%I A000583 M5004 N2154
%S A000583 0,1,16,81,256,625,1296,2401,4096,6561,10000,14641,20736,28561,38416,
%T A000583 50625,65536,83521,104976,130321,160000,194481,234256,279841,331776,
%U A000583 390625,456976,531441,614656,707281,810000,923521,1048576,1185921
%N A000583 Fourth powers: a(n) = n^4.
%C A000583 Figurate numbers based on 4-dimensional regular convex polytope called 
               the 4-measure polytope, 4-hypercube or tessaract with Schlaefli symbol 
               {4,3,3}. - Michael J. Welch (mjw1(AT)ntlworld.com), Apr 01 2004
%C A000583 Sum(k>0,1/a(k))=Pi^4/90 [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), 
               Sep 20 2009]
%C A000583 Totally multiplicative sequence with a(p) = p^4 for prime p. [From Jaroslav 
               Krizek (jaroslav.krizek(AT)atlas.cz), Nov 01 2009]
%D A000583 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A000583 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000583 S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques 
               Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%H A000583 T. D. Noe, <a href="b000583.txt">Table of n, a(n) for n = 0..1000</a>
%H A000583 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A000583 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Enumerative Formulas 
               for Some Functions on Finite Sets</a>
%H A000583 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
               Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
               a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%H A000583 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
               1031 Generating Functions and Conjectures</a>, Universit\'{e} du 
               Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H A000583 H. Bottomley, <a href="a583.gif">Illustration of initial terms</a>
%H A000583 H. Bottomley, <a href="http://www.gallup.unm.edu/~smarandache/math.htm">
               Some Smarandache-type multiplicative sequences</a>
%H A000583 Hyun Kwang Kim, <a href="http://com2mac.postech.ac.kr/papers/2001/01-22.pdf">
               On Regular Polytope Numbers</a>
%H A000583 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               BiquadraticNumber.html">Link to a section of The World of Mathematics.</
               a>
%H A000583 <a href="Sindx_Cor.html#core">Index entries for "core" sequences</a>
%F A000583 Multiplicative with a(p^e) = p^(4e). - David W. Wilson (davidwwilson(AT)comcast.net), 
               Aug 01, 2001.
%F A000583 G.f.: x*(1+11*x+11*x^2+x^3)/(1-x)^5. More generally, g.f. for n^m is 
               Euler(m, x)/(1-x)^(m+1), where Euler(m, x) is Eulerian polynomial 
               of degree m (cf. A008292).
%F A000583 Dirichlet generating function: zeta(s-4). - Franklin T. Adams-Watters, 
               Sep 11 2005.
%F A000583 E.g.f.: (x+7x^2+6x^3+x^4)*e^x. More generally, the general form for the 
               e.g.f. for n^m is phi_m(x)*e^x, where phi_m is the exponential polynomial 
               of order n. - Franklin T. Adams-Watters, Sep 11 2005.
%F A000583 a(n)=sum(sum(sum(n, j=1..n),k=1..n),m=1..n), n>=0 . - Zerinvary Lajos 
               (zerinvarylajos(AT)yahoo.com), May 09 2007
%F A000583 a(n) = {least common multiple of n and (n-1)^3}-(n-1)^3. E.g.: {least 
               common multiple of 1 and (1-1)^3}-(1-1)^3 = 0, {least common multiple 
               of 2 and (2-1)^3}-(2-1)^3 = 1, {least common multiple of 3 and (3-1)^3}-(3-1)^3 
               = 16, {least common multiple of 4 and (4-1)^3}-(4-1)^3 = 81, ... 
               - Mats Granvik (mgranvik(AT)abo.fi), Sep 24 2007
%F A000583 a(n) = C(n+3,4) + 11 C(n+2,4) + 11 C(n+1,4) + C(n,4)
%p A000583 A000583 := n->n^4;
%p A000583 a:=n->sum(sum(n^2, j=1..n),k=1..n): seq(a(n), n=0..33); - Zerinvary Lajos 
               (zerinvarylajos(AT)yahoo.com), May 09 2007
%p A000583 a:=n->sum(sum(sum(n, j=1..n),k=1..n),m=1..n): seq(a(n), n=0..33); - Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), May 09 2007
%p A000583 A000583:=-(z+1)*(z**2+10*z+1)/(z-1)**5; [S. Plouffe in his 1992 dissertation. 
               Gives sequence without initial zero.]
%p A000583 with (combinat):seq(fibonacci(3, n^2)-1, n=0..33); - Zerinvary Lajos 
               (zerinvarylajos(AT)yahoo.com), May 25 2008
%o A000583 (PARI) A000583(n) = n^4 [From Michael Porter (michael_b_porter(AT)yahoo.com), 
               Nov 09 2009]
%Y A000583 Cf. A000538, A005917.
%Y A000583 Cf. A000332, A014820, A092181, A092182, A092183.
%Y A000583 a(n) = A123865(n) + 1.
%Y A000583 Sequence in context: A017672 A055013 A080150 this_sequence A050751 A014188 
               A050463
%Y A000583 Adjacent sequences: A000580 A000581 A000582 this_sequence A000584 A000585 
               A000586
%K A000583 nonn,core,easy,nice,mult,new
%O A000583 0,3
%A A000583 N. J. A. Sloane (njas(AT)research.att.com).
%E A000583 More terms from Mats Granvik (mgranvik(AT)abo.fi), Sep 24 2007
%E A000583 Removed attribute "conjectured" from Plouffe g.f R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 
               Mar 11 2009