A100231 G.f. A(x) satisfies: 5^n - 1 = Sum_{k=0..n} [x^k]A(x)^n and also satisfies: (5+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.

1, 3, 4, -8, 0, 64, -192, -128, 2816, -7680, -13312, 157696, -352256, -1179648, 9748480, -16220160, -99614720, 630456320, -651427840, -8218214400, 41481666560, -13191086080, -667334737920, 2724661821440, 1460876083200, -53446942130176, 175607589634048, 286761410363392

Offset: 0

Author

Paul D. Hanna, Nov 29 2004

Keywords

• sign

Comments

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.

Examples

			From the table of powers of A(x) (A100232), we see that
5^n-1 = Sum of coefficients [x^0] through [x^n] in A(x)^n:
A^1=[1,3],4,-8,0,64,-192,-128,...
A^2=[1,6,17],8,-32,64,64,-896,...
A^3=[1,9,39,75],12,-72,256,-384,...
A^4=[1,12,70,220,321],16,-128,640,...
A^5=[1,15,110,470,1165,1363],20,-200,...
A^6=[1,18,159,852,2895,5922,5777],24,...
		

Crossrefs

Cf. A100232, A100233, A100228.

Programs

PARI

a(n)=if(n==0,1,(5^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)

PARI

a(n)=if(n==0,1,if(n==1,3,if(n==2,4,-((4*n-6)*a(n-1)+20*(n-3)*a(n-2))/n)))

PARI

a(n)=polcoeff((1+4*x+sqrt(1+4*x+20*x^2+x^2*O(x^n)))/2,n)

Formula

a(n)=-((4*n-6)*a(n-1)+20*(n-3)*a(n-2))/n for n>2, with a(0)=1, a(1)=3, a(3)=4. G.f.: A(x) = (1+4*x+sqrt(1+4*x+20*x^2))/2.