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 A059977 a(n) = binomial(n+2, 2)^4. 7
 1, 81, 1296, 10000, 50625, 194481, 614656, 1679616, 4100625, 9150625, 18974736, 37015056, 68574961, 121550625, 207360000, 342102016, 547981281, 855036081, 1303210000, 1944810000, 2847396321, 4097152081, 5802782976, 8100000000, 11156640625, 15178486401 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Number of 4-dimensional cage assemblies. See Chap. 61, "Hyperspace Prisons", of C. Pickover's book "Wonders of Numbers" for full explanation of "cage numbers." REFERENCES Clifford A. Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind and Meaning," Oxford University Press, 2001, p. 325. LINKS Harry J. Smith, Table of n, a(n) for n = 0..1000 Clifford A. Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind and Meaning," Zentralblatt review. Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1). FORMULA L(n) = ((n^m)(n + 1)^m)/(2^m) where m is the dimension, which in this case is 4. 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 a(n) = A000217(n+1)^4. - R. J. Mathar, Dec 13 2011 a(n) = (A000539(n+1) + A000541(n+1))/2. - Philippe Deléham, May 25 2015 From Amiram Eldar, May 15 2022: (Start) Sum_{n>=0} 1/a(n) = 160*Pi^2/3 + 16*Pi^4/45 - 560. Sum_{n>=0} (-1)^n/a(n) = 560 - 640*log(2) - 96*zeta(3). (End) EXAMPLE 1 = (1 + 1)/2, 81 = (33 + 129)/2, 1296 = (276 + 2316)/2, 10000 = (1300 + 18700)/2, ... - Philippe Deléham, May 25 2015 MAPLE with (combinat):seq(mul(stirling2(n+1, n), k=1..4), n=1..24); # Zerinvary Lajos, Dec 16 2007 MATHEMATICA m = 4; Table[ ( (n^m)(n + 1)^m )/(2^m), {n, 1, 30} ] PROG (Sage)[stirling_number2(n+1, n)^4for n in range(1, 25)] # Zerinvary Lajos, Mar 14 2009 (PARI) { for (n=0, 1000, write("b059977.txt", n, " ", ((n + 1)*(n + 2)/2)^4); ) } \\ Harry J. Smith, Jun 30 2009 CROSSREFS Cf. A000217, A000539, A000541, A059827, A059860. Sequence in context: A224355 A016768 A224014 * A231912 A116205 A237455 Adjacent sequences: A059974 A059975 A059976 * A059978 A059979 A059980 KEYWORD nonn AUTHOR Robert G. Wilson v, Mar 06 2001 EXTENSIONS Better definition from Zerinvary Lajos, May 23 2006 STATUS approved

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Last modified August 6 22:01 EDT 2024. Contains 374996 sequences. (Running on oeis4.)