Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I M5359 N2327 #69 Oct 02 2024 14:22:52
%S 1,82,707,3108,9669,24310,52871,103496,187017,317338,511819,791660,
%T 1182285,1713726,2421007,3344528,4530449,6031074,7905235,10218676,
%U 13044437,16463238,20563863,25443544,31208345,37973546,45864027,55014652,65570653,77688014
%N Sum of fourth powers of first n odd numbers.
%D F. E. Croxton and D. J. Cowden, Applied General Statistics. 2nd ed., Prentice-Hall, Englewood Cliffs, NJ, 1955, p. 742.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A002309/b002309.txt">Table of n, a(n) for n = 1..1000</a>
%H J. L. Bailey, <a href="/A002309/a002309.pdf">A table to facilitate the fitting of certain logistic curves</a>, Annals Math. Stat., 2 (1931), 355-359. [Annotated scanned copy]
%H J. L. Bailey, Jr., <a href="http://www.jstor.org/stable/2957534">A table to facilitate the fitting of certain logistic curves</a>, Annals Math. Stat., 2 (1931), 355-359.
%H F. E. Croxton and D. J. Cowden, <a href="/A000447/a000447.pdf">Applied General Statistics</a>, 2nd Ed., Prentice-Hall, Englewood Cliffs, NJ, 1955 [Annotated scans of just pages 742-743]
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).
%F a(n) = (48*n^5 - 40*n^3 + 7*n)/15. - _Ralf Stephan_, Jan 29 2003
%F 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
%F 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
%F From _Wolfdieter Lang_, Mar 11 2017: (Start)
%F G.f.: x*(1 + 76*x + 230*x^2 + 76*x^3 + 1*x^4)/(1-x)^6.
%F 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)
%e a(1) = 1^4 = 1.
%e a(2) = 1^4 + 3^4 = 82.
%e a(3) = 1^4 + 3^4 + 5^4 = 707.
%p A002309:=(1+76*z+230*z**2+76*z**3+z**4)/(z-1)**6; # conjectured by _Simon Plouffe_ in his 1992 dissertation
%t Table[(48*n^5 - 40*n^3 + 7*n)/15, {n, 0, 40}] (* _Stefan Steinerberger_, Apr 10 2006 *)
%t s = 0; lst = {s}; Do[s += n^4; AppendTo[lst, s], {n, 1, 60, 2}]; lst (* _Zerinvary Lajos_, Jul 12 2009 *)
%t Accumulate[Range[1,63,2]^4] (* _Harvey P. Dale_, Oct 24 2011 *)
%t LinearRecurrence[{6,-15,20,-15,6,-1},{1,82,707,3108,9669,24310},20] (* _Harvey P. Dale_, Sep 29 2015 *)
%o (PARI) a(n)=(48*n^5-40*n^3+7*n)/15 \\ _Charles R Greathouse IV_, Apr 07 2016
%o (Python)
%o def A002309(n): return n*(n**2*(6*n**2-5<<3)+7)//15 # _Chai Wah Wu_, Oct 02 2024
%Y Cf. A000027, A000290, A000447, A002593, A005408.
%K nonn,nice,easy
%O 1,2
%A _N. J. A. Sloane_
%E Definition changed by _David A. Corneth_, Mar 11 2017
%E Name clarified by _Mohammed Yaseen_, Jul 24 2023