A059977 -id:A059977 - cp's OEIS frontend

A168364 a(n) = n^5*(n^2 + 1)/2.

0, 1, 80, 1215, 8704, 40625, 143856, 420175, 1064960, 2421009, 5050000, 9824111, 18040320, 31559905, 52975664, 85809375, 134742016, 205879265, 307054800, 448173919, 641600000, 902586321, 1249755760, 1705630895, 2297217024

Offset: 0

N. J. A. Sloane, Dec 11 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A000578 (first differences of squares of triangular numbers), A357178 (of their cubes), A059977 (partial sums), A000217.

[n^5*(n^2+1)/2: n in [0..30]]; // Vincenzo Librandi, Aug 28 2011

Table[n^5(n^2+1)/2,{n,0,25}]  (* Harvey P. Dale, Apr 22 2011 *)
LinearRecurrence[{8, -28, 56, -70, 56, -28, 8, -1},{0, 1, 80, 1215, 8704, 40625, 143856, 420175}, 50] (* G. C. Greubel, Jul 19 2016 *)

G.f.: x*(1 + 72*x + 603*x^2 + 1168*x^3 + 603*x^4 + 72*x^5 + x^6)/(1-x)^8. - Harvey P. Dale, Apr 22 2011
a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8). - G. C. Greubel, Jul 19 2016
a(n) = A000217(n)^4 - A000217(n-1)^4. - Kelvin Voskuijl, Jan 16 2025

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

Original entry on oeis.org

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

Offset: 0

Robert G. Wilson v, Mar 06 2001

Number of 6-dimensional cage assemblies.

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

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).

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

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

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

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)

Better definition from Zerinvary Lajos, May 23 2006

A248619 a(n) = (n*(n+1))^4.

Original entry on oeis.org

0, 16, 1296, 20736, 160000, 810000, 3111696, 9834496, 26873856, 65610000, 146410000, 303595776, 592240896, 1097199376, 1944810000, 3317760000, 5473632256, 8767700496, 13680577296, 20851360000, 31116960000, 45558341136, 65554433296, 92844527616, 129600000000

Offset: 0

Eugene Chong, Oct 09 2014

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A016744, A059977; A002378: n*(n+1); A035287: n^2 *(n-1)^2; A060459: n^3*(n+1)^3.

Cf. A327773.

[(n*(n+1))^4: n in [0..30]]; // Vincenzo Librandi, Oct 16 2014

Maple
```
[ seq(n^4*(n+1)^4, n = 0..100) ];
```

Table[(n (n + 1))^4, {n, 0, 70}] (* or *) CoefficientList[Series[16 x (1 + 72 x + 603 x^2 + 1168 x^3 + 603 x^4 + 72 x^5 + x^6)/(1 - x)^9, {x, 0, 30}], x] (* Vincenzo Librandi, Oct 16 2014 *)
LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{0,16,1296,20736,160000,810000,3111696,9834496,26873856},30] (* Harvey P. Dale, Sep 09 2016 *)

a(n) = A002378(n)^4 = A016744(A000217(n)).
a(n) = 16*A059977(n) for n>0.
G.f.: 16*x*(1 + 72*x + 603*x^2 + 1168*x^3 + 603*x^4 + 72*x^5 + x^6)/(1 - x)^9. - Vincenzo Librandi, Oct 16 2014
Sum_{n>=1} 1/a(n) = A327773 = -35 + 10*Pi^2/3 + Pi^4/45. - Vaclav Kotesovec, Sep 25 2019

Terms a(76) and beyond corrected by Andrew Howroyd, Feb 20 2018

A059980 Number of 8-dimensional cage assemblies.

Original entry on oeis.org

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)

A271535 a(n) = ( n(n + 1)(2*n + 1)/6 )^2.

Original entry on oeis.org

0, 1, 25, 196, 900, 3025, 8281, 19600, 41616, 81225, 148225, 256036, 422500, 670761, 1030225, 1537600, 2238016, 3186225, 4447881, 6100900, 8236900, 10962721, 14402025, 18696976, 24010000, 30525625, 38452401, 48024900, 59505796, 73188025, 89397025, 108493056

Offset: 0

Vincenzo Librandi, Apr 20 2016

R. D. Carmichael and T. L. DeLand, Find the sum of the series 1^2 + 5^2 + 14^2 + 30^2 + ... + [n*(n+1)*(2*n+1)/6]^2, American Mathematical Monthly, Vol. 15, No. 6/7, Jun-Jul, 1908, pp. 132-133.
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A000290, A000330, A000537, A005563, A046092, A059977.

Magma
```
[(n*(n+1)*(2*n+1)/6)^2: n in [0..50]];
```

Table[(n (n + 1) (2 n + 1)/6)^2, {n, 0, 50}]

vector(100, n, n--; (n*(n + 1)*(2*n + 1)/6)^2) \\ Altug Alkan, Apr 21 2016

G.f.: x*(1 + 18*x + 42*x^2 + 18*x^3 + x^4)/(1 - x)^7.
a(n) = Sum_{j=1..n} Sum_{i=1..n} (i*j)^2. - Alexander Adamchuk, Oct 26 2004
E.g.f.: x*(36 + 414*x + 744*x^2 + 393*x^3 + 72*x^4 + 4*x^5)*exp(x)/36. - Ilya Gutkovskiy, Apr 21 2016
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7).
Sum_{i = 0..n} a(i) = n*(n + 1)*(n + 2)*(2*n + 1)*(2*n + 3)*(5*n^2 + 10*n - 1)/1260. [See Carmichael - DeLand in Links section, page 132.]
a(n) = A000330(n)^2. - Ray Chandler, Apr 21 2016
Sum_{n>=1} 1/a(n) = 84*Pi^2 - 828. - Amiram Eldar, Feb 25 2023

Edited by Bruno Berselli, Apr 22 2016

A202107 a(n) = n^4*(n+1)^4/8.

Original entry on oeis.org

2, 162, 2592, 20000, 101250, 388962, 1229312, 3359232, 8201250, 18301250, 37949472, 74030112, 137149922, 243101250, 414720000, 684204032, 1095962562, 1710072162, 2606420000, 3889620000, 5694792642, 8194304162, 11605565952, 16200000000, 22313281250, 30356972802

Offset: 1

Martin Renner, Dec 11 2011

A relation between fourth powers and the sum of fifth and seventh powers. See the first formula, which is from Beiler.

Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966, p. 161.

Temple Rice Hollcroft, On sums of powers of n consecutive integers, Bulletin of the American Mathematical Society 59 (1953), nr. 6, p. 526 (574t).
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A000217, A000539, A000541, A002117, A059977.

A202107:=n->(n^4)*(n+1)^4/8; seq(A202107(n), n=1..100); # Wesley Ivan Hurt, Nov 12 2013

Table[n^4 (n+1)^4/8, {n, 100}] (* Wesley Ivan Hurt, Nov 12 2013 *)

a(n) = 2*(Sum_{k=1..n} k)^4 = Sum_{k=1..n} (k^5 + k^7).
a(n) = 2*A059977(n-1).
a(n) = A000539(n) + A000541(n).
G.f.: -2*x*(1+72*x+603*x^2+1168*x^3+603*x^4+72*x^5+x^6) / (x-1)^9. - R. J. Mathar, Dec 13 2011
a(n) = 2*(A000217(n)^4). - Zak Seidov, Jan 21 2012
From Amiram Eldar, Apr 09 2024: (Start)
Sum_{n>=1} 1/a(n) = 8*Pi^4/45 + 80*Pi^2/3 - 280.
Sum_{n>=1} (-1)^(n+1)/a(n) = 280 - 320*log(2) - 48*zeta(3). (End)

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.

cp's OEIS Frontend

A168364 a(n) = n^5*(n^2 + 1)/2.

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

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A248619 a(n) = (n*(n+1))^4.

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A059980 Number of 8-dimensional cage assemblies.

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A271535 a(n) = ( n*(n + 1)*(2*n + 1)/6 )^2.

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A202107 a(n) = n^4*(n+1)^4/8.

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Maple

Mathematica

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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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References

Crossrefs

Formula

A271535 a(n) = ( n(n + 1)(2*n + 1)/6 )^2.