A023855 -id:A023855 - cp's OEIS frontend

A270531 a(n) = Sum_{i=1..floor(n/2)} (i*(n-i))!.

0, 0, 1, 2, 30, 744, 403320, 482631120, 22230943262640, 2439304469060699520, 16131709536027319923050880, 265557748777251180632423132716800, 382326737887135184960649117960539544556800, 1405822033408121123332642294795422193345577766681600

Offset: 0

Wesley Ivan Hurt, Mar 18 2016

Sum of the factorials of the products of the parts in each partition of n into two parts.

			a(4)=30; There are 2 partitions of 4 into two parts: (3,1) and (2,2). The sum of the factorials of the products of the parts in each partition is: (3*1)! + (2*2)! = 3! + 4! = 6 + 24 = 30.

G. C. Greubel, Table of n, a(n) for n = 0..40
Index entries for sequences related to partitions

Cf. A000142, A023855, A144895.

A270531:=n->add((i*(n-i))!, i=1..floor(n/2)): seq(A270531(n), n=0..15);

Table[Sum[(i*(n - i))!, {i, Floor[n/2]}], {n, 0, 15}]

a(n) = sum(k=1, n\2, (k*(n-k))!); \\ Michel Marcus, Mar 22 2016

a(n) ~ (n^2/4)! ~ sqrt(Pi) * n^(n^2/2+1) / (2^((n^2+1)/2) * exp(n^2/4)) if n is even and a(n) ~ ((n^2-1)/4)! ~ sqrt(Pi) * n^((n^2+1)/2) / (2^(n^2/2) * exp(n^2/4)) if n is odd. - Vaclav Kotesovec, Mar 18 2016

A293452 Triangle T(n,k) read by rows: T(n,k) is the number of iterations to reach a final state for an n X k lattice of sandpiles on a torus according to rules specified in A249872.

Original entry on oeis.org

0, 1, 7, 2, 14, 28, 7, 35, 65, 133, 10, 47, 86, 198, 316, 22, 86, 134, 331, 487, 913, 28, 106, 164, 399, 696, 1099, 1360, 50, 159, 288, 589, 930, 1518, 1798, 2987, 60, 187, 336, 681, 1070, 1966, 2320, 3432, 4340, 95, 265, 515, 1052, 1386, 2430, 3475, 4484, 5977, 7495, 110, 303, 584, 1184, 1556, 2718

Offset: 1

Joerg Arndt, Oct 09 2017

			Triangle begins:
0
1, 7
2, 14, 28
7, 35, 65, 133
10, 47, 86, 198, 316
22, 86, 134, 331, 487, 913
28, 106, 164, 399, 696, 1099, 1360
50, 159, 288, 589, 930, 1518, 1798, 2987
60, 187, 336, 681, 1070, 1966, 2320, 3432, 4340
95, 265, 515, 1052, 1386, 2430, 3475, 4484, 5977, 7495
...

Joerg Arndt, Table of n, a(n) for n = 1..5050 (rows 1..50)
Wikipedia, Abelian sandpile model

Cf. A249872.

T(n,n) = A249872(n).

Conjecture: T(n,1) = A023855(n).

A303120 Total area of all rectangles of size p X q such that p + q = n^2 and p <= q.

Original entry on oeis.org

0, 7, 60, 372, 1300, 4047, 9800, 22352, 44280, 84575, 147620, 251412, 402220, 632247, 949200, 1406272, 2011440, 2847447, 3920460, 5353300, 7147140, 9477567, 12336280, 15966672, 20345000, 25800047, 32284980, 40234292, 49568540, 60851175, 73958560, 89609472

Offset: 1

Wesley Ivan Hurt, Apr 18 2018

Sum of all the products formed using the corresponding largest and smallest parts of each partition of n^2 into two parts. - Wesley Ivan Hurt, Mar 26 2019

			a(3) = 60; The rectangles are 8 X 1, 7 X 2, 6 X 3 and 5 X 4. The total area is then 8*1 + 7*2 + 6*3 + 5*4 = 60.
a(4) = 372; The rectangles are 15 X 1, 14 X 2, 13 X 3, 12 X 4, 11 X 5, 10 X 6, 9 X 7 and 8 X 8. The total area of the rectangles is then 15*1 + 14*2 + 13*3 + 12*4 + 11*5 + 10*6 + 9*7 + 8*8 = 372.

Muniru A Asiru, Table of n, a(n) for n = 1..2000
Index entries for sequences related to partitions

Cf. A000290, A023855.

List([1..35],n->Sum([1..Int(n^2/2)],i->i*(n^2-i))); # Muniru A Asiru, Mar 15 2019

[&+[i*(n^2-i): i in [0..Floor(n^2/2)]]: n in [1..35]]; // Vincenzo Librandi, Apr 19 2018

A303120:=n->add(i*(n^2-i), i=1..floor(n^2/2)): seq(A303120(n), n=1..50); # Wesley Ivan Hurt, Mar 12 2019

Table[Sum[i*(n^2 - i), {i, Floor[n^2/2]}], {n, 50}]

a(n) = sum(i=1, n^2\2, i*(n^2-i)); \\ Michel Marcus, Mar 13 2019

a(n) = Sum_{i=1..floor(n^2/2)} i * (n^2 - i).
Conjectures from Colin Barker, Apr 19 2018 and Mar 19 2019: (Start)
G.f.: x^2*(7 + 46*x + 224*x^2 + 386*x^3 + 594*x^4 + 386*x^5 + 224*x^6 + 46*x^7 + 7*x^8) / ((1 - x)^7*(1 + x)^5).
a(n) = 2*a(n-1) + 4*a(n-2) - 10*a(n-3) - 5*a(n-4) + 20*a(n-5) - 20*a(n-7) + 5*a(n-8) + 10*a(n-9) - 4*a(n-10) - 2*a(n-11) + a(n-12) for n > 12.
a(n) = (n^2*(-4 + 3*(1+(-1)^n)*n^2 + 4*n^4)) / 48.
(End)

A368091 Triangle read by rows. T(n, k) = Sum_{p in P(n, k)} Product_{r in p} r, where P(n, k) are the partitions of n with length k.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 3, 2, 1, 0, 4, 7, 2, 1, 0, 5, 10, 7, 2, 1, 0, 6, 22, 18, 7, 2, 1, 0, 7, 28, 34, 18, 7, 2, 1, 0, 8, 50, 62, 50, 18, 7, 2, 1, 0, 9, 60, 121, 86, 50, 18, 7, 2, 1, 0, 10, 95, 182, 189, 118, 50, 18, 7, 2, 1

Offset: 0

Peter Luschny, Dec 11 2023

			Table T(n, k) starts:
  [0] [1]
  [1] [0, 1]
  [2] [0, 2,  1]
  [3] [0, 3,  2,   1]
  [4] [0, 4,  7,   2,  1]
  [5] [0, 5, 10,   7,  2,  1]
  [6] [0, 6, 22,  18,  7,  2,  1]
  [7] [0, 7, 28,  34, 18,  7,  2, 1]
  [8] [0, 8, 50,  62, 50, 18,  7, 2, 1]
  [9] [0, 9, 60, 121, 86, 50, 18, 7, 2, 1]

Cf. A368090, A074141, A023855, A006906 (row sums).

def T(n, k):
    return sum(product(r for r in p) for p in Partitions(n, length=k))
for n in range(10): print([T(n, k) for k in range(n + 1)])

A110422 a(n) = sum( (-1)^(r+1)(n-r)r, r = 1..floor(n/2) ).

Original entry on oeis.org

1, 2, -1, -2, 6, 8, -6, -8, 15, 18, -15, -18, 28, 32, -28, -32, 45, 50, -45, -50, 66, 72, -66, -72, 91, 98, -91, -98, 120, 128, -120, -128, 153, 162, -153, -162, 190, 200, -190, -200, 231, 242, -231, -242, 276, 288, -276, -288, 325, 338, -325, -338, 378, 392, -378, -392, 435, 450, -435, -450, 496, 512, -496, -512, 561

Offset: 2

Amarnath Murthy, Aug 01 2005

a(4n)=-a(4n-2); a(4n+1)=-a(4n-1). If sum in definition is not alternating one obtains A023855. - Emeric Deutsch, Aug 08 2005

			a(8) = -6 because 7*1-6*2+5*3-4*4 = -6.

Index entries for linear recurrences with constant coefficients, signature (2,-4,6,-6,6,-4,2,-1).

Cf. A023855.

a:=n->sum((-1)^(r+1)*(n-r)*r,r=1..floor(n/2)): seq(a(n),n=2..70); # Emeric Deutsch, Aug 08 2005

CoefficientList[Series[(2 x^3 - x^2 + 1)/((x - 1)^2 (x^2 + 1)^3), {x, 0, 70}], x] (* Vincenzo Librandi, Oct 30 2014 *)
LinearRecurrence[{2,-4,6,-6,6,-4,2,-1},{1,2,-1,-2,6,8,-6,-8},70] (* Harvey P. Dale, Apr 04 2020 *)

Vec(x^2*(2*x^3-x^2+1)/((x-1)^2*(x^2+1)^3) + O(x^100)) \\ Colin Barker, Oct 30 2014

a(2n) = (1/2)n-(-1)^n*(1/2)n^2; a(2n-1) = (1/2)n-(1/4)+(-1)^n*(1/4)(2n^2-2n+1). - Emeric Deutsch, Aug 08 2005
a(n) = (-1)^((2*n-5+(-1)^n)/4)*(2*n^2+1-(-1)^n+4*n*(-1)^((2*n-5+(-1)^n)/4))/16. - Luce ETIENNE, Oct 30 2014
G.f.: x^2*(2*x^3-x^2+1) / ((x-1)^2*(x^2+1)^3). - Colin Barker, Oct 30 2014

Corrected and extended by Emeric Deutsch, Aug 08 2005

A337173 a(n) = Sum_{k=1..floor(n/2)} k^2 * (n-k)^2.

Original entry on oeis.org

0, 1, 4, 25, 52, 170, 280, 674, 984, 1979, 2684, 4795, 6188, 10164, 12656, 19524, 23664, 34773, 41268, 58333, 68068, 93214, 107272, 143078, 162760, 212303, 239148, 306047, 341852, 430312, 477152, 592008, 652256, 799017, 875364, 1060257, 1155732, 1385746, 1503736

Offset: 1

Wesley Ivan Hurt, Jan 28 2021

			a(6) = 1^2*5^2 + 2^2*4^2 + 3^2*3^2 = 25 + 64 + 81 = 170.

Index entries for linear recurrences with constant coefficients, signature (1,5,-5,-10,10,10,-10,-5,5,1,-1).

Cf. A023855.

CoefficientList[Series[x (1 + 3 x + 16 x^2 + 12 x^3 + 23 x^4 + 5 x^5 + 4 x^6)/((1 - x)^6 (1 + x)^5), {x, 0, 80}], x]

G.f.: x^2*(1+3*x+16*x^2+12*x^3+23*x^4+5*x^5+4*x^6)/((1-x)^6*(1+x)^5).
a(n) = (2*n-1+(-1)^n)*(2*n+3+(-1)^n)*(16*n^3-n^2+10*n-4-(n^2+6*n+4)*(-1)^n)/3840.
a(n) = a(n-1) + 5*a(n-2) - 5*a(n-3) - 10*a(n-4) + 10*a(n-5) + 10*a(n-6) - 10*a(n-7) - 5*a(n-8) + 5*a(n-9) + a(n-10) - a(n-11).

A345026 Total area of all i X j rectangles where i and j are the final digits of positive integers r and s such that r + s = n.

Original entry on oeis.org

0, 1, 2, 7, 10, 22, 28, 50, 60, 95, 100, 131, 122, 152, 130, 162, 128, 165, 120, 165, 200, 226, 242, 252, 250, 247, 228, 215, 180, 260, 300, 356, 362, 397, 370, 387, 328, 330, 240, 330, 400, 451, 482, 497, 490, 472, 428, 380, 300, 425, 500, 581, 602, 642, 610, 612, 528, 495, 360

Offset: 1

Wesley Ivan Hurt, Jun 06 2021

			a(20) = 165; There are 10 ways to write 20 as the sum of two positive integers: (19,1), (18,2), (17,3), (16,4), (15,5), (14,6), (13,7), (12,8), (11,9), and (10,10). Using the final digits from each pair as the side lengths of the rectangles, the combined area is 9*1 + 8*2 + 7*3 + 6*4 + 5*5 + 4*6 + 3+7 + 2*8 + 1*9 + 0*0 = 165.

Cf. A010879, A023855.

Table[Sum[Mod[k, 10]*Mod[n - k, 10], {k, Floor[n/2]}], {n, 60}]

a(n) = Sum_{k=1..floor(n/2)} (k mod 10) * ((n-k) mod 10).

cp's OEIS Frontend

A270531 a(n) = Sum_{i=1..floor(n/2)} (i*(n-i))!.

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A293452 Triangle T(n,k) read by rows: T(n,k) is the number of iterations to reach a final state for an n X k lattice of sandpiles on a torus according to rules specified in A249872.

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A303120 Total area of all rectangles of size p X q such that p + q = n^2 and p <= q.

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A368091 Triangle read by rows. T(n, k) = Sum_{p in P(n, k)} Product_{r in p} r, where P(n, k) are the partitions of n with length k.

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A110422 a(n) = sum( (-1)^(r+1)*(n-r)*r, r = 1..floor(n/2) ).

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A337173 a(n) = Sum_{k=1..floor(n/2)} k^2 * (n-k)^2.

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A345026 Total area of all i X j rectangles where i and j are the final digits of positive integers r and s such that r + s = n.

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A110422 a(n) = sum( (-1)^(r+1)(n-r)r, r = 1..floor(n/2) ).