A073707 Coefficients of a power series whose convolution consists of only the even-indexed terms of the sequence.

1, 1, 2, 2, 5, 5, 8, 8, 18, 18, 28, 28, 50, 50, 72, 72, 129, 129, 186, 186, 301, 301, 416, 416, 664, 664, 912, 912, 1368, 1368, 1824, 1824, 2730, 2730, 3636, 3636, 5234, 5234, 6832, 6832, 9788, 9788, 12744, 12744, 17724, 17724, 22704, 22704, 31506, 31506

Offset: 0

Author

Paul D. Hanna, Aug 04 2002

Keywords

• easy
• nonn

Examples

			(1 + x + 2x^2 + 2x^3 + 5x^4 + 5x^5 + 8x^6 + 8x^7 + 28x^8 + 28x^9 + ...)^2 = (1 + 2x + 5x^2 + 8x^3 + 18x^4 + 28x^5 + 50x^6 + 72x^7 + 129x^8 + ...).
		

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.

Crossrefs

Cf. A073708, A073709.

Programs

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

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 *)

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)}

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

Extensions

Definition corrected by Paul D. Hanna, Feb 25 2010

Data corrected for n > 45 by Reinhard Zumkeller, Dec 21 2011