cp's OEIS Frontend

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

%I A073707 #33 Jul 19 2025 13:07:49
%S A073707 1,1,2,2,5,5,8,8,18,18,28,28,50,50,72,72,129,129,186,186,301,301,416,
%T A073707 416,664,664,912,912,1368,1368,1824,1824,2730,2730,3636,3636,5234,
%U A073707 5234,6832,6832,9788,9788,12744,12744,17724,17724,22704,22704,31506,31506
%N A073707 Coefficients of a power series whose convolution consists of only the even-indexed terms of the sequence.
%H A073707 Reinhard Zumkeller (confirmed by Paul D. Hanna), <a href="/A073707/b073707.txt">Table of n, a(n) for n = 0..10000</a>
%H A073707 Shane Chern, Dennis Eichhorn, Shishuo Fu, and James A. Sellers, <a href="https://arxiv.org/abs/2507.10965">Convolutive sequences, I: Through the lens of integer partition functions</a>, arXiv:2507.10965 [math.CO], 2025. See p. 18.
%F A073707 G.f.: A(x) satisfies A(x) = (1+x)*A(x^2)^2, with A(0)=1.
%F A073707 G.f.: A(x) = Product_{n>=0} (1 + x^(2^n))^(2^n).
%F A073707 G.f.: A(x) = (1/(1 - x)) * Product_{n>=0} 1/(1 - x^(2^(n+1)))^(2^n). - _Eitan Y. Levine_, Jun 24 2023
%e A073707 (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 + ...).
%t A073707 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 *)
%o A073707 (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))
%o A073707 (PARI) {a(n)=polcoeff(prod(k=0,#binary(n),(1+x^(2^k)+x*O(x^n))^(2^k)),n)}
%o A073707 (Haskell)
%o A073707 a073707 n = a073707_list !! n
%o A073707 a073707_list = 1 : f 0 0 [1] where
%o A073707    f x y zs = z : f (x + y) (1 - y) (z:zs) where
%o A073707      z = sum $ zipWith (*) hzs (reverse hzs) where hzs = drop x zs
%o A073707 -- _Reinhard Zumkeller_, Dec 21 2011
%Y A073707 Cf. A073708, A073709.
%K A073707 easy,nonn
%O A073707 0,3
%A A073707 _Paul D. Hanna_, Aug 04 2002
%E A073707 Definition corrected by _Paul D. Hanna_, Feb 25 2010
%E A073707 Data corrected for n > 45 by _Reinhard Zumkeller_, Dec 21 2011