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

0,3

Comments

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

Links

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

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]

Example

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 +...

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 *)

Prog

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

Crossrefs

Cf. A073707, A073708, A073710.

Sequence in context: A278832 A168376 A266221 * A085288 A124630 A321397

Adjacent sequences: A073706 A073707 A073708 * A073710 A073711 A073712

Keyword

easy,nice,nonn

Author

Paul D. Hanna, Aug 05 2002

Status

approved