A382314 G.f. satisfies A(x) = 1/(1-x) + 2*x*A(x^2) + 3*x^2*A(x^3).

1, 3, 4, 7, 1, 18, 1, 15, 13, 3, 1, 58, 1, 3, 4, 31, 1, 81, 1, 7, 4, 3, 1, 162, 1, 3, 40, 7, 1, 18, 1, 63, 4, 3, 1, 337, 1, 3, 4, 15, 1, 18, 1, 7, 13, 3, 1, 418, 1, 3, 4, 7, 1, 324, 1, 15, 4, 3, 1, 58, 1, 3, 13, 127, 1, 18, 1, 7, 4, 3, 1, 1161, 1, 3, 4, 7, 1, 18, 1, 31, 121, 3, 1, 58, 1, 3, 4, 15, 1, 81, 1, 7, 4, 3, 1, 1026

Offset: 0

Author

Paul D. Hanna, Apr 14 2025

Keywords

• nonn

Comments

Logarithmic derivative of A382126.

Examples

			G.f.: A(x) = 1 + 3*x + 4*x^2 + 7*x^3 + x^4 + 18*x^5 + x^6 + 15*x^7 + 13*x^8 + 3*x^9 + x^10 + 58*x^11 + x^12 + 3*x^13 + 4*x^14 + 31*x^15 + ...
where A(x) = 1/(1-x) + 2*x*A(x^2) + 3*x^2*A(x^3).
RELATED SERIES.
The logarithm of the g.f. B(x) for A382126 yields the series
log(B(x)) = x + 3*x^2/2 + 4*x^3/3 + 7*x^4/4 + x^5/5 + 18*x^6/6 + x^7/7 + 15*x^8/8 + 13*x^9/9 + 3*x^10/10 + x^11/11 + 58*x^12/12 + ... + a(n-1)*x^n/n + ...
where B(x) = B(x^2)*B(x^3)/(1-x) begins
B(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 6*x^5 + 11*x^6 + 13*x^7 + 20*x^8 + 26*x^9 + 36*x^10 + 44*x^11 + 66*x^12 + ... + A382126(n)*x^n + ...
so that A(x) = B'(x)/B(x).
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..1030

Crossrefs

Cf. A382126, A072079, A003586, A007814, A007949.

Programs

PARI

{a(n) = my(A=1+x +x*O(x^n)); for(i=1,#binary(n), A = 1/(1-x) + 2*x*subst(A,x,x^2) + 3*x^2*subst(A,x,x^3) + x*O(x^n)  ); polcoef(A,n)}
for(n=0,100,print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.

(1) A(x) = 1/(1-x) + 2*x*A(x^2) + 3*x^2*A(x^3).

(2) A(x) = (1-x)^2*(1+2*x)*(1+2*x+3*x^2)/((1-x)*(1-x^2)*(1-x^3)) + 4*x^3*A(x^4) + 12*x^5*A(x^6) + 9*x^8*A(x^9).

(3) A(x) = (1/x)*Sum_{n>=0} Sum_{k=0..n} binomial(n,k) * 2^(n-k)*3^k * x^(2^(n-k)*3^k) / (1 - x^(2^(n-k)*3^k)).

(4) A(x) = (1/x)*Sum_{n>=1} B(n) * A003586(n) * x^A003586(n)/(1 - x^A003586(n)) where B(n) = binomial(F2(n)+F3(n),F3(n)), with F2(n) = A007814(A003586(n)) and F3(n) = A007949(A003586(n)).

(5) A(x) = B'(x)/B(x) where B(x) = B(x^2)*B(x^3)/(1-x) is the g.f. of A382126.