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A002309
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Sum of fourth powers of first n odd numbers.
(Formerly M5359 N2327)
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18
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1, 82, 707, 3108, 9669, 24310, 52871, 103496, 187017, 317338, 511819, 791660, 1182285, 1713726, 2421007, 3344528, 4530449, 6031074, 7905235, 10218676, 13044437, 16463238, 20563863, 25443544, 31208345, 37973546, 45864027, 55014652, 65570653, 77688014
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OFFSET
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1,2
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REFERENCES
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F. E. Croxton and D. J. Cowden, Applied General Statistics. 2nd ed., Prentice-Hall, Englewood Cliffs, NJ, 1955, p. 742.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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F. E. Croxton and D. J. Cowden, Applied General Statistics, 2nd Ed., Prentice-Hall, Englewood Cliffs, NJ, 1955 [Annotated scans of just pages 742-743]
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FORMULA
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a(n) = (48*n^5 - 40*n^3 + 7*n)/15. - Ralf Stephan, Jan 29 2003
a(1)=1, a(2)=82, a(3)=707, a(4)=3108, a(5)=9669, a(6)=24310, a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6). - Harvey P. Dale, Oct 24 2011
a(n) = v(n,n-2) - v(n,n-1)*V(n,n-1), where v(n,k) and V(n,k) are the central factorial numbers of the first kind and the second kind, respectively, with odd indices. - Mircea Merca, Jan 25 2014
G.f.: x*(1 + 76*x + 230*x^2 + 76*x^3 + 1*x^4)/(1-x)^6.
E.g.f. (with offset 0): exp(x)*(1 + 81*x + 544*x^2/2! + 1232*x^3/3! + 1152*x^4/4! + 384*x^5/5!). (End)
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EXAMPLE
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a(1) = 1^4 = 1.
a(2) = 1^4 + 3^4 = 82.
a(3) = 1^4 + 3^4 + 5^4 = 707.
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MAPLE
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A002309:=(1+76*z+230*z**2+76*z**3+z**4)/(z-1)**6; # conjectured by Simon Plouffe in his 1992 dissertation
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MATHEMATICA
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s = 0; lst = {s}; Do[s += n^4; AppendTo[lst, s], {n, 1, 60, 2}]; lst (* Zerinvary Lajos, Jul 12 2009 *)
LinearRecurrence[{6, -15, 20, -15, 6, -1}, {1, 82, 707, 3108, 9669, 24310}, 20] (* Harvey P. Dale, Sep 29 2015 *)
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PROG
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CROSSREFS
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KEYWORD
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nonn,nice,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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