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A073709
First differences of A073708.
8
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
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
KEYWORD
easy,nice,nonn
AUTHOR
Paul D. Hanna, Aug 05 2002
STATUS
approved