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A100228 G.f. A(x) satisfies: 4^n - 1 = Sum_{k=0..n} [x^k] A(x)^n and also satisfies: (4+z)^n - (1+z)^n + z^n = Sum_{k=0..n} [x^k](A(x)+z*x)^n for all z, where [x^k] A(x)^n denotes the coefficient of x^k in A(x)^n.

%I A100228 #13 Jun 26 2015 22:24:34
%S A100228 1,2,3,-3,-6,24,3,-183,273,1131,-4407,-3081,48360,-54750,-396195,
%T A100228 1282551,1860186,-17122944,11240049,166745823,-432682314,-1054472016,
%U A100228 6822994737,-835915197,-76044224139,152526011235,587055710271,-2871405804783,-1378878506592,36081844133766
%N A100228 G.f. A(x) satisfies: 4^n - 1 = Sum_{k=0..n} [x^k] A(x)^n and also satisfies: (4+z)^n - (1+z)^n + z^n = Sum_{k=0..n} [x^k](A(x)+z*x)^n for all z, where [x^k] A(x)^n denotes the coefficient of x^k in A(x)^n.
%C A100228 More generally, if g.f. A(x) satisfies: m^n-b^n = Sum_{k=0..n} [x^k]A(x)^n, then A(x) also satisfies: (m+z)^n - (b+z)^n + z^n = Sum_{k=0..n} [x^k](A(x)+z*x)^n for all z and A(x)=(1+(m-1)*x+sqrt(1+2*(m-2*b-1)*x+(m^2-2*m+4*b+1)*x^2))/2.
%F A100228 a(n) = -((2*n-3)*a(n-1) + 13*(n-3)*a(n-2))/n for n>2, with a(0)=1, a(1)=2, a(3)=3.
%F A100228 G.f.: (1+3*x + sqrt(1 + 2*x + 13*x^2))/2.
%e A100228 From the table of powers of A(x) (A100229), we see that
%e A100228 4^n-1 = Sum of coefficients [x^0] through [x^n] in A(x)^n:
%e A100228 A^1=[1,2],3,-3,-6,24,3,-183,273,...
%e A100228 A^2=[1,4,10],6,-15,6,75,-174,-276,...
%e A100228 A^3=[1,6,21,35],9,-36,63,72,-612,...
%e A100228 A^4=[1,8,36,92,118],12,-66,192,-147,...
%e A100228 A^5=[1,10,55,185,380,392],15,-105,420,...
%e A100228 A^6=[1,12,78,322,879,1506,1297],18,-153,...
%t A100228 CoefficientList[Series[(1+3x+Sqrt[1+2x+13x^2])/2,{x,0,30}],x] (* or *) Join[{1},RecurrenceTable[{a[1]==2,a[2]==3,a[n]==-((2n-3)a[n-1]+ 13(n-3)a[n-2])/n},a,{n,30}]] (* _Harvey P. Dale_, Feb 29 2012 *)
%o A100228 (PARI) {a(n) = if(n==0,1,(4^n-1-sum(k=0,n,polcoeff(sum(j=0,min(k,n-1),a(j)*x^j)^n+x*O(x^k),k)))/n)}
%o A100228 for(n=0,20,print1(a(n),", "))
%o A100228 (PARI) {a(n) = if(n==0,1,if(n==1,2,if(n==2,3,-((2*n-3)*a(n-1)+13*(n-3)*a(n-2))/n)))}
%o A100228 for(n=0,30,print1(a(n),", "))
%o A100228 (PARI) {a(n) = polcoeff((1+3*x+sqrt(1+2*x+13*x^2+x^2*O(x^n)))/2,n)}
%o A100228 for(n=0,30,print1(a(n),", "))
%Y A100228 Cf. A100229, A100230.
%K A100228 sign
%O A100228 0,2
%A A100228 _Paul D. Hanna_, Nov 29 2004