A248619 -id:A248619 - cp's OEIS frontend

A248720 a(n) = (n*(n+1))^5.

0, 32, 7776, 248832, 3200000, 24300000, 130691232, 550731776, 1934917632, 5904900000, 16105100000, 40074642432, 92389579776, 199690286432, 408410100000, 796262400000, 1488827973632, 2682916351776, 4678757435232, 7923516800000, 13069123200000

Offset: 0

Eugene Chong, Oct 16 2014

This is the sequence (2^5)*A059860(n)= (2*binomial(n+1,2))^5, n >= 0. - Wolfdieter Lang, Nov 03 2014

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (11, -55, 165, -330, 462, -462, 330, -165, 55, -11, 1).

Cf. A059860, A002378 (n*(n+1)), A035287(n+1) ((n*(n+1))^2), A060459 ((n*(n+1))^3), A248619 ((n*(n+1))^4).

Magma
```
[(n*(n+1))^5: n in [0..30]];
```
Maple
```
[ seq(n^5*(n+1)^5, n = 0..100) ];
```

Table[(n (n + 1))^5, {n, 0, 70}] (* or *) CoefficientList[Series[32 x (x^8 + 232 x^7 + 5158 x^6 + 27664 x^5 + 47290 x^4 + 27664 x^3 + 5158 x^2 + 232 x + 1)/(1 - x)^11, {x, 0, 30}], x]
LinearRecurrence[{11,-55,165,-330,462,-462,330,-165,55,-11,1},{0,32,7776,248832,3200000,24300000,130691232,550731776,1934917632,5904900000,16105100000},20] (* Harvey P. Dale, Apr 23 2017 *)

a(n) = A002378(n)^5.
a(n) = 32*A059860(n) for n>0.
G.f.: 32*x*(x^8 + 232*x^7 + 5158*x^6 + 27664*x^5 + 47290*x^4 + 27664*x^3 + 5158*x^2 + 232*x + 1) / (1 - x)^11 (from A059860).
Sum_{n>=1} 1/a(n) = 126 - 35*Pi^2/3 - Pi^4/9. - Vaclav Kotesovec, Sep 25 2019
a(n) = 11*a(n-1) - 55*a(n-2) + 165*a(n-3) - 330*a(n-4) + 462*a(n-5) - 462*a(n-6) + 330*a(n-7) - 165*a(n-8) + 55*a(n-9) - 11*a(n-10) + a(n-11). - Wesley Ivan Hurt, Jan 20 2024

Terms a(32) and beyond corrected by Andrew Howroyd, Feb 20 2018

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

A327773 Decimal expansion of Sum_{k>=1} 1/(k*(k+1))^4.

Original entry on oeis.org

0, 6, 3, 3, 2, 7, 8, 0, 4, 3, 8, 6, 8, 0, 5, 1, 1, 2, 4, 8, 0, 3, 1, 0, 7, 2, 6, 0, 0, 2, 8, 3, 9, 5, 8, 9, 9, 2, 8, 4, 9, 9, 9, 2, 7, 9, 7, 3, 4, 2, 2, 5, 7, 0, 0, 7, 7, 1, 1, 7, 0, 1, 8, 2, 8, 8, 3, 9, 0, 6, 4, 0, 4, 3, 7, 9, 5, 5, 1, 6, 9, 7, 8, 6, 3, 9, 7, 2, 8, 4, 2, 7, 8, 5, 7, 3, 9, 4, 0, 5, 4, 6, 0, 6, 8, 0

Offset: 0

Vaclav Kotesovec, Sep 25 2019

Sum_{k>=1} 1/(k*(k+1)) = 1
Sum_{k>=1} 1/(k*(k+1))^2 = -3 + Pi^2/3
Sum_{k>=1} 1/(k*(k+1))^3 = 10 - Pi^2
Sum_{k>=1} 1/(k*(k+1))^4 = -35 + 10*Pi^2/3 + Pi^4/45
Sum_{k>=1} 1/(k*(k+1))^5 = 126 - 35*Pi^2/3 - Pi^4/9
Sum_{k>=1} 1/(k*(k+1))^6 = -462 + 42*Pi^2 + 7*Pi^4/15 + 2*Pi^6/945
Sum_{k>=1} 1/(k*(k+1))^7 = 1716 - 154*Pi^2 - 28*Pi^4/15 - 2*Pi^6/135
Sum_{k>=1} 1/(k*(k+1))^8 = -6435 + 572*Pi^2 + 22*Pi^4/3 + 8*Pi^6/105 + Pi^8/4725
Sum_{k>=1} 1/(k*(k+1))^9 = 24310 - 2145*Pi^2 - 143*Pi^4/5 - 22*Pi^6/63 - Pi^8/525
Sum_{k>=1} 1/(k*(k+1))^10 = -92378 + 24310*Pi^2/3 + 1001*Pi^4/9 + 286*Pi^6/189 + 11*Pi^8/945 + 2*Pi^10/93555
In general, for s > 1, Sum_{k>=1} 1/(k*(k+1))^s = (-1)^s * (-binomial(2*s-1, s-1) + Sum_{j=1..floor(s/2)} (-1)^(j+1) * binomial(2*s-2*j-1, s-1) * Bernoulli(2*j) * (2*Pi)^(2*j) / (2*j)!).
Equivalently, for s > 1, Sum_{k>=1} 1/(k*(k+1))^s = (-1)^s * (-binomial(2*s-1, s-1) + 2*Sum_{j=1..floor(s/2)} binomial(2*s-2*j-1, s-1) * zeta(2*j)).

			0.06332780438680511248031072600283958992849992797342257007711701828839...

R. J. Mathar, Tighly circumscribed regular polygons, arXiv:1301.6293 [math.MG], 2013, see Appendix.
Eric Weisstein's World of Mathematics, Bernoulli Number
Eric Weisstein's World of Mathematics, Riemann zeta function

Cf. A145426, A248619.

evalf(sum(1/(k*(k+1))^4, k=1..infinity), 120);

RealDigits[N[Sum[1/(k*(k + 1))^4, {k, 1, Infinity}], 105]][[1]]

PARI
```
suminf(k=1, 1/(k*(k+1))^4)
```

Equals Pi^4/45 + 10*Pi^2/3 - 35.

cp's OEIS Frontend

A248720 a(n) = (n*(n+1))^5.

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

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A327773 Decimal expansion of Sum_{k>=1} 1/(k*(k+1))^4.

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Keywords

Comments

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Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula