A073708 Generating function A(x) satisfies A(x) = (1+x)^2*A(x^2)^2, with A(0)=1.

1, 2, 5, 8, 18, 28, 50, 72, 129, 186, 301, 416, 664, 912, 1368, 1824, 2730, 3636, 5234, 6832, 9788, 12744, 17724, 22704, 31506, 40308, 54730, 69152, 93592, 118032, 156888, 195744, 259625, 323506, 423021, 522536, 681642, 840748, 1083402, 1326056, 1705665

Offset

0,2

Links

Paul D. Hanna, Table of n, a(n) for n = 0..1000

Formula

Equals the self-convolution of A073707.

Example

G.f.: A(x) = 1 + 2*x + 5*x^2 + 8*x^3 + 18*x^4 + 28*x^5 + 50*x^6 +...
where A(x)^2 = 1 + 4*x + 14*x^2 + 36*x^3 + 93*x^4 + 208*x^5 + 456*x^6 +...
This sequence equals the self-convolution of A073707, which begins:
[1, 1, 2, 2, 5, 5, 8, 8, 18, 18, 28, 28, 50, 50, ...].
The first differences of this sequence result in A073709:
[1, 1, 3, 3, 10, 10, 22, 22, 57, 57, 115, 115, ...];
the self-convolution of A073709 yields A073710:
[1, 2, 7, 12, 35, 58, 133, 208, ...],
which in turn equals the first differences of the unique terms of A073709.

Mathematica

A073708list[n_] := Module[{m = 1, A = 1}, While[m <= n, m = 2 m; A = ((1 + x)*(A /. x -> x^2))^2] + O[x]^m; CoefficientList[A, x][[1 ;; n]]]; A073708list[50] (* Jean-François Alcover, Apr 21 2016, adapted from PARI *)

Prog

(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))
(Haskell)
a073708 n = a073708_list !! n
a073708_list = conv a073707_list [] where
   conv (v:vs) ws = (sum $ zipWith (*) ws' $ reverse ws') : conv vs ws'
                    where ws' = v : ws
-- Reinhard Zumkeller, Jun 13 2013

Crossrefs

Cf. A073709, A073710. A073707(2n)=a(n).

Sequence in context: A050872 A086324 A293830 * A024460 A039658 A063675

Adjacent sequences: A073705 A073706 A073707 * A073709 A073710 A073711

Keyword

easy,nonn

Author

Paul D. Hanna, Aug 04 2002

Extensions

Edited by Michael Somos, May 03, 2003

Edited by Paul D. Hanna, Jan 04 2013

Status

approved