A014616 -id:A014616 - cp's OEIS frontend

A196416 Table read by antidiagonals: V(n,m) = solution to postage stamp problem with n stamps in set, m stamps on letter.

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 5, 4, 1, 1, 5, 8, 9, 5, 1, 1, 6, 11, 16, 13, 6, 1, 1, 7, 15, 27, 25, 17, 7, 1, 1, 8, 19, 36, 45, 37, 21, 8, 1, 1, 9, 24, 53, 72, 71, 53, 27, 9, 1, 1, 10, 29, 70, 115, 127, 109, 71, 33, 10, 1

Offset: 0

N. J. A. Sloane, Oct 01 2011

Given n, m, the postage stamp problem is to choose a set of n nonnegative integers such that the sums of m or fewer of these integers can realize the numbers 1, 2, ..., N-1, where N is as large as possible. V(n,m) denotes the value of N.

			Array begins:
m\n 0 1 2 3 4 5 6 ...
---------------------
0...1 1 1 1 1 1 1 ...
1...1 2 3 4 5 6 7  ...
2...1 3 5 9 13 17 21  ...
3...1 4 8 16 25 37 53 ...
4...1 5 11 27 45 71 109  ...
5...1 6 15 36 72 127 212  ...
6...1 7 19 53 115 217 389  ...
...

W. F. Lunnon, A postage stamp problem, Comput. J. 12 (1969) 377-380.

Rows and columns include: A014616/A024206, A195618/A001208, A196069/A001209, A001210, A001211, A196094/A001212, A001213, A001214, A001215, A001216.

A163255 An interspersion: the order array of A163254.

Original entry on oeis.org

1, 3, 2, 7, 5, 4, 13, 10, 8, 6, 21, 17, 14, 11, 9, 31, 26, 22, 18, 15, 12, 43, 37, 32, 27, 23, 19, 16, 57, 50, 44, 38, 33, 28, 24, 20, 73, 65, 58, 51, 45, 39, 34, 29, 25, 91, 82, 74, 66, 59, 52, 46, 40, 35, 30, 111, 101, 92, 83, 75, 67, 60, 53, 47, 41, 36

Offset: 1

Clark Kimberling, Jul 24 2009

A permutation of the natural numbers.
Except for initial terms, rows 1 to 4 are A002061, A002522, A014206, A059100 and columns 1 to 4 are A002620, A024206, A014616, A004116.
This is the interspersion of the fractal sequence A167430; i.e., row n of this array consists of the numbers k such that n=A167430(k). - Clark Kimberling, Nov 03 2009

			Corner:
1....3....7...13
2....5...10...17
4....8...14...22
To obtain A163255 from A163254, replace each term of A163254 by its rank when all the terms of A163254 are arranged in increasing order.

Clark Kimberling, Doubly interspersed sequences, double interspersions and fractal sequences, The Fibonacci Quarterly 48 (2010) 13-20.

Cf. A002061, A002522, A014206, A059100, A002620, A024206, A014616, A004116, A163253, A163254.

Cf. A167430. [From Clark Kimberling, Nov 03 2009]

A327117 Number T(n,k) of colored integer partitions of n using all colors of a k-set such that a color pattern for part i has i distinct colors in increasing order; 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, 4, 5, 0, 1, 7, 18, 15, 0, 1, 10, 45, 84, 52, 0, 1, 14, 94, 298, 415, 203, 0, 1, 18, 174, 844, 1995, 2178, 877, 0, 1, 23, 300, 2081, 7440, 13638, 12131, 4140, 0, 1, 28, 486, 4652, 23670, 64898, 95823, 71536, 21147, 0, 1, 34, 756, 9682, 67390, 259599, 566447, 694676, 445356, 115975

Offset: 0

Alois P. Heinz, Sep 13 2019

The sequence of column k satisfies a linear recurrence with constant coefficients of order k*2^(k-1) = A001787(k).

			T(3,2) = 4: 2ab1a, 2ab1b, 1a1a1b, 1a1b1b.
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 1,  2;
  0, 1,  4,   5;
  0, 1,  7,  18,   15;
  0, 1, 10,  45,   84,    52;
  0, 1, 14,  94,  298,   415,    203;
  0, 1, 18, 174,  844,  1995,   2178,    877;
  0, 1, 23, 300, 2081,  7440,  13638,  12131,   4140;
  0, 1, 28, 486, 4652, 23670,  64898,  95823,  71536,  21147;
  0, 1, 34, 756, 9682, 67390, 259599, 566447, 694676, 445356, 115975;
  ...

Alois P. Heinz, Rows n = 0..140, flattened
Wikipedia, Partition (number theory)

Columns k=0-3 give: A000007, A057427, A014616(n-1) for n>1, A327842.
Main diagonal gives A000110.
Row sums give A116540.
T(2n,n) gives A327843.
Cf. A001787, A255903, A326914, A326962, A327116, A327118.

C:= binomial:
b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      b(n-i*j, min(n-i*j, i-1), k)*C(C(k, i)+j-1, j), j=0..n/i)))
    end:
T:= (n, k)-> add(b(n$2, i)*(-1)^(k-i)*C(k, i), i=0..k):
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, Sum[b[n - i j, Min[n - i j, i - 1], k] Binomial[Binomial[k, i] + j - 1, j], {j, 0, n/i}]]];
T[n_, k_] := Sum[b[n, n, i] (-1)^(k - i) Binomial[k, i], {i, 0, k}];
Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Oct 04 2019, after Alois P. Heinz *)

Sum_{k=1..n} k * T(n,k) = A327118(n).

A112970 A generalized Stern sequence.

Original entry on oeis.org

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

Offset: 0

Paul Barry, Oct 07 2005

Conjectures: a(2^n)=a(2^(n+1)+1)=A033638(n); a(2^n-1)=a(3*2^n-1)=1.

The Gi1 and Gi2 triangle sums, see A180662 for their definitions, of Sierpinski's triangle A047999 equal this sequence. The Gi1 and Gi2 sums can also be interpreted as (i + 4*j = n) and (4*i + j = n) sums, see the Northshield reference. Some A112970(2^n-p) sequences, 0<=p<=32, lead to known sequences, see the crossrefs. - Johannes W. Meijer, Jun 05 2011

Sam Northshield, Sums across Pascal's triangle modulo 2, Congressus Numerantium, 200, pp. 35-52, 2010. - _Johannes W. Meijer_, Jun 05 2011

Cf. A002487, A112971.
Cf. A120562 (Northshield).
Cf. A033638 (p=0), A000012 (p=1), A004526 (p=2, p=3, p=5, p=9, p=17), A002620 (p=4, p=7, p=13, p=25), A000027 (p=6, p=11, p=21), A004116 (p=8, p=15, p=29), A035106 (p=10, p=19), A024206 (p=14, p=27), A007494 (p=18), A014616 (p=22), A179207 (p=26). - Johannes W. Meijer, Jun 05 2011

A112970:=proc(n) option remember; if n <0 then A112970(n):=0 fi: if (n=0 or n=1) then 1 elif n mod 2 = 0 then A112970(n/2) + A112970((n/2)-2) else A112970((n-1)/2); fi; end: seq(A112970(n),n=0..99); # Johannes W. Meijer, Jun 05 2011

a[n_] := a[n] = Which[n<0, 0, n==0 || n==1, 1, Mod[n, 2]==0, a[n/2] + a[n/2-2], True, a[(n-1)/2]];
Table[a[n], {n, 0, 99}] (* Jean-François Alcover, Aug 02 2022 *)

a(n) = Sum_{k=0..n} mod(sum{j=0..n, (-1)^(n-k)*C(j, n-j)*C(k, j-k)}, 2).
From Johannes W. Meijer, Jun 05 2011: (Start)
a(2*n+1) = a(n) and a(2*n) = a(n) + a(n-2) with a(0) = 1, a(1) = 1 and a(n)=0 for n<=-1.
G.f.: Product_{n>=0} (1 + x^(2^n) + x^(4*2^n)). (End)
G.f. A(x) satisfies: A(x) = (1 + x + x^4) * A(x^2). - Ilya Gutkovskiy, Jul 09 2019

A261242 Irregular triangle T(n, k) of number of connected bisymmetric n X n matrices B_n with 0 or 1 entries, B_n[1,1] = 1 = B_n[1,n], and k islands of 0's.

Original entry on oeis.org

1, 1, 2, 1, 4, 1, 4, 12, 18, 12, 8, 6, 2, 44, 56, 120, 28, 88, 4, 36, 0, 8

Offset: 1

Wolfdieter Lang and Kival Ngaokrajang, Aug 18 2015

The row length sequence is 1 for n = 1 and A000982(n-2) + 1 for n >= 2, that is:  1, 1, 2, 3, 6, 9, 14, 19, 26, 33, 42, ... = A261243.
This entry is motivated by A258643.
For bisymmetric matrices see the Wikipedia link.
For the number of independent entries of an n X n bisymmetric matrix B_n see a Jul 07 2015 comment on A002620(n+1), n >= 1. For the binary case (only 0 and 1 entries) see A060656(n+1), and the Dennis P. Walsh comment and link. If B_n[1,1] and B_n[1,n] is given then the four corners are fixed, and, for n >= 3, there are A002620(n+1) - 2 = A014616(n-2) entries free.
If the n X n bisymmetric matrix B_n of 0's and 1's with B_n[1, 1] = 1 = B_n[1, n] is considered as a grid of n^2 squares of length 1 (in some length unit) with the four corners filled with 1's and the other squares with 0 or 1 then a path between the centers of squares with step length 1 can be defined. No diagonal steps (length sqrt(2)) are allowed. B_n is called connected if there exists no path of 0's which dissects the grid into two parts.
An island of 0's (a 0-island) in B_n is defined as a set of 0's for which each pair is connected by a path of 0's, and a 0 entry at the coast of a 0-island has at least one entry 1 one step away. A single square filled with a 0 is a 0-island if all four neighbors 1 step (of length 1) apart are filled with 1's. If k=0 there exists no such 0-island. See the n=4 examples with k >=1 below. The k = 1 matrix has one simply connected 0-island of four squares. The four k = 2 matrices have two 0-islands consisting of one square each.
See the link with the figures by K. N. where red squares stand for 1 and empty squares for 0. Each matrix appears there rotated by 45 degrees in the counterclockwise direction. The mirror operation means row reversion in the matrix B_n. In the figures this is a mirror operation w.r.t. the middle NW-SE diagonal. 0-islands appear in the figures as holes.
For the row sums see A261244.

			The irregular triangle T(n, k) begins:
n\k   0   1    2   3   4  5   6   7   8  ...
1:    1
2:    1
3:    2   1
4:    4   1    4
5:   12  18   12   8   6  2
6:   44  56  120  28  88  4  36   0   8
...
n=4: k=0:
[[1,1,1,1], [1,1,1,1], [1,1,1,1], [1,1,1,1]],
[[1,0,0,1], [0,1,1,0], [0,1,1,0], [1,0,0,1]],
[[1,1,0,1], [1,1,1,0], [0,1,1,1], [1,0,1,1]],
[[1,0,1,1], [0,1,1,1], [1,1,1,0], [1,1,0,1]];
     k=1:
[[1,1,1,1], [1,0,0,1], [1,0,0,1], [1,1,1,1]];
     k=2:
[[1,1,1,1], [1,0,1,1], [1,1,0,1], [1,1,1,1]],
[[1,1,1,1], [1,1,0,1], [1,0,1,1], [1,1,1,1]],
[[1,1,0,1], [1,0,1,0], [0,1,0,1], [1,0,1,1]],
[[1,0,1,1], [0,1,0,1], [1,0,1,0], [1,1,0,1]].

Kival Ngaokrajang, Illustration of T(n,k) for n = 1..5, k >= 0, T(6,0), T(6,1), T(6,2), T(6,4), T(6,k) for k = 3, 5, 6, 8
Wikipedia, Bisymmetric Matrix.

Cf. A000982, A002620, A014616, A258643, A261243, A261244.

A360862 Triangle read by rows: T(n,k) is the number of unlabeled connected multigraphs with n edges on k nodes and degree >= 3 at each node, loops allowed, n >= 2, 1 <= k <= floor(2*n/3).

Original entry on oeis.org

1, 1, 2, 1, 4, 1, 7, 5, 1, 10, 20, 5, 1, 14, 48, 36, 1, 18, 99, 153, 30, 1, 23, 181, 481, 277, 17, 1, 28, 303, 1239, 1451, 323, 1, 34, 479, 2811, 5572, 2946, 193, 1, 40, 726, 5805, 17607, 17343, 3806, 71, 1, 47, 1055, 11148, 48401, 77708, 36872, 3188, 1, 54, 1492, 20219, 120018, 288476, 243007, 54386, 1496

Offset: 2

Andrew Howroyd, Feb 24 2023

Terms may be computed using the tools geng, vcolg and multig in nauty with some additional processing to check the degrees of nodes.

			Triangle begins:
  1;
  1,  2;
  1,  4;
  1,  7,    5;
  1, 10,   20,     5;
  1, 14,   48,    36;
  1, 18,   99,   153,     30;
  1, 23,  181,   481,    277,     17;
  1, 28,  303,  1239,   1451,    323;
  1, 34,  479,  2811,   5572,   2946,    193;
  1, 40,  726,  5805,  17607,  17343,   3806,    71;
  1, 47, 1055, 11148,  48401,  77708,  36872,  3188;
  1, 54, 1492, 20219, 120018, 288476, 243007, 54386, 1496;
  ...

Brendan McKay and Adolfo Piperno, nauty and Traces

Column 2 is A014616.
Row sums are A360863.
Diagonal sums are A360864.
Cf. A322115, A327615, A360866 (loopless).

A360870 Triangle read by rows: T(n,k) is the number of unlabeled connected multigraphs with n edges on k nodes, no cut-points and degree >= 3 at each node, loops allowed, n >= 2, 1 <= k <= floor(2*n/3).

Original entry on oeis.org

1, 1, 2, 1, 4, 1, 7, 2, 1, 10, 8, 2, 1, 14, 19, 11, 1, 18, 40, 48, 7, 1, 23, 77, 154, 70, 5, 1, 28, 132, 421, 392, 71, 1, 34, 217, 1008, 1638, 690, 35, 1, 40, 340, 2210, 5623, 4548, 767, 16, 1, 47, 510, 4477, 16745, 22657, 8594, 566, 1, 54, 742, 8557, 44698, 92844, 64716, 11247, 226

Offset: 2

Andrew Howroyd, Feb 25 2023

Columns k >= 3 correspond to the 2-connected graphs.

Terms may be computed using the tools geng, vcolg and multig in nauty with some additional processing to check the degrees of nodes.

			Triangle begins:
  1;
  1,  2;
  1,  4;
  1,  7,   2;
  1, 10,   8,    2;
  1, 14,  19,   11;
  1, 18,  40,   48,     7;
  1, 23,  77,  154,    70,     5;
  1, 28, 132,  421,   392,    71;
  1, 34, 217, 1008,  1638,   690,    35;
  1, 40, 340, 2210,  5623,  4548,   767,    16;
  1, 47, 510, 4477, 16745, 22657,  8594,   566;
  1, 54, 742, 8557, 44698, 92844, 64716, 11247, 226;
  ...

Brendan McKay and Adolfo Piperno, nauty and Traces.

Column 2 is A014616.
Row sums are A360882.
Row sums except first column are A360871.
Cf. A046752, A322115, A360862, A360866.

A055130 Triangle T(n,k) of numbers of k-covers of an unlabeled n-set, k=1..2^n-1.

Original entry on oeis.org

1, 1, 2, 1, 1, 4, 9, 10, 6, 3, 1, 1, 7, 29, 87, 181, 287, 364, 365, 290, 187, 97, 39, 13, 4, 1, 1, 10, 72, 417, 1973, 7745, 25830, 74017, 183420, 395311, 744495, 1229807, 1787135, 2289925, 2591162, 2591163, 2289929, 1787148, 1229846, 744592, 395498

Offset: 1

Vladeta Jovovic, Jun 14 2000

			Triangle begins:
[1] 1;
[2] 1, 2,  1;
[3] 1, 4,  9, 10,   6,   3,   1;
[4] 1, 7, 29, 87, 181, 287, 364, 365, 290, 187, 97, 39, 13, 4, 1;
  ...
There are 9 3-covers of an unlabeled 3-set: {{1,2},{2,3},{1,2,3}}, {{1,2},{2,3},{1,3}}, {{1,2},{3},{1,2,3}}, {{1},{1,2},{1,2,3}}, {{1,2},{2,3},{3}}, {{1,2},{2},{2,3}}, {{1},{2},{1,2,3}}, {{1},{2},{1,3}} and {{1},{2},{3}}.

Andrew Howroyd, Table of n, a(n) for n = 1..2036 (rows 1..10)

Row sums give A055621.
Columns k=1..3 are A000012, A014616(n-1), A055195.
Cf. A052265, A368186.

\\ G(n,m) defined in A368186.
row(n)={my(m=2^n-1); Vec(G(n,m) - G(n-1,m))} \\ Andrew Howroyd, Jan 03 2024

T(n,n) = A368186(n). - Andrew Howroyd, Jan 03 2024

A339334 Triangle read by rows, 1 <= k <= n: T(n,k) is the sum of the minimal number of coins needed for amounts 1..n with an optimal greedy k-coin system of denominations.

Original entry on oeis.org

1, 3, 2, 6, 4, 3, 10, 6, 5, 4, 15, 9, 7, 6, 5, 21, 11, 9, 8, 7, 6, 28, 14, 11, 10, 9, 8, 7, 36, 18, 14, 12, 11, 10, 9, 8, 45, 21, 17, 14, 13, 12, 11, 10, 9, 55, 25, 19, 16, 15, 14, 13, 12, 11, 10, 66, 30, 22, 19, 17, 16, 15, 14, 13, 12, 11

Offset: 1

Pontus von Brömssen, Nov 30 2020

An optimal greedy k-coin system of denominations for amounts 1..n is a set of k coin denominations such that the sum of the number of coins needed for each of the amounts 1, ..., n is as small as possible when the coins are chosen greedily, i.e., the largest coin value less than or equal to the remaining amount is always chosen.

			Triangle begins:
  n\k|  1  2  3  4  5  6  7  8  9 10 11 12
  ---|-------------------------------------
   1 |  1
   2 |  3  2
   3 |  6  4  3
   4 | 10  6  5  4
   5 | 15  9  7  6  5
   6 | 21 11  9  8  7  6
   7 | 28 14 11 10  9  8  7
   8 | 36 18 14 12 11 10  9  8
   9 | 45 21 17 14 13 12 11 10  9
  10 | 55 25 19 16 15 14 13 12 11 10
  11 | 66 30 22 19 17 16 15 14 13 12 11
  12 | 78 33 25 22 19 18 17 16 15 14 13 12
For n = 8, one of the optimal greedy 3-coin systems is (1,2,4), with the representations
  1 = 1
  2 = 2
  3 = 2 + 1
  4 = 4
  5 = 4 + 1
  6 = 4 + 2
  7 = 4 + 2 + 1
  8 = 4 + 4
with a total of 14 = T(8,3) terms.
Shallit (2003) shows that T(99,k) is 4950, 900, 526, 410, 346, 313, 286 for k = 1..7.

Pontus von Brömssen, Rows n = 1..40, flattened
Jeffrey Shallit, What this country needs is an 18¢ piece, The Mathematical Intelligencer 25 (2003), issue 2, 20-23.
Jeffrey Shallit, What this country needs is an 18¢ piece.
Wikipedia, Change-making problem.

Cf. A014616, A339333.

T(n,k) = A339333(n,k) for all k when 1 <= n <= 7 or n = 10.
T(n,k) = A339333(n,k) for all n when k = 1 or k = 2.
T(n,k) >= A339333(n,k).
T(n,k) >= 2n - k, with equality if and only if n <= A014616(k).

A184674 a(n) = n+floor((n/2-1/(2*n))^2); complement of A184675.

Original entry on oeis.org

1, 2, 4, 7, 10, 14, 18, 23, 28, 34, 40, 47, 54, 62, 70, 79, 88, 98, 108, 119, 130, 142, 154, 167, 180, 194, 208, 223, 238, 254, 270, 287, 304, 322, 340, 359, 378, 398, 418, 439, 460, 482, 504, 527, 550, 574, 598, 623, 648, 674, 700, 727, 754, 782, 810, 839, 868, 898, 928, 959, 990, 1022, 1054, 1087, 1120, 1154, 1188, 1223, 1258, 1294, 1330, 1367, 1404

Offset: 1

Clark Kimberling, Jan 19 2011

Conjecture: a(n) = A014616(n-1). - R. J. Mathar, Jan 29 2011

The above conjecture is true. - Stefano Spezia, Apr 04 2023

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A184675, A184676, A014616.

[n+Floor((n/2-1/(2*n))^2): n in [1..80]]; // Vincenzo Librandi, Jul 10 2011

A184674:=n->n+floor((n/2-1/(2*n))^2): seq(A184674(n), n=1..100); # Wesley Ivan Hurt, Feb 22 2017

a[n_]:=n+Floor[(n/2-1/(2n))^2];
b[n_]:=n+Floor[n^(1/2)+(n+1)^(1/2)];
Table[a[n],{n,1,120}]   (* A184674 *)
Table[b[n],{n,1,120}]   (* A184675 *)
FindLinearRecurrence[Table[a[n],{n,1,120}]]
Join[{1},LinearRecurrence[{2,0,-2,1},{2,4,7,10},72]] (* Ray Chandler, Aug 02 2015 *)

a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4) for n>=6.
G.f.: x*(x^4 - x^3 - 1)/((x + 1)*(x - 1)^3). - Álvar Ibeas, Jul 20 2021
a(n) = (2*n^2 + 8*n - 9 + (-1)^n)/8 for n > 1. - Stefano Spezia, Apr 04 2023

cp's OEIS Frontend

A196416 Table read by antidiagonals: V(n,m) = solution to postage stamp problem with n stamps in set, m stamps on letter.

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A163255 An interspersion: the order array of A163254.

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A327117 Number T(n,k) of colored integer partitions of n using all colors of a k-set such that a color pattern for part i has i distinct colors in increasing order; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A112970 A generalized Stern sequence.

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A261242 Irregular triangle T(n, k) of number of connected bisymmetric n X n matrices B_n with 0 or 1 entries, B_n[1,1] = 1 = B_n[1,n], and k islands of 0's.

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A360862 Triangle read by rows: T(n,k) is the number of unlabeled connected multigraphs with n edges on k nodes and degree >= 3 at each node, loops allowed, n >= 2, 1 <= k <= floor(2*n/3).

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A360870 Triangle read by rows: T(n,k) is the number of unlabeled connected multigraphs with n edges on k nodes, no cut-points and degree >= 3 at each node, loops allowed, n >= 2, 1 <= k <= floor(2*n/3).

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A055130 Triangle T(n,k) of numbers of k-covers of an unlabeled n-set, k=1..2^n-1.

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A339334 Triangle read by rows, 1 <= k <= n: T(n,k) is the sum of the minimal number of coins needed for amounts 1..n with an optimal greedy k-coin system of denominations.

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