A073712 Self-convolution of A073711.

1, 2, 3, 6, 7, 12, 16, 26, 31, 42, 59, 72, 104, 116, 184, 186, 303, 282, 497, 406, 783, 612, 1224, 840, 1856, 1232, 2784, 1656, 4136, 2376, 6008, 3138, 8735, 4362, 12345, 5754, 17693, 7756, 24432, 10170, 34471, 13302, 46771, 17688, 65144, 22296, 87008

Offset: 0

Author

Paul D. Hanna, Aug 05 2002

Keywords

• easy
• nice
• nonn
• look

Comments

The g.f. G(x) of A073711 satisfies: G(x) = G(x^2) + x*G(x^2)^2.

The terms of this sequence found at odd-indexed positions are equal to twice that of A194279, which equals the self-convolution cube of A073711.

Links

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

Crossrefs

Cf. A073711, A194279.

Programs

Haskell

a073712 n = a073712_list !! n
a073712_list = map (g a073711_list) [1..] where
g xs k = sum $ zipWith (*) xs $ reverse $ take k xs
-- Reinhard Zumkeller, Dec 20 2012

Mathematica

nmax = 46; max = 2*nmax+1; f[x_] := Sum[ a[k]*x^k, {k, 0, max}]; a[0] = a[1] = a[2] = 1; coes = CoefficientList[ Series[ f[x] - f[x^2] - x*f[x^2]^2, {x, 0, max}], x]; sol = Solve[ Thread[ coes == 0]] // First; Table[ a[2*n+1], {n, 0, nmax}] /. sol (* Jean-François Alcover, Mar 06 2013 *)

PARI

a(n)=local(A=1); for(i=0,#binary(n), A=subst(A,x,x^2+x*O(x^n))+x*subst(A,x,x^2+x*O(x^n))^2);polcoeff(A^2,n)
for(n=0,65,print1(a(n),", ")) \\ Paul D. Hanna, Dec 21 2012

Formula

a(n) = A073711(2*n+1) for n>=0.

a(2*n+1) = 2*A194279(n) for n>=0, where A194279 equals the self-convolution cube of A073711.

Extensions

Name changed and entry revised by Paul D. Hanna, Dec 21 2012