

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
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OFFSET

0,2


COMMENTS

More generally, if g.f. A(x) satisfies: m^nb^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+(m1)*x+sqrt(1+2*(m2*b1)*x+(m^22*m+4*b+1)*x^2))/2.


LINKS



FORMULA

a(n)=(3*(2*n3)*a(n1)+29*(n3)*a(n2))/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^n1 = 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^n1sum(k=0, n, polcoeff(sum(j=0, min(k, n1), 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*n3)*a(n1)+29*(n3)*a(n2))/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



