|
| |
|
|
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.
|
|
2
|
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,2
|
|
|
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.
|
|
|
LINKS
|
|
|
|
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.
|
|
|
EXAMPLE
|
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,...
|
|
|
PROG
|
(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)
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
sign
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
