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

0,2

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, ...}.

Links

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

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).

Example

(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,...}.

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]&

Prog

(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

Crossrefs

Cf. A073707, A073708, A073709.

Sequence in context: A059053 A032025 A088662 * A326026 A092831 A055257

Adjacent sequences: A073707 A073708 A073709 * A073711 A073712 A073713

Keyword

easy,nice,nonn

Author

Paul D. Hanna, Aug 05 2002

Status

approved