A073709 First differences of A073708.

1, 1, 3, 3, 10, 10, 22, 22, 57, 57, 115, 115, 248, 248, 456, 456, 906, 906, 1598, 1598, 2956, 2956, 4980, 4980, 8802, 8802, 14422, 14422, 24440, 24440, 38856, 38856, 63881, 63881, 99515, 99515, 159106, 159106, 242654, 242654, 379609, 379609

Offset: 0

Author

Paul D. Hanna, Aug 05 2002

Keywords

• easy
• nice
• nonn

Comments

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

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 3*x^3 + 10*x^4 + 10*x^5 + 22*x^6 + 22*x^7 +...
where A(x) =  A(x^2)^2/(1-x) and thus
A(x) = 1 / [(1-x)*(1-x^2)^2*(1-x^4)^4*(1-x^8)^8*(1-x^16)^16*...].
Compare A(x)*(1-x) to A(x)^2:
A(x)*(1-x) = 1 + 2*x^2 + 7*x^4 + 12*x^6 + 35*x^8 + 58*x^10 + 133*x^12 +...
A(x)^2 = 1 + 2*x + 7*x^2 + 12*x^3 + 35*x^4 + 58*x^5 + 133*x^6 + 208*x^7 +...
Also note that
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 +...
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A073707, A073708, A073710.

Programs

Haskell

a073709 n = a073709_list !! n
a073709_list = 1 : zipWith (-) (tail a073708_list) a073708_list
--- Reinhard Zumkeller, Jun 13 2013

Mathematica

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 *)

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

Formula

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

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