A163102 -id:A163102 - cp's OEIS frontend

A163276 a(n) = n^6*(n+1)^2/2.

0, 2, 288, 5832, 51200, 281250, 1143072, 3764768, 10616832, 26572050, 60500000, 127552392, 252315648, 473027282, 847072800, 1458000000, 2424307712, 3910286178, 6139206432, 9409176200, 14112000000, 20755401282, 29988984608

Offset: 0

Omar E. Pol, Jul 24 2009

G. C. Greubel, 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. A006002, A099903, A163102, A163274, A163275, A163277.

[n^6*(n+1)^2/2: n in [0..30]]; // Vincenzo Librandi, Dec 13 2016

seq((1/2)*n^6*(n+1)^2, n = 0 .. 25); # Emeric Deutsch, Aug 01 2009

Table[(1/2)*n^6*(n + 1)^2, {n,0,50}] (* or *) LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1}, {0, 2, 288, 5832, 51200, 281250, 1143072, 3764768, 10616832}, 50] (* G. C. Greubel, Dec 12 2016 *)

concat([0], Vec(2*x*(1 + 135*x +1656*x^2 +4456*x^3 +3231*x^4 +585*x^5 +16*x^6)/(1-x)^9 + O(x^50))) \\ G. C. Greubel, Dec 12 2016

G.f.: 2*x*(1+135*x+1656*x^2+4456*x^3+3231*x^4+585*x^5+16*x^6)/(1-x)^9. - Colin Barker, May 05 2012
From Amiram Eldar, May 14 2022: (Start)
Sum_{n>=1} 1/a(n) = 2*Pi^2 + Pi^4/15 + 2*Pi^6/945 - 14 - 8*zeta(3) - 4*zeta(5).
Sum_{n>=1} (-1)^(n+1)/a(n) = 14 + 2*Pi^2/3 + 7*Pi^4/120 + 31*Pi^6/15120 - 24*log(2) - 6*zeta(3) - 15*zeta(5)/4. (End)

Extended by Emeric Deutsch, Aug 01 2009

A163277 a(n) = n^7*(n+1)^2/2.

Original entry on oeis.org

0, 2, 576, 17496, 204800, 1406250, 6858432, 26353376, 84934656, 239148450, 605000000, 1403076312, 3027787776, 6149354666, 11859019200, 21870000000, 38788923392, 66474865026, 110505715776, 178774347800, 282240000000

Offset: 0

Omar E. Pol, Jul 24 2009

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Cf. A006002, A099903, A163102, A163274, A163275, A163276.

[n^7*(n+1)^2/2: n in [0..30]]; // Vincenzo Librandi, Dec 13 2016

A163277 := proc(n) n^7*(n+1)^2/2 ; end proc: seq(A163277(n),n=0..60) ; \\ R. J. Mathar, Feb 05 2010

Table[(1/2)*n^7*(n + 1)^2, {n,0,50}] (* G. C. Greubel, Dec 12 2016 *)

concat([0], Vec(2*x*(1 +278*x +5913*x^2 +27760*x^3 +38435*x^4 +16434*x^5 +1867*x^6 +32*x^7)/(x-1)^10 + O(x^50))) \\ G. C. Greubel, Dec 12 2016

From R. J. Mathar, Feb 05 2010: (Start)
a(n) = n^2*A163275(n).
G.f.: 2*x*(1 +278*x +5913*x^2 +27760*x^3 +38435*x^4 +16434*x^5 +1867*x^6 +32*x^7)/(x-1)^10. (End)
From Amiram Eldar, May 14 2022: (Start)
Sum_{n>=1} 1/a(n) = 16 - 7*Pi^2/3 - 4*Pi^4/45 - 4*Pi^6/945 + 10*zeta(3) + 6*zeta(5) + 2*zeta(7).
Sum_{n>=1} (-1)^(n+1)/a(n) = 28*log(2) + 15*zeta(3)/2 + 45*zeta(5)/8 + 63*zeta(7)/32 - 16 - 5*Pi^2/6 - 7*Pi^4/90 - 31*Pi^6/7560. (End)

Extended by R. J. Mathar, Feb 05 2010

A254371 Sum of cubes of the first n even numbers (A016743).

Original entry on oeis.org

0, 8, 72, 288, 800, 1800, 3528, 6272, 10368, 16200, 24200, 34848, 48672, 66248, 88200, 115200, 147968, 187272, 233928, 288800, 352800, 426888, 512072, 609408, 720000, 845000, 985608, 1143072, 1318688, 1513800, 1729800, 1968128, 2230272, 2517768, 2832200, 3175200

Offset: 0

Luciano Ancora, Mar 16 2015

Property: for n >= 2, each (a(n), a(n)+1, a(n)+2) is a triple of consecutive terms that are the sum of two nonzero squares; precisely: a(n) = (n*(n + 1))^2 + (n*(n + 1))^2, a(n)+1 = (n^2+2n)^2 + (n^2-1)^2 and a(n)+2 = (n^2+n+1)^2 + (n^2+n-1)^2 (see Diophante link). - Bernard Schott, Oct 05 2021

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Luciano Ancora, The Square Pyramidal Number and other figurate numbers, ch. 3.
Diophante, Une miniature avec trois entiers consécutifs (in French).
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000537 (sum of first n cubes); A002593 (sum of first n odd cubes).
Cf. A060300 (2*a(n)).
First bisection of A105636; second bisection of A212892.
Cf. A000217, A000415, A002378, A035287, A163102.

List([0..35],n->2*(n*(n+1))^2); # Muniru A Asiru, Oct 24 2018

[2*n^2*(n+1)^2: n in [0..40]]; // Bruno Berselli, Mar 23 2015

A254371:=n->2*n^2*(n + 1)^2: seq(A254371(n), n=0..50); # Wesley Ivan Hurt, Apr 28 2017

Table[2 n^2 (n+1)^2, {n, 0, 40}] (* or *) LinearRecurrence[{5, -10, 10, -5, 1}, {0, 8, 72, 288, 800}, 40]
Accumulate[Range[0,80,2]^3] (* Harvey P. Dale, Jun 26 2017 *)

a(n)=sum(i=0, n, 8*i^3); \\ Michael B. Porter, Mar 16 2015

G.f.: 8*x*(1 + 4*x + x^2)/(1 - x)^5.
a(n) = 2*n^2*(n + 1)^2.
a(n) = 2*A035287(n+1) = 2*A002378(n)^2 = 8*A000217(n)^2. - Bruce J. Nicholson, Apr 23 2017
a(n) = 8*A000537(n). - Michel Marcus, Apr 23 2017
From Amiram Eldar, Aug 25 2022: (Start)
Sum_{n>=1} 1/a(n) = Pi^2/6 - 3/2.
Sum_{n>=1} (-1)^(n+1)/a(n) = 3/2 - 2*log(2). (End)
From Elmo R. Oliveira, Aug 14 2025: (Start)
E.g.f.: 2*x*(2 + x)*(2 + 6*x + x^2)*exp(x).
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
a(n) = 4*A163102(n) = A060300(n)/2. (End)

A269946 Triangle read by rows, Lah numbers of order 3, T(n,n) = 1, T(n,k) = 0 if k<0 or k>n, otherwise T(n,k) = T(n-1,k-1)+((n-1)^3+k^3)*T(n-1, k), for n>=0 and 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 18, 18, 1, 0, 504, 648, 72, 1, 0, 32760, 47160, 7200, 200, 1, 0, 4127760, 6305040, 1141560, 45000, 450, 1, 0, 895723920, 1416456720, 283704120, 13741560, 198450, 882, 1, 0, 308129028480, 498072032640, 106386981120, 5876519040, 106616160, 691488, 1568, 1

Offset: 0

Peter Luschny, Mar 22 2016

			Triangle starts:
[1]
[0, 1]
[0, 2,       1]
[0, 18,      18,      1]
[0, 504,     648,     72,      1]
[0, 32760,   47160,   7200,    200,   1]
[0, 4127760, 6305040, 1141560, 45000, 450, 1]

Peter Luschny, The P-transform.

Cf. A038207 (order 0), A111596 (order 1), A268434 (order 2).

Cf. A255433, A163102, A269947, A269948.

T := proc(n, k) option remember;
    `if`(n=k, 1,
    `if`(k<0 or k>n, 0,
     T(n-1, k-1) + ((n-1)^3+k^3) * T(n-1, k) )) end:
for n from 0 to 6 do seq(T(n,k), k=0..n) od;

T[n_, n_] = 1; T[, 0] = 0; T[n, k_] /; 0 < k < n := T[n, k] = T[n-1, k-1] + ((n-1)^3 + k^3)*T[n-1, k]; T[, ] = 0;
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 20 2017 *)

T(n,k) = Sum_{j=k..n} A269947(n,j)*A269948(j,k).
T(n,1) = Product_{k=1..n} (k-1)^3+1 for n>=1 (cf. A255433).
T(n,n-1) = (n-1)^2*n^2/2 for n>=1 (cf. A163102).

A307304 Number of inequivalent ways of placing 2 nonattacking rooks on n X n board up to rotations and reflections of the board.

Original entry on oeis.org

0, 1, 4, 13, 31, 66, 123, 214, 346, 535, 790, 1131, 1569, 2128, 2821, 3676, 4708, 5949, 7416, 9145, 11155, 13486, 16159, 19218, 22686, 26611, 31018, 35959, 41461, 47580, 54345, 61816, 70024, 79033, 88876, 99621, 111303, 123994, 137731, 152590, 168610, 185871

Offset: 1

Mo Li, Apr 19 2019

			For n = 4 the a(4) = 13 solutions are
{{1,0,0,0}}  {{1,0,0,0}}  {{1,0,0,0}}
{{0,1,0,0}}  {{0,0,1,0}}  {{0,0,0,1}}
{{0,0,0,0}}  {{0,0,0,0}}  {{0,0,0,0}}
{{0,0,0,0}}  {{0,0,0,0}}  {{0,0,0,0}}
—————————————————————————————————————
{{1,0,0,0}}  {{1,0,0,0}}  {{1,0,0,0}}
{{0,0,0,0}}  {{0,0,0,0}}  {{0,0,0,0}}
{{0,0,1,0}}  {{0,0,0,1}}  {{0,0,0,0}}
{{0,0,0,0}}  {{0,0,0,0}}  {{0,0,0,1}}
—————————————————————————————————————
{{0,1,0,0}}  {{0,1,0,0}}  {{0,1,0,0}}
{{1,0,0,0}}  {{0,0,1,0}}  {{0,0,0,1}}
{{0,0,0,0}}  {{0,0,0,0}}  {{0,0,0,0}}
{{0,0,0,0}}  {{0,0,0,0}}  {{0,0,0,0}}
—————————————————————————————————————
{{0,1,0,0}}  {{0,1,0,0}}  {{0,1,0,0}}
{{0,0,0,0}}  {{0,0,0,0}}  {{0,0,0,0}}
{{0,0,1,0}}  {{0,0,0,1}}  {{0,0,0,0}}
{{0,0,0,0}}  {{0,0,0,0}}  {{0,0,1,0}}
—————————————————————————————————————
{{0,0,0,0}}
{{0,1,0,0}}
{{0,0,1,0}}
{{0,0,0,0}}

Leisure Maths Entertainment Forum, 2 nonattacking rooks on n X n board, Chinese blog.

Cf. A000903, A163102, A035287, A179058, A144084.

Table[
Piecewise[{{(n (n^3 - 2 n^2 + 6 n - 4))/16, Mod[n, 2] == 0},
{((n - 1) (n^3 - n^2 + 5 n - 1))/16, Mod[n, 2] == 1}}],{n, 20}]

a(n) = (1/16)*n*(n^3-2n^2+6n-4) if n is even;
a(n) = (1/16)*(n-1)*(n^3-n^2+5n-1) if n is odd.
G.f.: -x^2*(x^2+1)*(x^2+x+1)/((x+1)^2*(x-1)^5). - Alois P. Heinz, Apr 26 2019

A331528 a(n) = n^2 * (n+1)^2 * (n^2+n+1) / 12.

Original entry on oeis.org

0, 1, 21, 156, 700, 2325, 6321, 14896, 31536, 61425, 111925, 193116, 318396, 505141, 775425, 1156800, 1683136, 2395521, 3343221, 4584700, 6188700, 8235381, 10817521, 14041776, 18030000, 22920625, 28870101, 36054396, 44670556, 54938325, 67101825, 81431296, 98224896

Offset: 0

Seiichi Manyama, Jan 19 2020

Let b(n,k) = Sum_{j=0..n} (-1)^(n-j)* j^k * binomial(n,j) * binomial(n+j,j).
b(n,0) = 1.
b(n,1) = 1/1! * n   * (n+1).
b(n,2) = 1/2! * n^2 * (n+1)^2.
b(n,3) = 1/3! * n^2 * (n+1)^2 * (n^2+n+1) (= 2*a(n)).
b(n,4) = 1/4! * n^3 * (n+1)^3 * (n^2+n+4).
b(n,5) = 1/5! * n^2 * (n+1)^2 * (n^6+3*n^5+13*n^4+21*n^3+18*n^2+8*n-4).
b(n,6) = 1/6! * n^3 * (n+1)^3 * (n^2+n+4) * (n^4+2*n^3+17*n^2+16*n-6).

Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A002378 (b(n,1)), A163102 (b(n,2)), A168178 (first differences).

[n^2*(n+1)^2*(n^2+n+1)/12:n in [0..32]]; // Marius A. Burtea, Jan 19 2020

R:=PowerSeriesRing(Integers(), 33); [0] cat (Coefficients(R!( x*(x^4+14*x^3+30*x^2+14*x+1)/(1-x)^7))); // Marius A. Burtea, Jan 19 2020

a[n_] := (n*(n+1))^2 * (n^2+n+1) / 12; Array[a, 33, 0] (* Amiram Eldar, May 05 2021 *)

PARI
```
{a(n) = n^2*(n+1)^2*(n^2+n+1)/12}
```

G.f.: x * (x^4+14*x^3+30*x^2+14*x+1)/(1-x)^7.

cp's OEIS Frontend

A163276 a(n) = n^6*(n+1)^2/2.

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A163277 a(n) = n^7*(n+1)^2/2.

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A254371 Sum of cubes of the first n even numbers (A016743).

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A269946 Triangle read by rows, Lah numbers of order 3, T(n,n) = 1, T(n,k) = 0 if k<0 or k>n, otherwise T(n,k) = T(n-1,k-1)+((n-1)^3+k^3)*T(n-1, k), for n>=0 and 0<=k<=n.

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A307304 Number of inequivalent ways of placing 2 nonattacking rooks on n X n board up to rotations and reflections of the board.

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A331528 a(n) = n^2 * (n+1)^2 * (n^2+n+1) / 12.

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