A059978 -id:A059978 - cp's OEIS frontend

A059980 Number of 8-dimensional cage assemblies.

1, 6561, 1679616, 100000000, 2562890625, 37822859361, 377801998336, 2821109907456, 16815125390625, 83733937890625, 360040606269696, 1370114370683136, 4702525276151521, 14774554437890625, 42998169600000000, 117033789351264256, 300283484326400961

Offset: 1

Robert G. Wilson v, Mar 06 2001

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

Clifford A. Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind and Meaning," Zentralblatt review.
Index entries for linear recurrences with constant coefficients, signature (17,-136,680,-2380,6188,-12376,19448,-24310,24310,-19448,12376,-6188,2380,-680,136,-17,1).

Cf. A059827, A059860, A059977, A059978, A091043.

m = 8; Table[n^m (n + 1)^m/2^m, {n, 1, 18}]

G.f.: -x*(x^14 +6544*x^13 +1568215*x^12 +72338144*x^11 +1086859301*x^10 +6727188848*x^9 +19323413187*x^8 +27306899520*x^7 +19323413187*x^6 +6727188848*x^5 +1086859301*x^4 +72338144*x^3 +1568215*x^2 +6544*x +1)/(x-1)^17. - Colin Barker, Jul 09 2012
From Peter Bala, Jul 02 2019 (Start)
a(n) = (n*(n + 1)/2)^8.
a(n) = (1/16)*( S(9,n) + 7*S(11,n) + 7*S(13,n) + S(15,n) ), where S(r,n) = Sum_{k = 1..n} k^r. Cf. A059977 and A059978. (End)
From Amiram Eldar, May 15 2022: (Start)
Sum_{n>=1} 1/a(n) = 146432*Pi^2 + 5632*Pi^4/3 + 2048*Pi^6/105 + 256*Pi^8/4725 - 1647360.
Sum_{n>=1} (-1)^(n+1)/a(n) = 1647360 - 1757184*log(2) - 304128*zeta(3) - 57600*zeta(5) - 4032*zeta(7). (End)

A249076 a(n) = (n*(n+1))^6.

Original entry on oeis.org

0, 64, 46656, 2985984, 64000000, 729000000, 5489031744, 30840979456, 139314069504, 531441000000, 1771561000000, 5289852801024, 14412774445056, 36343632130624, 85766121000000, 191102976000000, 404961208827904, 820972403643456, 1600135042849344, 3010936384000000, 5489031744000000

Offset: 0

Jiwoo Lee, Oct 20 2014

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

Cf. A059978; A002378: n*(n+1); A035282: n^2 *(n+1)^2; A060459: n^3 *(n+1)^3; A248619: n^4 *(n+1)^4;

Magma
```
[(n*(n+1))^6: n in [0..30]];
```
Maple
```
[ seq(n^6*(n+1)^6, n = 0..100) ];
```

Table[(n (n + 1))^6, {n, 0, 70}] (* or *)
CoefficientList[Series[64*x*(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, {x, 0, 30}], x]

a(n)=(n*(n+1))^6 \\ Charles R Greathouse IV, Oct 21 2014

a(n) = A002378(n)^6.
a(n) = 64*A059978(n) for n>0.
G.f.: 64*x*(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. [corrected by Georg Fischer, May 10 2019]
Sum_{n>=1} 1/a(n) = -462 + 42*Pi^2 + 7*Pi^4/15 + 2*Pi^6/945. - Vaclav Kotesovec, Sep 25 2019

Incorrect term corrected by Colin Barker, Oct 21 2014

Terms a(21) and beyond corrected by Andrew Howroyd, Feb 22 2018

A091480 Table of multigraphs (by antidiagonals) with n (>=1) nodes and k (>=0) edges. Each type of object labeled from its own label set.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 9, 6, 1, 0, 1, 27, 36, 10, 1, 0, 1, 81, 216, 100, 15, 1, 0, 1, 243, 1296, 1000, 225, 21, 1, 0, 1, 729, 7776, 10000, 3375, 441, 28, 1, 0, 1, 2187, 46656, 100000, 50625, 9261, 784, 36, 1, 0, 1, 6561, 279936, 1000000, 759375

Offset: 1

Christian G. Bower, Jan 13 2004

			1  0   0    0     0 ...
1  1   1    1     1 ...
1  3   9   27    81 ...
1  6  36  216  1296 ...
1 10 100 1000 10000 ...

F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, p. 114 (2.4.44).

Columns 0-8: A000012, A000217(n-1), A000537(n-1), A059827(n-1), A059977(n-1), A059860(n-1), A059978(n-1), A059979(n-1), A059980(n-1).
Rows 1-10: A000007, A000012, A000244, A000400, A011557, A001024, A009965, A009972, A009980, A009989.
Cf. A091478.

a(n, k) = binomial(n, 2)^k.

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A059980 Number of 8-dimensional cage assemblies.

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A249076 a(n) = (n*(n+1))^6.

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A091480 Table of multigraphs (by antidiagonals) with n (>=1) nodes and k (>=0) edges. Each type of object labeled from its own label set.

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Formula