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%I #34 May 15 2022 07:31:17
%S 1,81,1296,10000,50625,194481,614656,1679616,4100625,9150625,18974736,
%T 37015056,68574961,121550625,207360000,342102016,547981281,855036081,
%U 1303210000,1944810000,2847396321,4097152081,5802782976,8100000000,11156640625,15178486401
%N a(n) = binomial(n+2, 2)^4.
%C Number of 4-dimensional cage assemblies.
%C See Chap. 61, "Hyperspace Prisons", of C. Pickover's book "Wonders of Numbers" for full explanation of "cage numbers."
%D Clifford A. Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind and Meaning," Oxford University Press, 2001, p. 325.
%H Harry J. Smith, <a href="/A059977/b059977.txt">Table of n, a(n) for n = 0..1000</a>
%H Clifford A. Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind and Meaning," <a href="http://www.zentralblatt-math.org/zmath/en/search/?q=an:0983.00008&format=complete">Zentralblatt review</a>.
%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (9,-36,84,-126,126,-84,36,-9,1).
%F L(n) = ((n^m)(n + 1)^m)/(2^m) where m is the dimension, which in this case is 4.
%F O.g.f.: -(1+72*x+603*x^2+1168*x^3+603*x^4+72*x^5+x^6)/(-1+x)^9. - _R. J. Mathar_, Mar 31 2008
%F a(n) = A000217(n+1)^4. - _R. J. Mathar_, Dec 13 2011
%F a(n) = (A000539(n+1) + A000541(n+1))/2. - _Philippe Deléham_, May 25 2015
%F From _Amiram Eldar_, May 15 2022: (Start)
%F Sum_{n>=0} 1/a(n) = 160*Pi^2/3 + 16*Pi^4/45 - 560.
%F Sum_{n>=0} (-1)^n/a(n) = 560 - 640*log(2) - 96*zeta(3). (End)
%e 1 = (1 + 1)/2, 81 = (33 + 129)/2, 1296 = (276 + 2316)/2, 10000 = (1300 + 18700)/2, ... - _Philippe Deléham_, May 25 2015
%p with (combinat):seq(mul(stirling2(n+1,n),k=1..4),n=1..24); # _Zerinvary Lajos_, Dec 16 2007
%t m = 4; Table[ ( (n^m)(n + 1)^m )/(2^m), {n, 1, 30} ]
%o (Sage)[stirling_number2(n+1,n)^4for n in range(1,25)] # _Zerinvary Lajos_, Mar 14 2009
%o (PARI) { for (n=0, 1000, write("b059977.txt", n, " ", ((n + 1)*(n + 2)/2)^4); ) } \\ _Harry J. Smith_, Jun 30 2009
%Y Cf. A000217, A000539, A000541, A059827, A059860.
%K nonn
%O 0,2
%A _Robert G. Wilson v_, Mar 06 2001
%E Better definition from _Zerinvary Lajos_, May 23 2006
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