A059978 a(n) = binomial(n+2,n)^6.

1, 729, 46656, 1000000, 11390625, 85766121, 481890304, 2176782336, 8303765625, 27680640625, 82653950016, 225199600704, 567869252041, 1340095640625, 2985984000000, 6327518887936, 12827693806929, 25002110044521, 47045881000000, 85766121000000, 151939915084881

Offset: 0

Author

Robert G. Wilson v, Mar 06 2001

Keywords

• nonn
• easy

Comments

Number of 6-dimensional cage assemblies.

References

• Clifford A. Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind and Meaning," Oxford University Press, 2001, p. 325.

Links

• T. D. Noe, 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 (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

Crossrefs

Cf. A059827, A059860, A000217, A059977, A059980, A091043.

Programs

Maple

with (combinat):seq(mul(stirling2(n+1,n),k=1..6),n=1..18); # Zerinvary Lajos, Dec 14 2007

Mathematica

m = 6; Table[n^m (n + 1)^m/2^m, {n, 1, 24}]

Formula

G.f.: (x^10 + 716*x^9 + 37257*x^8 + 450048*x^7 + 1822014*x^6 +2864328*x^5 + 1822014*x^4 + 450048*x^3 + 37257*x^2 + 716*x + 1)/(1-x)^13. - Colin Barker, Jul 09 2012

G.f.: 6F5([3,3,3,3,3,3], [1,1,1,1,1], z). - Benedict W. J. Irwin, Mar 14 2016

a(n) = (1/16)*( 3*S(7,n+1) + 10*S(9,n+1) + 3*S(11,n+1) ), where S(r,n) = Sum_{k = 1..n} k^r. Cf. A059977 and A059980. - Peter Bala, Jul 02 2019

From Amiram Eldar, May 15 2022: (Start)

Sum_{n>=0} 1/a(n) = 2688*Pi^2 + 448*Pi^4/15 + 128*Pi^6/945 - 29568.

Sum_{n>=0} (-1)^n/a(n) = 29568 - 32256*log(2) - 5376*zeta(3) - 720*zeta(5). (End)

Extensions

Better definition from Zerinvary Lajos, May 23 2006