 16   + 36  + 16 + 1  = 7110.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• E. Weisstein, Goursat's Surface, Mathworld--A Wolfram Web Resource.

Crossrefs

Cf. A046816, A186376, A186377, A186378. Periods: A318245, A318417.

Programs

Maple

A186375 := n -> 4^n*hypergeom([1/2,-n,-n], [1,1], 1):
seq(simplify(A186375(n)), n=0..17); # Peter Luschny, May 24 2017

Mathematica

Table[Sum[Binomial[n,k]^2*Binomial[2k,k]*4^(n-k),{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 20 2012 *)
(* From Bradley Klee, Aug 22 2018: Start *)PyramidLevel[n_]:=If[n==0, {{1}}, Table[Coefficient[(2*x+y+z)^n,x^j*y^k*z^(n-j-k)]^2, {j,0,n}, {k,0,n-j}]]; a1[n_]:= Total[Flatten[PyramidLevel[n]]];
a1 /@ Range[0, 10]
RecurrenceTable[{4*(4*n-5)*(4*n-3)*a[n-2]-2*(10*n^2-10*n+3)*a[n-1]+n^2*a[n]==0, a[0]==1, a[1]==6},a,{n,0,1000}] (*  End *)
a[ n_] := If[ n < 0, 0, Block[ {x, y, z}, Expand[ (x + y + 2 z)^n] /. {t_Integer -> t^2, x -> 1, y -> 1, z -> 1}]]; (* Michael Somos, Aug 27 2018 *)

PARI

{a(n)=sum(k=0,n,binomial(n,k)^2*binomial(2*(n-k),n-k)*4^k)}

PARI

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

PARI

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

Formula

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

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

(2) A(x) = B(x)^2 * B(2^2*x)

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

Recurrence: n^2*a(n) = 2*(10*n^2-10*n+3)*a(n-1) - 4*(4*n-5)*(4*n-3)*a(n-2). - Vaclav Kotesovec, Oct 20 2012

a(n) ~ 2^(4*n+1/2)/(Pi*n). - Vaclav Kotesovec, Oct 20 2012

a(n) = 4^n*hypergeom([1/2,-n,-n], [1,1], 1). - Peter Luschny, May 24 2017

G.f.: G(x)=Sum_{n>=0}a(n)x^n, 6*(10*x-1)*G + (192*x^2-40*x+1)*G' + x*(16*x-1)*(4*x-1)*G''=0. - Bradley Klee, Aug 22 2018

Extensions

Name edited by Bradley Klee, Aug 22 2018