A059977 a(n) = binomial(n+2, 2)^4.

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

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