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

0,3

Links

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

Example

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 + ...

Prog

(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), ", "))

Crossrefs

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

Sequence in context: A291443 A210145 A020956 * A164393 A164391 A164153

Adjacent sequences: A384817 A384818 A384819 * A384821 A384822 A384823

Keyword

nonn

Author

Paul D. Hanna, Jun 18 2025

Status

approved