A033487 -id:A033487 - cp's OEIS frontend

A180118 a(n) = Sum_{k=1..n} (k+2)!/k! = Sum_{k=1..n} (k+2)*(k+1).

0, 6, 18, 38, 68, 110, 166, 238, 328, 438, 570, 726, 908, 1118, 1358, 1630, 1936, 2278, 2658, 3078, 3540, 4046, 4598, 5198, 5848, 6550, 7306, 8118, 8988, 9918, 10910, 11966, 13088, 14278, 15538, 16870, 18276, 19758, 21318, 22958, 24680, 26486, 28378, 30358

Offset: 0

Gary Detlefs, Aug 10 2010

In general, sequences of the form a(n) = sum((k+x+2)!/(k+x)!,k=1..n) have a closed form a(n) = n*(11+12*x+3*x^2+3*x*n+6*n+n^2)/3.
This sequence is related to A033487 by A033487(n) = n*a(n)-sum(a(i), i=0..n-1). - Bruno Berselli, Jan 24 2011
The minimal number of multiplications (using schoolbook method) needed to compute the matrix chain product of a sequence of n+1 matrices having dimensions 1 X 2, 2 X 3, ..., (n+1) X (n+2), respectively. - Alois P. Heinz, Jan 27 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
B. Berselli, A description of the recursive method in Comments lines: website Matem@ticamente (in Italian).
Wikipedia, Matrix chain multiplication
Wikipedia, Schoolbook matrix multiplication
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A005718, A033487, A050534.

[n*(n^2+6*n+11)/3: n in [0..45]]; // Vincenzo Librandi, Jun 15 2011

f[n_]:=n*(n^2 + 6 n + 11)/3; f[Range[0,60]] (* Vladimir Joseph Stephan Orlovsky, Feb 10 2011*)
CoefficientList[Series[2*x*(3 - 3*x + x^2)/(1 - x)^4, {x, 0, 50}], x] (* Vaclav Kotesovec, May 10 2019 *)
Table[Sum[(k+1)(k+2),{k,n}],{n,0,50}] (* or *) LinearRecurrence[{4,-6,4,-1},{0,6,18,38},50] (* Harvey P. Dale, Apr 21 2020 *)

a(n) = +4*a(n-1)-6*a(n-2)+4*a(n-3)-1*a(n-4) for n>=4.
a(n) = n*(n^2+6*n+11)/3.
From Bruno Berselli, Jan 24 2011:  (Start)
G.f.: 2*x*(3-3*x+x^2)/(1-x)^4. [corrected by Georg Fischer, May 10 2019]
Sum(a(k), k=0..n) = 2*A005718(n) for n>0. (End)

A188667 Ordered (2,2)-selections from the multiset {1,1,2,2,3,3,...,n,n}.

Original entry on oeis.org

0, 0, 3, 21, 72, 180, 375, 693, 1176, 1872, 2835, 4125, 5808, 7956, 10647, 13965, 18000, 22848, 28611, 35397, 43320, 52500, 63063, 75141, 88872, 104400, 121875, 141453, 163296, 187572, 214455, 244125, 276768, 312576, 351747, 394485, 441000

Offset: 0

Thomas Wieder, Apr 07 2011

Number of ordered (2,2)-selections which can be taken from the first 2n elements of A008619, the positive integers repeated.  Order does count among subselections, e.g. [[1,1],[2,2]] and [[2,2],[1,1]] are different (2,2)-selections. Order does not count within a subselection, e.g. [1,3] is equivalent to [3,1].
Many thanks to Alois P. Heinz, Joerg Arndt, and Olivier Gérard for pointing out bugs in earlier versions of this sequence and for their comments!
The number of (not ordered) (2,2)-selections from natural numbers repeated = A008619 is equal to A086602 (observed by Alois P. Heinz).
The number of ordered (1,1)-selections from natural numbers repeated = A008619 is equal to the squares = A000290.
The number of ordered (1,1)-selections from the natural numbers = A000027 ("[1,2,3,...,n]-multiset") is equal to the Oblong numbers = A002378.
The number of ordered (2,2)-selections from the natural numbers = A000027 ("[1,2,3,...,n]-multiset") is equal to A033487.
The number of (not ordered) (1,1)-selections from the natural numbers = A000027 ("[1,2,3,...,n]-multiset") is equal to the triangular numbers = A000217.
The number of (not ordered) (2,2)-selections from the natural numbers = A000027 ("[1,2,3,...,n]-multiset") is equal to the tritriangular numbers = A050534.
For n>0, the terms of this sequence are related to A014209 by a(n) = sum( i*A014209(i), i=0..n-1 ). [Bruno Berselli, Dec 20 2013]

			Example: For n=3 there are 21 ordered selections of the type (2,2):
[[1,1],[2,2]], [[1,2],[1,2]], [[2,2],[1,1]], [[1,2],[2,3]],
[[1,3],[2,2]], [[2,2],[1,3]], [[2,3],[1,2]], [[1,1],[2,3]],
[[1,2],[1,3]], [[1,3],[1,2]], [[2,3],[1,1]], [[1,1],[3,3]],
[[1,3],[1,3]], [[3,3],[1,1]], [[1,2],[3,3]], [[1,3],[2,3]],
[[2,3],[1,3]], [[3,3],[1,2]], [[2,2],[3,3]], [[2,3],[2,3]],
[[3,3],[2,2]].

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Quang T. Bach, Roshil Paudyal, Jeffrey B. Remmel, A Fibonacci analogue of Stirling numbers, arXiv:1510.04310 [math.CO], 2015 (see p. 28).
T. Wieder, Generation of all possible multiselections from a multiset, Progress in Applied Mathematics, 2(1) (2011), 61-66, DOI:10.3968/j.pam.1925252820110201.010. - Thomas Wieder, Oct 15 2011

Cf. A014209.

Table[n*(n + 4)*(n - 1)^2/4, {n, 0, 100}] (* Vincenzo Librandi, Oct 18 2012 *)

a(n) = n*(n+4)*(n-1)^2/4.

G.f.: 3*x^2*(x^2-2*x-1) / (x-1)^5.

A286030 Irregular triangle T(n,k) read by rows: Let S be a 3-member set of integers {f,g,h} where f >= g >= h >= 0 and f+g+h = n. Let S(n,k) be an irregular triangle composed of all S listed in reverse lexicographic order by row n. Then T(n,k) = n!Q/(3f!g!h!), where Q is the number of permutations of S(n,k). (See "Comments" and "Examples" for additional explanation.)

Original entry on oeis.org

1, 1, 2, 1, 6, 2, 1, 8, 6, 12, 1, 10, 20, 20, 30, 1, 12, 30, 30, 20, 120, 30, 1, 14, 42, 42, 70, 210, 140, 210, 1, 16, 56, 56, 112, 336, 70, 560, 420, 560, 1, 18, 72, 72, 168, 504, 252, 1008, 756, 630, 2520, 560

Offset: 1

Bob Selcoe, Apr 30 2017

See "Example" below for the starting construction of S(n,k) and T(n,k).
To understand S(n,k) and Q, consider example S(5,k), i.e., f+g+h = 5, and S(n,k) are listed in reverse lexicographic order. So S(5,k) = {5,0,0}, {4,1,0}, {3,2,0}, {3,1,1}, {2,2,1} k=1..5, respectively. Q is the number of permutations of S(n,k). So Q=3 when S(n,k) = {5,0,0}, {3,1,1} and {2,2,1}; and Q=6 when S(n,k) = {4,1,0} and {3,2,0}.
In general, by definition: Q=1 when all members of S(n,k) are equal, Q=3 when S(n,k) contains a pair, and Q=6 when none of the members of S(n,k) is equal.
Suppose three equally-matched players are playing a tournament of n games; and for each game there is one winner and two losers. Then S(n,k) is the "overall win record" (where player order does not matter) after n games. Let p be the probability that any S(n,k) occurs after n games. Then p = T(n,k)/3^(n-1). (See also "Example" section.)
Generally, when S(n,k) is a z-member set {f,g,h,i..,y}, then Q is the number of permutations of S(n,k), T(n,k) = n!*Q/(z*f!*g!*h!..*y!) and p = T(n,k)/z^(n-1). So when z=2 we get A008314. (Observation prompted by query from Linda Rogers.)
For triangle T(n,k):
Row sums are 3^(n-1).
Row lengths are A001399(n).
Final terms in each row are A199127(n).
For n >= 3: T(n,2) = 2*n.
For n >= 5: T(n,3) = T(n,4) = A002378(n-1) (oblong numbers).
For n >= 6: T(n,6) = A007531(n).
For n >= 8: T(n,9) = A033487(n-3).

			Triangle T(n,k) begins:
n/k 1    2    3    4    5   6    7     8    9    10    11    12    13    14
1:  1
2:  1,   2
3:  1,   6,   2
4:  1,   8,   6,  12
5:  1,  10,  20,  20,  30
6:  1,  12,  30,  30,  20, 120,  30
7:  1,  14,  42,  42,  70, 210, 140,  210
8:  1,  16,  56,  56, 112, 336,  70,  560,  420, 560
9:  1,  18,  72,  72, 168, 504, 252, 1008,  756, 630, 2520, 560
10: 1,  20,  90,  90, 240, 720, 420, 1680, 1260, 252, 2520, 5040, 3150, 4200
Triangle S(n,k) begins:
n/k    1        2        3        4        5        6        7
1:  {1,0,0}
2:  {2,0,0}  {1,1,0}
3:  {3,0,0}  {2,1,0}  {1,1,1}
4:  {4,0,0}  {3,1,0}  {2,2,0}  {2,1,1}
5:  {5,0,0}  {4,1,0}  {3,2,0}  {3,1,1}  {2,2,1}
6:  {6,0,0}  {5,1,0}  {4,2,0}  {4,1,1}  {3,3,0}  {3,2,1}  {2,2,2}
T(4,3) = 6 because n=4 and S(4,3) = {2,2,0}; so Q=3 and 3*4!/(3*2!*2!*0!) = 6. Therefore p = 6/27 = 2/9 that the overall win record = {2,2,0} after playing 4 tournament games.

Nicolas Behr, Pawel Sobocinski, Rule Algebras for Adhesive Categories, arXiv:1807.00785 [cs.LO], 2018, also LIPIcs 27th EACSL Annual Conference on Computer Science Logic (CSL 2018), Vol. 119, pp. 11:1-11:21.

Cf. A001399, A002378, A007531, A008314, A033487, A199127.

Cf. A000041 (partition numbers).

A356251 a(n) = n(n+1)(n+2)(n+3)(2*n+1)/12.

Original entry on oeis.org

0, 6, 50, 210, 630, 1540, 3276, 6300, 11220, 18810, 30030, 46046, 68250, 98280, 138040, 189720, 255816, 339150, 442890, 570570, 726110, 913836, 1138500, 1405300, 1719900, 2088450, 2517606, 3014550, 3587010, 4243280, 4992240, 5843376, 6806800, 7893270, 9114210

Offset: 0

Edward Krogius, Jul 31 2022

Sum of all numbers squared in ordered triples (x,y,z) where 0 <= x <= y <= z <= n.

			a(1) = 6 because we have the triples (0,0,0), (0,0,1), (0,1,1), (1,1,1).

Edward Krogius, Table of n, a(n) for n = 0..999
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A033487.

Table[n*(n + 1)*(n + 2)*(n + 3)*(2*n + 1)/12, {n, 0, 35}] (* Amiram Eldar, Sep 11 2022 *)
Table[Sum[x^2 + y^2 + z^2, {x, 0, g}, {y, x, g}, {z, y, g}], {g, 0, 30}] (* Horst H. Manninger, Jun 19 2025 *)

G.f.: 2*x*(7*x+3)/(x-1)^6.
From Amiram Eldar, Sep 11 2022: (Start)
Sum_{n>=1} 1/a(n) = 136/15 - 64*log(2)/5.
Sum_{n>=1} (-1)^(n+1)/a(n) = 16*Pi/5 - 32*log(2)/5 - 82/15. (End)

A370373 T(n, k) is the total number of non-symmetric peaks in all partitions of n with exactly k blocks, n >= 4, 3 <= k <= n-1.

Original entry on oeis.org

1, 6, 3, 27, 30, 6, 108, 205, 90, 10, 405, 1188, 870, 210, 15, 1458, 6279, 6888, 2730, 420, 21, 5103, 31306, 48622, 28308, 7070, 756, 28, 17496, 149985, 318726, 256914, 92988, 16002, 1260, 36, 59049, 698256, 1984950, 2136150, 1054305, 260316, 32760, 1980, 45, 196830

Offset: 4

Noor Kezil, Jun 07 2024

			The triangle T(n, k) begins:
   4|    1
   5|    6      3
   6|   27     30      6
   7|  108    205     90     10
   8|  405   1188    870    210     15
   9| 1458   6279   6888   2730    420     21
  10| 5103  31306  48622  28308   7070    756   28
.
T(5,3) represents the partitions of the set {1,2,3,4,5} into 3 blocks:
The canonical form of a set partition of [n] is an n-tuple indicating the block in which each integer occurs. The non-symmetric peaks in the canonical sequential form are listed:
  (1, 2, 3, 1, 1) -> 1 non-symmetric peak,  (2, 3, 1)
  (1, 2, 3, 1, 2) -> 1 non-symmetric peak,  (2, 3, 1)
  (1, 2, 3, 1, 3) -> 1 non-symmetric peak,  (2, 3, 1)
  (1, 2, 2, 3, 1) -> 1 non-symmetric peak,  (2, 3, 1)
  (1, 1, 2, 3, 1) -> 1 non-symmetric peak,  (2, 3, 1)
  (1, 2, 1, 3, 2) -> 1 non-symmetric peak,  (1, 3, 2)

W. Asakly and Noor Kezil, Counting symmetric and non-symmetric peaks in a set partition, arXiv:2401.01687 [math.CO], 2024.

Cf. A008277 (Stirling2).
Cf. A373288.
Cf. A027471 (1st column), A033487 (subdiagonal).

T := (n, k) -> binomial(k-1, 2) * Stirling2(n-1, k) + 2 * add(binomial(j, 3) * add(j^(i-3) * Stirling2(n-i, k), i=3..n-k), j = 3..k): seq(print(seq(T(n, k), k = 3..n-1)), n = 4..10);

T[n_, k_] := Binomial[k-1, 2] * StirlingS2[n-1, k] + 2 * Sum[Binomial[j, 3] * Sum[j^(i-3) * StirlingS2[n-i, k], {i, 3, n-k}], {j, 3, k}];Table[T[n, k], {n, 4, 12}, {k, 3, n-1}]

T(n, k) = binomial(k-1, 2) * stirling(n-1, k, 2) + 2 * sum(j=3, k, binomial(j, 3) * sum(i=3, n-k, j^(i-3) * stirling(n-i, k, 2)));

T(n,k) = binomial(k-1, 2) * Stirling2(n-1, k) + 2 * Sum_{j=3..k} binomial(j, 3) * Sum_{i=3..n-k} j^(i-3) * Stirling2(n-i, k).

A385577 Array read by ascending antidiagonals: A(n,m) = n*Pochhammer(n+1,m+1)/(m+2).

Original entry on oeis.org

0, 1, 0, 3, 2, 0, 6, 8, 6, 0, 10, 20, 30, 24, 0, 15, 40, 90, 144, 120, 0, 21, 70, 210, 504, 840, 720, 0, 28, 112, 420, 1344, 3360, 5760, 5040, 0, 36, 168, 756, 3024, 10080, 25920, 45360, 40320, 0, 45, 240, 1260, 6048, 25200, 86400, 226800, 403200, 362880, 0

Offset: 0

Stefano Spezia, Jul 03 2025

			Array begins as:
   0,  0,   0,    0,     0,     0,      0, ...
   1,  2,   6,   24,   120,   720,   5040, ...
   3,  8,  30,  144,   840,  5760,  45360, ...
   6, 20,  90,  504,  3360, 25920, 226800, ...
  10, 40, 210, 1344, 10080, 86400, 831600, ...
  ...

James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 20.

Paul W. Haggard and Bonnie L. Sadler, A Generalization of Triangular Numbers, International Journal of Mathematical Education in Science and Technology 25 (2): 195-202, (1994).

Cf. A000142, A001048, A002740, A006231, A007290, A068424, A238474.
Cf. A000217 (m=0), A033487 (m=2), A158874 (m=3).
Cf. A000004 (n=0).

A[n_,m_]:=n*Pochhammer[n+1,m+1]/(m+2); Table[A[n-m,m],{n,0,9},{m,0,n}]//Flatten

Sum_{m=0..n} A(n-m,m) = A006231(n+1).
A(n,1) = A007290(n+2).
A(1,n) = A000142(n+1).
A(2,n) = A001048(n+2).
A(3,n) = abs(A238474(n+1)).
A(n,n) = n!*A002740(n+2)

cp's OEIS Frontend

A180118 a(n) = Sum_{k=1..n} (k+2)!/k! = Sum_{k=1..n} (k+2)*(k+1).

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A188667 Ordered (2,2)-selections from the multiset {1,1,2,2,3,3,...,n,n}.

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A356251 a(n) = n*(n+1)*(n+2)*(n+3)*(2*n+1)/12.

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A370373 T(n, k) is the total number of non-symmetric peaks in all partitions of n with exactly k blocks, n >= 4, 3 <= k <= n-1.

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A385577 Array read by ascending antidiagonals: A(n,m) = n*Pochhammer(n+1,m+1)/(m+2).

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A356251 a(n) = n(n+1)(n+2)(n+3)(2*n+1)/12.