A073709 - OEIS

%I #23 Apr 23 2016 14:37:11

%S 1,1,3,3,10,10,22,22,57,57,115,115,248,248,456,456,906,906,1598,1598,

%T 2956,2956,4980,4980,8802,8802,14422,14422,24440,24440,38856,38856,

%U 63881,63881,99515,99515,159106,159106,242654,242654,379609,379609

%N First differences of A073708.

%C The convolution of this sequence results in A073710 and is equal to the first differences of the unique terms of this sequence.

%H Reinhard Zumkeller, <a href="/A073709/b073709.txt">Table of n, a(n) for n = 0..10000</a>

%F G.f. satisfies: A(x) = A(x^2)^2/(1-x).

%F G.f.: Product_{n>=0} 1/(1-x^(2^n))^(2^n). [_Paul D. Hanna_, May 01 2010]

%e G.f.: A(x) = 1 + x + 3*x^2 + 3*x^3 + 10*x^4 + 10*x^5 + 22*x^6 + 22*x^7 +...

%e where A(x) =  A(x^2)^2/(1-x) and thus

%e A(x) = 1 / [(1-x)*(1-x^2)^2*(1-x^4)^4*(1-x^8)^8*(1-x^16)^16*...].

%e Compare A(x)*(1-x) to A(x)^2:

%e A(x)*(1-x) = 1 + 2*x^2 + 7*x^4 + 12*x^6 + 35*x^8 + 58*x^10 + 133*x^12 +...

%e A(x)^2 = 1 + 2*x + 7*x^2 + 12*x^3 + 35*x^4 + 58*x^5 + 133*x^6 + 208*x^7 +...

%e Also note that

%e A(x)^2/(1-x) = 1 + 3*x + 10*x^2 + 22*x^3 + 57*x^4 + 115*x^5 + 248*x^6 + 456*x^7 +...

%t terms = 42; For[m = 1; A = 1, m <= 2*terms, m = 2*m, A = ((1+x)*(Normal[A] /. x -> x^2))^2 + O[x]^m]; Join[{1}, Differences[CoefficientList[A, x] ]][[1 ;; terms]] (* _Jean-François Alcover_, Mar 06 2013, updated Apr 23 2016 *)

%o (PARI) {a(n)=polcoeff(prod(j=0,#binary(n),1/(1-x^(2^j)+x*O(x^n))^(2^j)),n)} \\ _Paul D. Hanna_, May 01 2010

%o (Haskell)

%o a073709 n = a073709_list !! n

%o a073709_list = 1 : zipWith (-) (tail a073708_list) a073708_list

%o --- _Reinhard Zumkeller_, Jun 13 2013

%Y Cf. A073707, A073708, A073710.

%K easy,nice,nonn

%O 0,3

%A _Paul D. Hanna_, Aug 05 2002