OFFSET
0,4
EXAMPLE
G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 14*x^4 + 81*x^5 + 554*x^6 + 4175*x^7 + 33894*x^8 + 292482*x^9 + 2658803*x^10 + 25312031*x^11 + 251337905*x^12 + ...
such that the following series are equal
B(x) = 1 + x*A(x) + x^2*A(x)^3 + x^3*A(x)^6 + x^4*A(x)^10 + x^5*A(x)^15 + x^6*A(x)^21 + x^7*A(x)^28 + x^8*A(x)^36 + x^9*A(x)^45 + ...
B(x) = 1 + x/(1-x) + x^2/((1-x)*(1-2*x)) + x^3/((1-x)*(1-2*x)*(1-3*x)) + x^4/((1-x)*(1-2*x)*(1-3*x)*(1-4*x)) + x^5/((1-x)*(1-2*x)*(1-3*x)*(1-4*x)*(1-5*x)) + x^6/((1-x)*(1-2*x)*(1-3*x)*(1-4*x)*(1-5*x)*(1-6*x)) + ....
where B(x) is the o.g.f. of the Bell numbers:
B(x) = 1 + x + 2*x^2 + 5*x^3 + 15*x^4 + 52*x^5 + 203*x^6 + 877*x^7 + 4140*x^8 + 21147*x^9 + 115975*x^10 + ... + A000110(n)*x^n + ...
PROG
(PARI) {a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0);
A[#A] = polcoeff( sum(m=0, #A, x^m*( 1/prod(k=1, m, 1-k*x +x*O(x^#A)) - Ser(A)^(m*(m+1)/2)) ), #A) ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Apr 30 2019
STATUS
approved