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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
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
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
KEYWORD
nonn
AUTHOR
Robert G. Wilson v, Mar 06 2001
EXTENSIONS
Better definition from Zerinvary Lajos, May 23 2006
STATUS
approved