OFFSET
0,3
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..500
FORMULA
O.g.f. A(x) satisfies:
(1) A(x) = 1/(1 - x * (d/dx x + x^2*A(x)) ).
(2) 0 = [x^n] exp( n * (x + x^2*A(x)) ) / A(x), for n > 0.
EXAMPLE
O.g.f.: A(x) = 1 + x + 3*x^2 + 8*x^3 + 29*x^4 + 106*x^5 + 438*x^6 + 1869*x^7 + 8499*x^8 + 39926*x^9 + 195332*x^10 + 983262*x^11 + 5099640*x^12 + ...
where
1/A(x) = 1 - x - 2*x^2 - 3*x^3 - 12*x^4 - 40*x^5 - 174*x^6 - 742*x^7 - 3504*x^8 - 16821*x^9 - 84990*x^10 - ...
and
d/dx x + x^2*A(x) = 1 + 2*x + 3*x^2 + 12*x^3 + 40*x^4 + 174*x^5 + 742*x^6 + 3504*x^7 + 16821*x^8 + + 84990*x^9 + ...
PROG
(PARI) /* A(x) = 1/(1 - x*( d/dx x + x^2*A(x) )) */
{a(n) = my(A=1+x); for(i=1, n, A=1/(1 - x*deriv(x + x^2*A+x*O(x^n)))); polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) /* [x^n] exp( n * (x + x^2*A(x)') ) / A(x) = 0 */
{a(n) = my(A=[1], m); for(i=1, n, A=concat(A, 0); m=#A; A[m] = Vec( exp( (m-1)*(x+x^2*Ser(A)) ) / Ser(A) )[m] ); A[n+1]}
for(n=0, 20, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Oct 21 2020
STATUS
approved