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A073710
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Convolution of A073709, which is also the first differences of the unique terms of A073709.
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4
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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
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,2
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COMMENTS
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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, ...}.
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LINKS
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FORMULA
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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).
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EXAMPLE
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(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,...}.
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MATHEMATICA
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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]&
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PROG
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(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
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CROSSREFS
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KEYWORD
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easy,nice,nonn
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AUTHOR
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STATUS
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approved
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