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.

1, 1, 2, 4, 8, 14, 25, 43, 74, 124, 205, 335, 543, 869, 1379, 2170, 3388, 5249, 8079, 12353, 18776, 28375, 42651, 63782, 94923, 140614, 207384, 304578, 445528, 649200, 942495, 1363447, 1965697, 2824676, 4046190, 5778273, 8227533, 11681632, 16540183, 23357053, 32898242

Offset: 0

Author

Paul D. Hanna, Jun 18 2025

Keywords

• nonn

Examples

			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 + ...
where log(A(x)) = x/(1-x)^2 - D(x) and D(x) is the l.g.f. of A384819:
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 + ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..520

Crossrefs

Cf. A384819, A082579 (exp(x/(1-x)^2)).

Programs

PARI

{a(n) = my(L=[1],A=1); for(i=1,n, L = concat(L,t);
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)}
for(n=0,40, print1(a(n),", "))