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

Author

Robert G. Wilson v, Mar 06 2001

Keywords

• nonn

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."

Examples

			1 = (1 + 1)/2, 81 = (33 + 129)/2, 1296 = (276 + 2316)/2, 10000 = (1300 + 18700)/2, ... - _Philippe Deléham_, May 25 2015
		

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).

Crossrefs

Cf. A000217, A000539, A000541, A059827, A059860.

Cf. A168364 (first differences).

Programs

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} ]

PARI

a(n) = { ((n + 1)*(n + 2)/2)^4 } \\ Harry J. Smith, Jun 30 2009

Sage

[stirling_number2(n+1,n)^4for n in range(1,25)] # Zerinvary Lajos, Mar 14 2009

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)

Extensions

Better definition from Zerinvary Lajos, May 23 2006