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A384820 G.f. A(x) = exp( Sum_{n>=1} (n^2 - A384819(n))*x^n/n ) where A384819(k) < k for k >= 1 such that A(x) is a power series with integral coefficients.

%I A384820 #12 Jun 22 2025 00:51:52
%S A384820 1,1,2,4,8,14,25,43,74,124,205,335,543,869,1379,2170,3388,5249,8079,
%T A384820 12353,18776,28375,42651,63782,94923,140614,207384,304578,445528,
%U A384820 649200,942495,1363447,1965697,2824676,4046190,5778273,8227533,11681632,16540183,23357053,32898242
%N A384820 G.f. A(x) = exp( Sum_{n>=1} (n^2 - A384819(n))*x^n/n ) where A384819(k) < k for k >= 1 such that A(x) is a power series with integral coefficients.
%H A384820 Paul D. Hanna, <a href="/A384820/b384820.txt">Table of n, a(n) for n = 0..520</a>
%e A384820 G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 8*x^4 + 14*x^5 + 25*x^6 + 43*x^7 + 74*x^8 + 124*x^9 + 205*x^10 + 335*x^11 + 543*x^12 + 869*x^13 + 1379*x^14 + 2170*x^15 + 3388*x^16 + ...
%e A384820 where log(A(x)) = x/(1-x)^2 - D(x) and D(x) is the l.g.f. of A384819:
%e A384820 D(x) = 0*x + 1*x^2/2 + 2*x^3/3 + 1*x^4/4 + 4*x^5/5 + 3*x^6/6 + 6*x^7/7 + 1*x^8/8 + 2*x^9/9 + 7*x^10/10 + 10*x^11/11 + 3*x^12/12 + 12*x^13/13 + 11*x^14/14 + 3*x^15/15 + 1*x^16/16 + ... + A384819(n)*x^n/n + ...
%o A384820 (PARI) {a(n) = my(L=[1],A=1); for(i=1,n, L = concat(L,t);
%o A384820 for(t=1,(#L)^2+1, if( denominator( eval(polcoef( A = exp( intformal(Ser(L)) ),#L)) )==1, L[#L] = t + (#L)*(#L-1); break)) ); polcoef(A,n)}
%o A384820 for(n=0,40, print1(a(n),", "))
%Y A384820 Cf. A384819, A082579 (exp(x/(1-x)^2)).
%K A384820 nonn
%O A384820 0,3
%A A384820 _Paul D. Hanna_, Jun 18 2025