OFFSET
0,3
LINKS
Reinhard Zumkeller (confirmed by Paul D. Hanna), Table of n, a(n) for n = 0..10000
Shane Chern, Dennis Eichhorn, Shishuo Fu, and James A. Sellers, Convolutive sequences, I: Through the lens of integer partition functions, arXiv:2507.10965 [math.CO], 2025. See p. 18.
FORMULA
G.f.: A(x) satisfies A(x) = (1+x)*A(x^2)^2, with A(0)=1.
G.f.: A(x) = Product_{n>=0} (1 + x^(2^n))^(2^n).
G.f.: A(x) = (1/(1 - x)) * Product_{n>=0} 1/(1 - x^(2^(n+1)))^(2^n). - Eitan Y. Levine, Jun 24 2023
G.f.: A(x) satisfies B(x/A(x)) = x where B(x) = x*A(B(x)) is the g.f. of A389473. - Paul D. Hanna, Oct 27 2025
EXAMPLE
G.f.: A(x) = 1 + x + 2*x^2 + 2*x^3 + 5*x^4 + 5*x^5 + 8*x^6 + 8*x^7 + 18*x^8 + 18*x^9 + 28*x^10 + 28*x^11 + 50*x^12 + ...
where A(x) = (1 + x) * A(x^2)^2 as can be seen from the coefficients in A(x^2)^2 = 1 + 2*x^2 + 5*x^4 + 8*x^6 + 18*x^8 + 28*x^10 + 50*x^12 + 72*x^14 + ...
MATHEMATICA
nmax = 49; CoefficientList[ Series[ Product[ (1+x^(2^n))^(2^n), {n, 0, Log[nmax]/Log[2]}], {x, 0, nmax}], x] (* Jean-François Alcover, Jan 04 2013, from 2nd formula, modified by Vaclav Kotesovec, Oct 23 2020 *)
PROG
(PARI) a(n)=local(A, m); if(n<0, 0, m=1; A=1+O(x); while(m<=n, m*=2; A=(1+x)*subst(A, x, x^2)^2); polcoeff(A, n))
(PARI) {a(n)=polcoeff(prod(k=0, #binary(n), (1+x^(2^k)+x*O(x^n))^(2^k)), n)}
(Haskell)
a073707 n = a073707_list !! n
a073707_list = 1 : f 0 0 [1] where
f x y zs = z : f (x + y) (1 - y) (z:zs) where
z = sum $ zipWith (*) hzs (reverse hzs) where hzs = drop x zs
-- Reinhard Zumkeller, Dec 21 2011
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul D. Hanna, Aug 04 2002
EXTENSIONS
Definition corrected by Paul D. Hanna, Feb 25 2010
Data corrected for n > 45 by Reinhard Zumkeller, Dec 21 2011
STATUS
approved