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A100076 E.g.f. A(x) satisfies: Sum_{k=0..n} (A(x)^n)_k/k! = [sqrt(5)^n] for all n>=0, where (A(x)^n)_k/k! is the coefficient of x^k in A(x)^n. 1

%I #3 Mar 30 2012 18:36:43

%S 1,1,1,-3,9,-33,513,-10917,155313,-869697,-27095391,1126973331,

%T -25370851671,400873570911,-3945969886815,-19472448499317,

%U 3355787673885537,-205870807636111233,10635145244261722305,-447262563680813504349,13896854240554592685081

%N E.g.f. A(x) satisfies: Sum_{k=0..n} (A(x)^n)_k/k! = [sqrt(5)^n] for all n>=0, where (A(x)^n)_k/k! is the coefficient of x^k in A(x)^n.

%C See triangle A100075 of initial coefficients of successive powers of the e.g.f. for this sequence.

%e List the coefficients of powers of e.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n!:

%e A(x)^0: [1,__0,0,0,0,0,0,0,0,...],

%e A(x)^1: [1,1,__1,-3,9,-33,513,-10917,155313,...],

%e A(x)^2: [1,2,4,__0,0,-36,1080,-17928,181440,...],

%e A(x)^3: [1,3,9,15,__9,-99,1521,-17631,112401,...],

%e A(x)^4: [1,4,16,48,96,__-72,1296,-11664,31104,...],

%e A(x)^5: [1,5,25,105,345,555,__1305,-6705,-6255,...],

%e A(x)^6: [1,6,36,192,864,2772,6408,__648,-15552,...],...

%e then for each row n, Sum_{k=0..n} (A(x)^n)_k/k! = [sqrt(5)^n]:

%e [sqrt(5)^0] = 1 = 1

%e [sqrt(5)^1] = 1+1 = 2

%e [sqrt(5)^2] = 1+2+4/2! = 5

%e [sqrt(5)^3] = 1+3+9/2!+15/3! = 11

%e [sqrt(5)^4] = 1+4+16/2!+48/3!+96/4! = 25

%e [sqrt(5)^5] = 1+5+25/2!+105/3!+345/4!+555/5! = 55

%e [sqrt(5)^6] = 1+6+36/2!+192/3!+864/4!+2772/5!+6408/6! = 125

%o (PARI) {a(n)=local(A,C,F,G);if(n==0,A=1,F=sum(k=0,n-1,a(k)*x^k/k!); C=floor(sqrt(5)^n+1/10^15)-sum(k=0,n-1,polcoeff(F^n+x*O(x^k),k,x)); G=sum(k=0,n-1,polcoeff(F^n+x*O(x^k),k,x)*x^k); A=n!*polcoeff((G+C*x^n)^(1/n)+x*O(x^n),n,x));A}

%Y Cf. A100075, A100064.

%K sign

%O 0,4

%A _Paul D. Hanna_, Nov 03 2004

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