A028310 -id:A028310 - cp's OEIS frontend

A340584 Irregular triangle read by rows T(n,k) in which row n lists sigma(n) + sigma(n-1) together with the first n - 2 terms of A000203 in reverse order, with T(1,1) = 1, n >= 1.

1, 4, 7, 1, 11, 3, 1, 13, 4, 3, 1, 18, 7, 4, 3, 1, 20, 6, 7, 4, 3, 1, 23, 12, 6, 7, 4, 3, 1, 28, 8, 12, 6, 7, 4, 3, 1, 31, 15, 8, 12, 6, 7, 4, 3, 1, 30, 13, 15, 8, 12, 6, 7, 4, 3, 1, 40, 18, 13, 15, 8, 12, 6, 7, 4, 3, 1, 42, 12, 18, 13, 15, 8, 12, 6, 7, 4, 3, 1, 38, 28, 12, 18, 13, 15, 8, 12, 6, 7, 4, 3, 1

Offset: 1

Omar E. Pol, Jan 12 2021

T(n,k) is the total area (or number of cells) of the terraces that are in the k-th level that contains terraces starting from the base of the symmetric tower (a polycube) described in A221529 which has A000041(n-1) levels in total. The terraces of the polycube are the symmetric representation of sigma. The terraces are in the levels that are the partition numbers A000041 starting from the base. Note that for n >= 2 there are n - 1 terraces because the first terrace of the tower is formed by two symmetric representations of sigma in the same level. The volume (or the number of cubes) equals A066186(n), the sum of all parts of all partitions of n. The volume is also the sum of all divisors of all terms of the first n rows of A336811. That is due to the correspondence between divisors and partitions (cf. A336811). The growth of the volume (A066186) represents the convolution of A000203 and A000041.

			Triangle begins:
   1;
   4;
   7,  1;
  11,  3,  1;
  13,  4,  3,  1;
  18,  7,  4,  3,  1;
  20,  6,  7,  4,  3,  1;
  23, 12,  6,  7,  4,  3,  1;
  28,  8, 12,  6,  7,  4,  3,  1;
  31, 15,  8, 12,  6,  7,  4,  3,  1;
  30, 13, 15,  8, 12,  6,  7,  4,  3,  1;
  40, 18, 13, 15,  8, 12,  6,  7,  4,  3,  1;
  42, 12, 18, 13, 15,  8, 12,  6,  7,  4,  3,  1;
  38, 28, 12, 18, 13, 15,  8, 12,  6,  7,  4,  3,  1;
...
For n = 7, sigma(7) = 1 + 7 = 8 and sigma(6) = 1 + 2 + 3 + 6 = 12, and 8 + 12 = 20, so the first term of row 7 is T(7,1) = 20. The other terms in row 7 are the first five terms of A000203 in reverse order, that is [6, 7, 4, 3, 1] so the 7th row of the triangle is [20, 6, 7, 4, 3, 1].
From _Omar E. Pol_, Jul 11 2021: (Start)
For n = 7 we can see below the top view and the lateral view of the pyramid described in A245092 (with seven levels) and the top view and the lateral view of the tower described in A221529 (with 11 levels).
                                           _
                                          | |
                                          | |
                                          | |
        _                                 |_|_
       |_|_                               |   |
       |_ _|_                             |_ _|_
       |_ _|_|_                           |   | |
       |_ _ _| |_                         |_ _|_|_
       |_ _ _|_ _|_                       |_ _ _| |_
       |_ _ _ _| | |_                     |_ _ _|_ _|_ _
       |_ _ _ _|_|_ _|                    |_ _ _ _|_|_ _|
.
         Figure 1.                           Figure 2.
        Lateral view                       Lateral view
       of the pyramid.                     of the tower.
.
.       _ _ _ _ _ _ _                      _ _ _ _ _ _ _
       |_| | | | | | |                    |_| | | | |   |
       |_ _|_| | | | |                    |_ _|_| | |   |
       |_ _|  _|_| | |                    |_ _|  _|_|   |
       |_ _ _|    _|_|                    |_ _ _|    _ _|
       |_ _ _|  _|                        |_ _ _|  _|
       |_ _ _ _|                          |       |
       |_ _ _ _|                          |_ _ _ _|
.
          Figure 3.                          Figure 4.
          Top view                           Top view
       of the pyramid.                     of the tower.
.
Both polycubes have the same base which has an area equal to A024916(7) = 41 equaling the sum of the 7th row of triangle.
Note that in the top view of the tower the symmetric representation of sigma(6) and the symmetric representation of sigma(7) appear unified in the level 1 of the structure as shown above in the figure 4 (that is due to the first two partition numbers A000041 are [1, 1]), so T(7,1) = sigma(7) + sigma(6) = 8 + 12 = 20. (End)

The length of row n is A028310(n-1).
Row sums give A024916.
Column 1 gives 1 together with A092403.
Other columns give A000203.
Cf. A175254 (volume of the pyramid).
Cf. A066186 (volume of the tower).
Cf. A346533 (mirror).
Cf. A000041, A221529, A237270, A237593, A245092, A272172 (similar), A336811, A338156, A339106, A340035.

Table[If[n <= 2, {Total@ #}, Prepend[#2, Total@ #1] & @@ TakeDrop[#, 2]] &@ DivisorSigma[1, Range[n, 1, -1]], {n, 14}] // Flatten (* Michael De Vlieger, Jan 13 2021 *)

A073273 a(n) = floor(sqrt(prime(n)*prime(n+2))).

Original entry on oeis.org

3, 4, 7, 9, 13, 15, 19, 23, 26, 32, 35, 39, 43, 47, 52, 56, 62, 65, 69, 74, 77, 83, 89, 94, 99, 103, 105, 109, 117, 121, 131, 134, 142, 144, 152, 156, 161, 167, 172, 176, 184, 186, 193, 195, 203, 210, 218, 225, 229, 233, 236, 244, 248, 256, 262, 266, 272, 275

Offset: 1

Reinhard Zumkeller, Jul 22 2002

nonn

A000040(n) < a(n) < A000040(n+2).

			prime(10)*prime(12) = 29*37 = 1073 = 32*32+49, therefore a(10)=32; A073274(10) = prime(11)-a(10) = 31-32 = -1.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000040, A073271, A073274, A078444.

[Floor(Sqrt(NthPrime(n)*NthPrime(n+2))): n in [1..60]]; // Vincenzo Librandi, Dec 12 2015

Table[Floor[Sqrt[Prime[n] Prime[n + 2]]], {n, 60}] (* Vincenzo Librandi, Dec 12 2015 *)

a(n) = sqrtint(prime(n)*prime(n+2)); \\ Michel Marcus, Dec 12 2015

a(n) = A098090(A028310(n - 1)) + A089038(n). - Miko Labalan, Dec 12 2015

A182938 If n = Product (p_j^e_j) then a(n) = Product (binomial(p_j, e_j)).

Original entry on oeis.org

1, 2, 3, 1, 5, 6, 7, 0, 3, 10, 11, 3, 13, 14, 15, 0, 17, 6, 19, 5, 21, 22, 23, 0, 10, 26, 1, 7, 29, 30, 31, 0, 33, 34, 35, 3, 37, 38, 39, 0, 41, 42, 43, 11, 15, 46, 47, 0, 21, 20, 51, 13, 53, 2, 55, 0, 57, 58, 59, 15, 61, 62, 21, 0, 65, 66

Offset: 1

Peter Luschny, Jan 16 2011

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Plot of Sum_{k=1..n} a(k) / n^2 for n = 1..1000000

Cf. A000026, A001414, A008473, A008474, A008475, A008476, A008477, A028310, A069799.

Cf. A027748, A124010, A007318, A328745.

a182938 n = product $ zipWith a007318'
   (a027748_row n) (map toInteger $ a124010_row n)
-- Reinhard Zumkeller, Feb 18 2012

A182938 := proc(n) local e,j; e := ifactors(n)[2]:
mul (binomial(e[j][1], e[j][2]), j=1..nops(e)) end:
seq (A182938(n), n=1..100);

a[n_] := Times @@ (Map[Binomial @@ # &, FactorInteger[n], 1]);
Table[a[n], {n, 1, 100}] (* Kellen Myers, Jan 16 2011 *)

a(n)=prod(i=1,#n=factor(n)~,binomial(n[1,i],n[2,i])) \\ M. F. Hasler

for(n=1, 100, print1(direuler(p=2, n, (1 + X)^p)[n], ", ")) \\ Vaclav Kotesovec, Mar 28 2025

a(A185359(n)) = 0. - Reinhard Zumkeller, Feb 18 2012
Dirichlet g.f.: Product_{p prime} (1 + p^(-s))^p. - Ilya Gutkovskiy, Oct 26 2019
Conjecture: Sum_{k=1..n} a(k) ~ c * n^2, where c = 0.33754... - Vaclav Kotesovec, Mar 28 2025

Given terms checked with new PARI code by M. F. Hasler, Jan 16 2011

A256116 Number T(n,k) of length 2n k-ary words, either empty or beginning with the first letter of the alphabet and using each letter at least once, that can be built by repeatedly inserting doublets into the initially empty word; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 1, 9, 10, 0, 1, 34, 112, 84, 0, 1, 125, 930, 1800, 1008, 0, 1, 461, 7018, 26400, 35640, 15840, 0, 1, 1715, 51142, 334152, 816816, 840840, 308880, 0, 1, 6434, 368464, 3944220, 15550080, 27824160, 23063040, 7207200

Offset: 0

Alois P. Heinz, Mar 15 2015

			T(3,2) = 9: aaaabb, aaabba, aabaab, aabbaa, aabbbb, abaaba, abbaaa, abbabb, abbbba.
T(3,3) = 10: aabbcc, aabccb, aacbbc, aaccbb, abbacc, abbcca, abccba, acbbca, accabb, accbba.
T(4,2) = 34: aaaaaabb, aaaaabba, aaaabaab, aaaabbaa, aaaabbbb, aaabaaba, aaabbaaa, aaabbabb, aaabbbba, aabaaaab, aabaabaa, aabaabbb, aababbab, aabbaaaa, aabbaabb, aabbabba, aabbbaab, aabbbbaa, aabbbbbb, abaaaaba, abaabaaa, abaababb, abaabbba, ababbaba, abbaaaaa, abbaaabb, abbaabba, abbabaab, abbabbaa, abbabbbb, abbbaaba, abbbbaaa, abbbbabb, abbbbbba.
T(4,4) = 84: aabbccdd, aabbcddc, aabbdccd, aabbddcc, aabccbdd, aabccddb, aabcddcb, aabdccdb, aabddbcc, aabddccb, aacbbcdd, aacbbddc, aacbddbc, aaccbbdd, aaccbddb, aaccdbbd, aaccddbb, aacdbbdc, aacddbbc, aacddcbb, aadbbccd, aadbbdcc, aadbccbd, aadcbbcd, aadccbbd, aadccdbb, aaddbbcc, aaddbccb, aaddcbbc, aaddccbb, abbaccdd, abbacddc, abbadccd, abbaddcc, abbccadd, abbccdda, abbcddca, abbdccda, abbddacc, abbddcca, abccbadd, abccbdda, abccddba, abcddcba, abdccdba, abddbacc, abddbcca, abddccba, acbbcadd, acbbcdda, acbbddca, acbddbca, accabbdd, accabddb, accadbbd, accaddbb, accbbadd, accbbdda, accbddba, accdbbda, accddabb, accddbba, acdbbdca, acddbbca, acddcabb, acddcbba, adbbccda, adbbdacc, adbbdcca, adbccbda, adcbbcda, adccbbda, adccdabb, adccdbba, addabbcc, addabccb, addacbbc, addaccbb, addbbacc, addbbcca, addbccba, addcbbca, addccabb, addccbba.
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 1,    2;
  0, 1,    9,    10;
  0, 1,   34,   112,     84;
  0, 1,  125,   930,   1800,   1008;
  0, 1,  461,  7018,  26400,  35640,  15840;
  0, 1, 1715, 51142, 334152, 816816, 840840, 308880;

Alois P. Heinz, Rows n = 0..140, flattened

Columns k=0-2 give: A000007, A057427, A010763(n-1) for n>0.
Main diagonal gives A065866(n-1) (for n>0).
Row sums give A294603.
Cf. A183135, A256117.

A:= proc(n, k) option remember; `if`(n=0, 1, k/n*
      add(binomial(2*n, j) *(n-j) *(k-1)^j, j=0..n-1))
    end:
T:= (n, k)-> add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k)/
    `if`(k=0, 1, k):
seq(seq(T(n, k), k=0..n), n=0..12);

Unprotect[Power]; 0^0 = 1; A[n_, k_] := A[n, k] = If[n==0, 1, k/n*Sum[ Binomial[2*n, j]*(n-j)*(k-1)^j, {j, 0, n-1}]];
T[n_, k_] := Sum[A[n, k-i]*(-1)^i*Binomial[k, i], {i, 0, k}]/If[k==0, 1, k]; Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 22 2017, translated from Maple *)

T(n,k) = (Sum_{i=0..k} (-1)^i * C(k,i) * A183135(n,k-i)) / A028310(k).

T(n,k) = (k-1)! * A256117(n,k) for k > 0.

A330640 a(n) is the number of partitions of n with Durfee square of size <= 2.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 22, 29, 40, 51, 67, 83, 105, 127, 156, 185, 222, 259, 305, 351, 407, 463, 530, 597, 676, 755, 847, 939, 1045, 1151, 1272, 1393, 1530, 1667, 1821, 1975, 2147, 2319, 2510, 2701, 2912, 3123, 3355, 3587, 3841, 4095, 4372, 4649, 4950, 5251, 5577, 5903, 6255, 6607, 6986

Offset: 0

Omar E. Pol, Dec 22 2019

This is an easy sequence since A006918 is the partial sums of A008805 (triangular numbers repeated).

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Cf. A000041, A006918, A008805, A028310, A115994, A115720.

Vec((1 - x - x^2 + 2*x^3 - x^5 + x^6) / ((1 - x)^4*(1 + x)^2) + O(x^60)) \\ Colin Barker, Dec 31 2019

a(n) = A028310(n), 0 <= n <= 2.
a(n) = A028310(n) + A006918(n-3), n >= 3.
Or without A028310:
a(0) = 1, a(1) = 1, a(2) = 2.
a(n) = n + A006918(n-3), n >= 3.
From Colin Barker, Dec 31 2019: (Start)
G.f.: (1 - x - x^2 + 2*x^3 - x^5 + x^6) / ((1 - x)^4*(1 + x)^2).
a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6) for n>6.
a(n) = (3 - 3*(-1)^n + (49+3*(-1)^n)*n - 6*n^2 + 2*n^3) / 48.
(End)

A346533 Irregular triangle read by rows in which row n lists the first n - 2 terms of A000203 together with the sum of A000203(n-1) and A000203(n), with a(1) = 1.

Original entry on oeis.org

1, 4, 1, 7, 1, 3, 11, 1, 3, 4, 13, 1, 3, 4, 7, 18, 1, 3, 4, 7, 6, 20, 1, 3, 4, 7, 6, 12, 23, 1, 3, 4, 7, 6, 12, 8, 28, 1, 3, 4, 7, 6, 12, 8, 15, 31, 1, 3, 4, 7, 6, 12, 8, 15, 13, 30, 1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 40, 1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 42, 1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, 38

Offset: 1

Omar E. Pol, Jul 22 2021

T(n,k) is the total area (or number of cells) of the terraces that are in the k-th level that contains terraces starting from the top of the symmetric tower (a polycube) described in A221529.
The height of the tower equals A000041(n-1).
The terraces of the tower are the symmetric representation of sigma.
The terraces are in the levels that are the partition numbers A000041 starting from the base.
Note that for n >= 2 there are n - 1 terraces because the lower terrace of the tower is formed by two symmetric representations of sigma in the same level.

			Triangle begins:
  1;
  4;
  1, 7;
  1, 3, 11;
  1, 3,  4, 13;
  1, 3,  4,  7, 18;
  1, 3,  4,  7,  6, 20;
  1, 3,  4,  7,  6, 12, 23;
  1, 3,  4,  7,  6, 12,  8, 28;
  1, 3,  4,  7,  6, 12,  8, 15, 31;
  1, 3,  4,  7,  6, 12,  8, 15, 13, 30;
  1, 3,  4,  7,  6, 12,  8, 15, 13, 18, 40;
  1, 3,  4,  7,  6, 12,  8, 15, 13, 18, 12, 42;
  1, 3,  4,  7,  6, 12,  8, 15, 13, 18, 12, 28, 38;
  ...
For n = 7, sigma(7) = 1 + 7 = 8 and sigma(6) = 1 + 2 + 3 + 6 = 12, and 8 + 12 = 20, so the last term of row 7 is T(7,6) = 20. The other terms in row 7 are the first five terms of A000203, so the 7th row of the triangle is [1, 3, 4, 7, 6, 20].
For n = 7 we can see below the top view and the lateral view of the pyramid described in A245092 (with seven levels) and the top view and the lateral view of the tower described in A221529 (with 11 levels).
                                           _
                                          | |
                                          | |
                                          | |
        _                                 |_|_
       |_|_                               |   |
       |_ _|_                             |_ _|_
       |_ _|_|_                           |   | |
       |_ _ _| |_                         |_ _|_|_
       |_ _ _|_ _|_                       |_ _ _| |_
       |_ _ _ _| | |_                     |_ _ _|_ _|_ _
       |_ _ _ _|_|_ _|                    |_ _ _ _|_|_ _|
.
         Figure 1.                           Figure 2.
        Lateral view                       Lateral view
       of the pyramid.                     of the tower.
.
.       _ _ _ _ _ _ _                      _ _ _ _ _ _ _
       |_| | | | | | |                    |_| | | | |   |
       |_ _|_| | | | |                    |_ _|_| | |   |
       |_ _|  _|_| | |                    |_ _|  _|_|   |
       |_ _ _|    _|_|                    |_ _ _|    _ _|
       |_ _ _|  _|                        |_ _ _|  _|
       |_ _ _ _|                          |       |
       |_ _ _ _|                          |_ _ _ _|
.
          Figure 3.                          Figure 4.
          Top view                           Top view
       of the pyramid.                     of the tower.
.
Both polycubes have the same base which has an area equal to A024916(7) = 41 equaling the sum of the 7th row of triangle.
Note that in the top view of the tower the symmetric representation of sigma(6) and the symmetric representation of sigma(7) appear unified in the level 1 of the structure as shown above in the figure 4 (that is due the first two partition numbers A000041 are [1, 1]), so T(7,6) = sigma(7) + sigma(6) = 8 + 12 = 20.
.
Illustration of initial terms:
   Row 1    Row 2      Row 3      Row 4        Row 5          Row 6
.
    1        4         1 7        1 3 11       1 3 4 13       1 3 4 7 18
.   _        _ _       _ _ _      _ _ _ _      _ _ _ _ _      _ _ _ _ _ _
   |_|      |   |     |_|   |    |_| |   |    |_| | |   |    |_| | | |   |
            |_ _|     |    _|    |_ _|   |    |_ _|_|   |    |_ _|_| |   |
                      |_ _|      |      _|    |_ _|  _ _|    |_ _|  _|   |
                                 |_ _ _|      |     |        |_ _ _|    _|
                                              |_ _ _|        |        _|
                                                             |_ _ _ _|
.

Paolo Xausa, Table of n, a(n) for n = 1..11176 (rows 1..150 of the triangle, flattened)
Omar E. Pol, Illustration of the partitions of 10, prism and tower

Mirror of A340584.
The length of row n is A028310(n-1).
Row sums give A024916.
Leading diagonal gives A092403.
Other diagonals give A000203.
Companion of A346562.
Cf. A175254 (volume of the pyramid).
Cf. A066186 (volume of the tower).
Cf. A000041, A221529, A237270, A237593, A245092, A245093 (similar), A336811, A336812, A338156, A339106, A340035.

A346533row[n_]:=If[n==1,{1},Join[DivisorSigma[1,Range[n-2]],{Total[DivisorSigma[1,{n-1,n}]]}]];Array[A346533row,15] (* Paolo Xausa, Oct 23 2023 *)

A101052 Number of preferential arrangements of n labeled elements when only k <= 3 ranks are allowed.

Original entry on oeis.org

1, 1, 3, 13, 51, 181, 603, 1933, 6051, 18661, 57003, 173053, 523251, 1577941, 4750203, 14283373, 42915651, 128878021, 386896203, 1161212893, 3484687251, 10456158901, 31372671003, 94126401613, 282395982051, 847221500581

Offset: 0

Thomas Wieder, Nov 28 2004

nonn

The (labeled) case for k <= 2 is given by A000225. The unlabeled analog for k <= 2 is given by A028310 (A000027). The unlabeled analog for k <= 3 is given by A000124.

Alice and Bob went out for dinner; Alice paid 10 euros for the taxi, Bob paid 20 euros for the dinner; if they have to equally divide the expenses Alice will have to give 5 euros to Bob. With two people, Alice and Bob, there are three possible cases: Alice has to give money to Bob; Bob has to give money to Alice; they paid the same amount, so no debtors nor creditors. With three people, there are 13 cases; with four people, there are 51 cases, and so on. - Alessandro Gentilini (alessandro.gentilini(AT)gmail.com), Aug 10 2006

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Hongyi Cao, Gang Xu, Renshu Gu, Jinlan Xu, Xiaoyu Zhang, Timon Rabczuk, Yuzhe Luo, and Xifeng Gao, Robust and Feature-Preserving Offset Meshing, arXiv:2412.15564 [cs.GR], 2024. See p. 14.
Samuele Giraudo, Combinatorial operads from monoids, arXiv preprint arXiv:1306.6938, 2013
Index entries for linear recurrences with constant coefficients, signature (6,-11,6).

Cf. A000670, A000225, A000124, A028310, A097237.

Cf. A000918.

A101052 := n -> 3^n+2-2*2^n; [ seq(3^n+2-2*2^n,n=0..30) ];

a = Exp[x] - 1;CoefficientList[Series[1+a+a^2+a^3,{x,0,20}],x]*Table[n!,{n,0,20}]
LinearRecurrence[{6,-11,6},{1,1,3},30] (* Harvey P. Dale, Mar 13 2013 *)

E.g.f. = 2*exp(z) - 2*exp(z)^2 + exp(z)^3;
o.g.f. = -(-1+3*z-6*z^2)/(11*z^2+1-6*z-6*z^3).
a(n) = 3^n + 2 - 2*2^n; recurrence: a(n+3) - 6*a(n+2) + 11*a(n+1) - 6*a(n), a(0) = 1, a(1) = 1, a(2) = 3.
G.f.: Sum_{n>=0} a(n)*log(1+x)^n/n! = (1-x^4)/(1-x). - Paul D. Hanna, Feb 18 2012
Binomial transform of A000918 in which the first term is changed from -1 to 1 as: (1, 0, 2, 6, 14, 30, 62, ...). - Gary W. Adamson, Mar 23 2012

A237447 Infinite square array: row 1 is the positive integers 1, 2, 3, ..., and on any subsequent row n, n is moved to the front: n, 1, ..., n-1, n+1, n+2, ...

Original entry on oeis.org

1, 2, 2, 3, 1, 3, 4, 3, 1, 4, 5, 4, 2, 1, 5, 6, 5, 4, 2, 1, 6, 7, 6, 5, 3, 2, 1, 7, 8, 7, 6, 5, 3, 2, 1, 8, 9, 8, 7, 6, 4, 3, 2, 1, 9, 10, 9, 8, 7, 6, 4, 3, 2, 1, 10, 11, 10, 9, 8, 7, 5, 4, 3, 2, 1, 11, 12, 11, 10, 9, 8, 7, 5, 4, 3, 2, 1, 12, 13, 12, 11, 10, 9, 8, 6, 5, 4, 3, 2, 1, 13, 14, 13, 12, 11, 10, 9, 8, 6, 5, 4, 3, 2, 1, 14, 15, 14, 13, 12, 11, 10, 9, 7, 6, 5, 4, 3, 2, 1, 15

Offset: 1

Antti Karttunen, Feb 10 2014

Row n is the lexicographically earliest permutation of positive integers beginning with n. This also holds for the reverse colexicographic order, thus A007489(n-1) gives the position of n-th row of this array (which is one-based) in zero-based arrays A195663 & A055089.
The finite n X n square matrices in sequence A237265 converge towards this infinite square array.
Rows can be constructed also simply as follows: The first row is A000027 (natural numbers, also known as positive integers). For the n-th row, n=2, ..., pick n out from the terms of A000027 and move it to the front. This will create a permutation with one cycle of length n, in cycle notation: (1 n n-1 n-2 ... 3 2), which is the inverse of (1 2 ... n-1 n).
There are A000110(n) ways to choose n permutations from the n first rows of this table so that their composition is identity (counting all the different composition orders). This comment is essentially the same as my May 01 2006 comment on A000110, please see there for more information. - Antti Karttunen, Feb 10 2014
Also, for n > 1, the whole symmetric group S_n can be generated with just two rows, row 2, which is transposition (1 2), and row n, which is the inverse of cycle (1 ... n). See Rotman, p. 24, Exercise 2.9 (iii).

			The top left 9 X 9 corner of this infinite square array:
  1 2 3 4 5 6 7 8 9
  2 1 3 4 5 6 7 8 9
  3 1 2 4 5 6 7 8 9
  4 1 2 3 5 6 7 8 9
  5 1 2 3 4 6 7 8 9
  6 1 2 3 4 5 7 8 9
  7 1 2 3 4 5 6 8 9
  8 1 2 3 4 5 6 7 9
  9 1 2 3 4 5 6 7 8
Note how this is also the 9th finite subsquare of the sequence A237265, which can be picked from its terms A237265(205) .. A237265(285), where 205 = 1+A000330(9-1), the starting offset for that 9th subsquare in A237265.

Joseph J. Rotman, An Introduction to the Theory of Groups, 4th ed., Springer-Verlag, New York, 1995. First chapter, pp. 1-19 [For a general introduction], and from chapter 2, problem 2.9, p. 24.

Transpose: A237448.
Topmost row and the leftmost column: A000027. Second column: A054977. Central diagonal: A028310 (note the different starting offsets).
Antidiagonal sums: A074148.
This array is the infinite limit of the n X n square matrices in A237265.
Cf. also A000330, A002260, A004736, A002024, A010054, A007489, A055089, A195663.

T:= proc(r,c) if c > r then c elif c=1 then r else c-1 fi end proc:
seq(seq(T(r,n-r),r=1..n-1),n=1..20); # Robert Israel, May 09 2017

Table[Function[n, If[k == 1, n, k - Boole[k <= n]]][m - k + 1], {m, 15}, {k, m, 1, -1}] // Flatten (* Michael De Vlieger, May 09 2017 *)

A237447(n,k=0)=if(k, if(k>1, k-(k<=n), n), A237447(A002260(n), A004736(n))) \\ Yields the element [n,k] of the matrix, or the n-th term of the "linearized" sequence if no k is given. - M. F. Hasler, Mar 09 2014

(define (A237447 n) (+ (* (A010054 n) (A002024 n)) (* (- 1 (A010054 n)) (- (A004736 n) (if (>= (A002260 n) (A004736 n)) 1 0)))))
;; Another variant based on Cano's A237265.
(define (A237447 n) (let* ((row (A002260 n)) (col (A004736 n)) (sss (max row col)) (sof (+ 1 (A000330 (- sss 1))))) (A237265 (+ sof (* sss (- row 1)) (- col 1)))))

When col > row, T(row,col) = col, when 1 < col <= row, T(row,col) = col-1, and when col=1, T(row,1) = row.
a(n) = A010054(n) * A002024(n) + (1-A010054(n)) * (A004736(n) - [A002260(n) >= A004736(n)]). [This gives the formula for this entry represented as a one-dimensional sequence. Here the expression inside Iverson brackets results 1 only when the row index (A002260) is greater than or equal to the column index (A004736), otherwise zero. A010054 is the characteristic function for the triangular numbers, A000217.]
T(row,col) = A237265((A000330(max(row,col)-1)+1) + (max(row,col)*(row-1)) + (col-1)). [Takes the infinite limit of n X n matrices of A237265.]
G.f. as array: g(x,y) = (1 - 4*x*y + 3*x*y^2 + x^2*y - x*y^3)*x*y/((1-x*y)*(1-x)^2*(1-y)^2). - Robert Israel, May 09 2017

A245687 Number T(n,k) of endofunctions on [n] such that the minimal cardinality of the nonempty preimages equals k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 2, 2, 0, 24, 0, 3, 0, 216, 36, 0, 4, 0, 2920, 200, 0, 0, 5, 0, 44100, 2250, 300, 0, 0, 6, 0, 799134, 22932, 1470, 0, 0, 0, 7, 0, 16429504, 342608, 3136, 1960, 0, 0, 0, 8, 0, 382625856, 4638384, 147168, 9072, 0, 0, 0, 0, 9, 0, 9918836100, 79610850, 1522800, 18900, 11340, 0, 0, 0, 0, 10

Offset: 0

Alois P. Heinz, Jul 29 2014

T(0,0) = 1 by convention.

			Triangle T(n,k) begins:
  1;
  0,        1;
  0,        2,      2;
  0,       24,      0,    3;
  0,      216,     36,    0,    4;
  0,     2920,    200,    0,    0,  5;
  0,    44100,   2250,  300,    0,  0,  6;
  0,   799134,  22932, 1470,    0,  0,  0,  7;
  0, 16429504, 342608, 3136, 1960,  0,  0,  0,  8;
  ...

Alois P. Heinz, Rows n = 0..100, flattened

T(n,1) = n*A241581(n) for n>0.
Rows sums give A000312.
Main diagonal gives A028310.
T(2n,n) gives A273325.
Cf. A019575 (the same for maximal cardinality).

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1, k) +add(b(n-j, i-1, k)/j!, j=k..n)))
    end:
T:= (n, k)-> `if`(k=0, `if`(n=0, 1, 0), `if`(k=n, n,
    `if`(k>=(n+1)/2, 0, n!*(b(n$2, k)-b(n$2, k+1))))):
seq(seq(T(n, k), k=0..n), n=0..12);

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i<1, 0, b[n, i-1, k] + Sum[b[n-j, i-1, k]/j!, {j, k, n}]]]; T[n_, k_] := If[k == 0, If[n == 0, 1, 0], If[k == n, n, If[k >= (n+1)/2, 0, n!*(b[n, n, k] - b[n, n, k+1])]]]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Feb 02 2015, after Alois P. Heinz *)

A330641 a(n) is the number of partitions of n with Durfee square of size <= 3.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 230, 295, 380, 480, 607, 758, 943, 1161, 1426, 1733, 2100, 2525, 3023, 3595, 4261, 5017, 5888, 6874, 7996, 9258, 10687, 12281, 14073, 16066, 18288, 20747, 23478, 26482, 29801, 33442, 37441, 41811, 46596, 51801, 57478, 63639, 70329, 77567

Offset: 0

Omar E. Pol, Dec 22 2019

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-3,0,6,0,-3,-2,1,2,-1).

Cf. A000041, A006918, A008805, A028310, A115994, A115720, A117485, A330640.

seq(n)={Vec((1 - x - x^2 + 2*x^4 + x^5 - 2*x^6 - x^7 + x^8 + x^9 - x^11 + x^12) / ((1 - x)^6*(1 + x)^2*(1 + x + x^2)^2) + O(x*x^n))} \\ Andrew Howroyd, Dec 27 2024

a(n) = A000041(n), 0 <= n <= 15.
a(n) = A330640(n), 0 <= n <= 8.
a(n) = A330640(n) + A117485(n), n >= 9.
a(n) = n + A006918(n-3) + A117485(n), n >= 9.
From Colin Barker, Dec 31 2019: (Start)
G.f.: (1 - x - x^2 + 2*x^4 + x^5 - 2*x^6 - x^7 + x^8 + x^9 - x^11 + x^12) / ((1 - x)^6*(1 + x)^2*(1 + x + x^2)^2).
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) - 3*a(n-4) + 6*a(n-6) - 3*a(n-8) - 2*a(n-9) + a(n-10) + 2*a(n-11) - a(n-12) for n>12. (End)

cp's OEIS Frontend

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A073273 a(n) = floor(sqrt(prime(n)*prime(n+2))).

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A182938 If n = Product (p_j^e_j) then a(n) = Product (binomial(p_j, e_j)).

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A256116 Number T(n,k) of length 2n k-ary words, either empty or beginning with the first letter of the alphabet and using each letter at least once, that can be built by repeatedly inserting doublets into the initially empty word; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A330640 a(n) is the number of partitions of n with Durfee square of size <= 2.

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A346533 Irregular triangle read by rows in which row n lists the first n - 2 terms of A000203 together with the sum of A000203(n-1) and A000203(n), with a(1) = 1.

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A101052 Number of preferential arrangements of n labeled elements when only k <= 3 ranks are allowed.

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A237447 Infinite square array: row 1 is the positive integers 1, 2, 3, ..., and on any subsequent row n, n is moved to the front: n, 1, ..., n-1, n+1, n+2, ...