A002309 - 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