A100234 G.f. A(x) satisfies: 6^n - 1 = Sum_{k=0..n} [x^k]A(x)^n and also satisfies: (6+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, 4, 5, -15, 20, 90, -695, 1785, 3895, -53985, 196255, 121635, -4907130, 23332140, -13181145, -470127465, 2866898820, -4455872910, -44776087145, 356263904235, -873534120380, -3988869806010, 44179467566755, -147200296896765, -293052319462105, 5409366658571715

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) (A100235), we see that
6^n-1 = Sum of coefficients [x^0] through [x^n] in A(x)^n:
A^1=[1,4],5,-15,20,90,-695,1785,...
A^2=[1,8,26],10,-55,190,-245,-1690,...
A^3=[1,12,63,139],15,-120,635,-2130,...
A^4=[1,16,116,436,726],20,-210,1480,...
A^5=[1,20,185,965,2830,3774],25,-325,...
A^6=[1,24,270,1790,7335,17634,19601],30,...
		

Crossrefs

Cf. A100235, A100236, A100231.

Programs

PARI

a(n)=if(n==0,1,(6^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==1,4,if(n==2,5,-(3*(2*n-3)*a(n-1)+29*(n-3)*a(n-2))/n)))

PARI

a(n)=polcoeff((1+5*x+sqrt(1+6*x+29*x^2+x^2*O(x^n)))/2,n)

Formula

a(n)=-(3*(2*n-3)*a(n-1)+29*(n-3)*a(n-2))/n for n>2, with a(0)=1, a(1)=4, a(2)=5. G.f.: A(x) = (1+5*x+sqrt(1+6*x+29*x^2))/2.