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A309336
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a(n) = n^4 if n odd, 15*n^4/16 if n even.
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2
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0, 1, 15, 81, 240, 625, 1215, 2401, 3840, 6561, 9375, 14641, 19440, 28561, 36015, 50625, 61440, 83521, 98415, 130321, 150000, 194481, 219615, 279841, 311040, 390625, 428415, 531441, 576240, 707281, 759375, 923521, 983040, 1185921, 1252815, 1500625, 1574640, 1874161, 1954815
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OFFSET
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0,3
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COMMENTS
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LINKS
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Index entries for linear recurrences with constant coefficients, signature (0,5,0,-10,0,10,0,-5,0,1).
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FORMULA
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G.f.: x * (1 + 15*x + 76*x^2 + 165*x^3 + 230*x^4 + 165*x^5 + 76*x^6 + 15*x^7 + x^8)/(1 - x^2)^5.
G.f.: Sum_{k>=1} J_4(k) * x^k/(1 - x^(2*k)), where J_4() is the Jordan function (A059377).
Dirichlet g.f.: zeta(s-4) * (1 - 1/2^s).
a(n) = n^4 * (31 - (-1)^n)/32.
a(n) = Sum_{d|n, n/d odd} J_4(d).
Sum_{n>=1} 1/a(n) = 241*Pi^4/21600 = 1.086832913851601267313987...
Multiplicative with a(2^e) = 15*2^(4*e-4), and a(p^e) = p^(4*e) for odd primes p. - Amiram Eldar, Oct 26 2020
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MATHEMATICA
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a[n_] := If[OddQ[n], n^4, 15 n^4/16]; Table[a[n], {n, 0, 38}]
nmax = 38; CoefficientList[Series[x (1 + 15 x + 76 x^2 + 165 x^3 + 230 x^4 + 165 x^5 + 76 x^6 + 15 x^7 + x^8)/(1 - x^2)^5, {x, 0, nmax}], x]
LinearRecurrence[{0, 5, 0, -10, 0, 10, 0, -5, 0, 1}, {0, 1, 15, 81, 240, 625, 1215, 2401, 3840, 6561}, 39]
Table[n^4 (31 - (-1)^n)/32, {n, 0, 38}]
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CROSSREFS
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KEYWORD
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nonn,easy,mult
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AUTHOR
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STATUS
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approved
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