A073710 Convolution of A073709, which is also the first differences of the unique terms of A073709.

1, 2, 7, 12, 35, 58, 133, 208, 450, 692, 1358, 2024, 3822, 5620, 10018, 14416, 25025, 35634, 59591, 83548, 136955, 190362, 303725, 417088, 655128, 893168, 1374632, 1856096, 2820456, 3784816, 5658968, 7533120, 11144042, 14754964, 21542374

Offset: 0

Author

Paul D. Hanna, Aug 05 2002

Keywords

• easy
• nice
• nonn

Comments

First differences consist of duplicated terms: {1, 1, 5, 5, 23, 23, 75, 75, 242, 242, 666, 666, 1798, 1798, ...}; the convolution of these first differences results in: {1, 2, 11, 20, 81, 142, 451, 760, 2143, 3526, 8965, ...}, which in turn has first differences that consist of duplicated terms: {1, 1, 9, 9, 61, 61, 309, 309, ...}.

Examples

			(1 +x +3x^2 +3x^3 +10x^4 +10x^5 +22x^6 +22x^7 +57x^8 +57x^9 +...)^2 = (1 +2x +7x^2 +12x^3 +35x^4 +58x^5 +133x^6 +208x^7 +450x^8 +...) and the first differences of the unique terms {1,3,10,22,57,...} is {1,2,7,12,35,...}.
		

Links

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

Crossrefs

Cf. A073707, A073708, A073709.

Programs

Haskell

a073710 n = a073710_list !! n
a073710_list = conv a073709_list [] where
   conv (v:vs) ws = (sum $ zipWith (*) ws' $ reverse ws') : conv vs ws'
                    where ws' = v : ws
-- Reinhard Zumkeller, Jun 13 2013

Mathematica

max = 70; f[x_] := Sum[ a[k]*x^k, {k, 0, max}]; a[0] = a[1] = 1; coes = CoefficientList[ Series[ f[x^2]^2 - (1 - x)*f[x], {x, 0, max}], x]; A073709 = Table[a[k], {k, 0, max}] /. Solve[ Thread[coes == 0]] // First; A073709 // Union // Differences // Prepend[#, 1]&

Formula

Let f(x) = sum_{n=0..inf} A073709(n) x^n, then f(x) satisfies f(x)^2 = sum_{n=0..inf} a(n) x^n, as well as the functional equation f(x^2)^2 = (1 - x)*f(x).