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A209444
a(n) = Pell(n)*A000143(n) for n>=1 with a(0)=1, where A000143(n) is the number of ways of writing n as a sum of 8 squares.
4
1, 16, 224, 2240, 13632, 58464, 219520, 930176, 3805824, 11930320, 33558336, 122352192, 440858880, 1176756448, 3112368896, 11008771200, 35248366848, 89371035936, 232665100640, 727171963840, 2289378446208, 5950875374080, 13907284255872, 43816224486528
OFFSET
0,2
COMMENTS
Compare g.f. to the Lambert series of A000143: 1 + 16*Sum_{n>=1} n^3*x^n/(1 - (-x)^n).
LINKS
FORMULA
G.f.: 1 + 16*Sum_{n>=1} Pell(n)*n^3*x^n/(1 - A002203(n)*(-x)^n + (-1)^n*x^(2*n)), where A002203(n) = Pell(n-1) + Pell(n+1).
EXAMPLE
G.f.: A(x) = 1 + 16*x + 112*x^2 + 896*x^3 + 3408*x^4 + 10080*x^5 +...
where A(x) = 1 + 1*16*x + 2*112*x^2 + 5*448*x^3 + 12*1136*x^4 + 29*2016*x^5 + 70*3136*x^6 + 169*5504*x^7 + 408*9328*x^8 +...+ Pell(n)*A000143(n)*x^n +...
The g.f. is also given by the identity:
A(x) = 1 + 16*( 1*1*x/(1+2*x-x^2) + 2*8*x^2/(1-6*x^2+x^4) + 5*27*x^3/(1+14*x^3-x^6) + 12*64*x^4/(1-34*x^4+x^8) + 29*125*x^5/(1+82*x^5-x^10) + 70*216*x^6/(1-198*x^6+x^12) + 169*343*x^7/(1+478*x^7-x^14) +...).
MATHEMATICA
A000143:= Table[SquaresR[8, n], {n, 0, 200}]; Join[{1}, Table[Fibonacci[n, 2]*A000143[[n + 1]], {n, 1, 50}]] (* G. C. Greubel, Jan 02 2018 *)
PROG
(PARI) {Pell(n)=polcoeff(x/(1-2*x-x^2+x*O(x^n)), n)}
{A002203(n)=Pell(n-1)+Pell(n+1)}
{a(n)=polcoeff(1+16*sum(m=1, n, Pell(m)*m^3*x^m/(1-A002203(m)*(-x)^m+(-1)^m*x^(2*m)+x*O(x^n))), n)}
for(n=0, 31, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Mar 09 2012
STATUS
approved