A002309 Sum of fourth powers of first n odd numbers.
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
Offset: 1
Examples
a(1) = 1^4 = 1. a(2) = 1^4 + 3^4 = 82. a(3) = 1^4 + 3^4 + 5^4 = 707.
References
- 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).
Links
- T. D. Noe, Table of n, a(n) for n = 1..1000
- J. L. Bailey, A table to facilitate the fitting of certain logistic curves, Annals Math. Stat., 2 (1931), 355-359. [Annotated scanned copy]
- J. L. Bailey, Jr., A table to facilitate the fitting of certain logistic curves, Annals Math. Stat., 2 (1931), 355-359.
- 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]
- Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
- Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
- Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).
Programs
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Maple
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
Table[(48*n^5 - 40*n^3 + 7*n)/15, {n, 0, 40}] (* Stefan Steinerberger, Apr 10 2006 *) s = 0; lst = {s}; Do[s += n^4; AppendTo[lst, s], {n, 1, 60, 2}]; lst (* Zerinvary Lajos, Jul 12 2009 *) Accumulate[Range[1,63,2]^4] (* Harvey P. Dale, Oct 24 2011 *) LinearRecurrence[{6,-15,20,-15,6,-1},{1,82,707,3108,9669,24310},20] (* Harvey P. Dale, Sep 29 2015 *) -
PARI
a(n)=(48*n^5-40*n^3+7*n)/15 \\ Charles R Greathouse IV, Apr 07 2016
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Python
def A002309(n): return n*(n**2*(6*n**2-5<<3)+7)//15 # Chai Wah Wu, Oct 02 2024
Formula
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
From Wolfdieter Lang, Mar 11 2017: (Start)
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)
Extensions
Definition changed by David A. Corneth, Mar 11 2017
Name clarified by Mohammed Yaseen, Jul 24 2023