A048887 -id:A048887 - cp's OEIS frontend

A126198 Triangle read by rows: T(n,k) (1 <= k <= n) = number of compositions of n into parts of size <= k.

1, 1, 2, 1, 3, 4, 1, 5, 7, 8, 1, 8, 13, 15, 16, 1, 13, 24, 29, 31, 32, 1, 21, 44, 56, 61, 63, 64, 1, 34, 81, 108, 120, 125, 127, 128, 1, 55, 149, 208, 236, 248, 253, 255, 256, 1, 89, 274, 401, 464, 492, 504, 509, 511, 512, 1, 144, 504, 773, 912, 976, 1004, 1016, 1021, 1023, 1024

Offset: 1

N. J. A. Sloane, Mar 09 2007

Also has an interpretation as number of binary vectors of length n-1 in which the length of the longest run of 1's is <= k (see A048004). - N. J. A. Sloane, Apr 03 2011

Higher Order Fibonacci numbers: A126198(n,k) = Sum_{h=0..k} A048004(n,h); for example, A126198(7,3) = Sum_{h=0..3} A048004(7,h) or A126198(7,3) = 1 + 33 + 47 + 27 = 108, the 7th tetranacci number. A048004 row(7) produces A126198 row(7) list of 1,34,81,108,120,125,127,128 which are 1, the 7th Fibonacci, the 7th tribonacci, ... 7th octanacci numbers. - Richard Southern, Aug 04 2017

			Triangle begins:
  1;
  1,  2;
  1,  3,  4;
  1,  5,  7,  8;
  1,  8, 13, 15, 16;
  1, 13, 24, 29, 31, 32;
  1, 21, 44, 56, 61, 63, 64;
Could also be extended to a square array:
  1  1  1  1  1  1  1 ...
  1  2  2  2  2  2  2 ...
  1  3  4  4  4  4  4 ...
  1  5  7  8  8  8  8 ...
  1  8 13 15 16 16 16 ...
  1 13 24 29 31 32 32 ...
  1 21 44 56 61 63 64 ...
which when read by antidiagonals (downwards) gives A048887.

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, pp. 154-155.

Alois P. Heinz, Rows n = 1..141

Rows are partial sums of rows of A048004. Cf. A048887, A092921 for other versions.
2nd column = Fibonacci numbers, next two columns are A000073, A000078; last three diagonals are 2^n, 2^n-1, 2^n-3.
Cf. A082267.

A126198 := proc(n,k) coeftayl( x*(1-x^k)/(1-2*x+x^(k+1)),x=0,n); end: for n from 1 to 11 do for k from 1 to n do printf("%d, ",A126198(n,k)); od; od; # R. J. Mathar, Mar 09 2007
# second Maple program:
T:= proc(n, k) option remember;
      if n=0 or k=1 then 1
    else add(T(n-j, k), j=1..min(n, k))
      fi
    end:
seq(seq(T(n, k), k=1..n), n=1..15);  # Alois P. Heinz, Oct 23 2011

rows = 11; t[n_, k_] := Sum[ (-1)^i*2^(n-i*(k+1))*Binomial[ n-i*k, i], {i, 0, Floor[n/(k+1)]}] - Sum[ (-1)^i*2^((-i)*(k+1)+n-1)*Binomial[ n-i*k-1, i], {i, 0, Floor[(n-1)/(k+1)]}]; Flatten[ Table[ t[n, k], {n, 1, rows}, {k, 1, n}]](* Jean-François Alcover, Nov 17 2011, after Max Alekseyev *)

G.f. for column k: (x-x^(k+1))/(1-2*x+x^(k+1)). [Riordan]
T(n,3) = A008937(n) - A008937(n-3) for n>=3. T(n,4) = A107066(n-1) - A107066(n-5) for n>=5. T(n,5) = A001949(n+4) - A001949(n-1) for n>=5. - R. J. Mathar, Mar 09 2007
T(n,k) = A181695(n,k) - A181695(n-1,k). - Max Alekseyev, Nov 18 2010
Conjecture: Sum_{k=1..n} T(n,k) = A039671(n), n>0. - L. Edson Jeffery, Nov 29 2013

More terms from R. J. Mathar, Mar 09 2007

A125127 Array L(k,n) read by antidiagonals: k-step Lucas numbers.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 3, 4, 1, 1, 3, 7, 7, 1, 1, 3, 7, 11, 11, 1, 1, 3, 7, 15, 21, 18, 1, 1, 3, 7, 15, 26, 39, 29, 1, 1, 3, 7, 15, 31, 51, 71, 47, 1, 1, 3, 7, 15, 31, 57, 99, 131, 76, 1, 1, 3, 7, 15, 31, 63, 113, 191, 241, 123, 1

Offset: 1

Jonathan Vos Post, Nov 21 2006

			Table begins:
1 | 1  1  1   1   1   1    1    1    1    1
2 | 1  3  4   7  11  18   29   47   76  123
3 | 1  3  7  11  21  39   71  131  241  443
4 | 1  3  7  15  26  51   99  191  367  708
5 | 1  3  7  15  31  57  113  223  439  863
6 | 1  3  7  15  31  63  120  239  475  943
7 | 1  3  7  15  31  63  127  247  493  983
8 | 1  3  7  15  31  63  127  255  502 1003
9 | 1  3  7  15  31  63  127  255  511 1013

Freddy Barrera, Table of n, a(n) for n = 1..5050
C. A. Charalambides, Lucas numbers and polynomials of order k and the length of the longest circular success run, The Fibonacci Quarterly, 29 (1991), 290-297.
Eric Weisstein's World of Mathematics, Lucas n-Step Number

n-step Lucas number analog of A092921 Array F(k, n) read by antidiagonals: k-generalized Fibonacci numbers (and see related A048887, A048888). L(1, n) = "1-step Lucas numbers" = A000012. L(2, n) = 2-step Lucas numbers = A000204. L(3, n) = 3-step Lucas numbers = A001644. L(4, n) = 4-step Lucas numbers = A001648 Tetranacci numbers A073817 without the leading term 4. L(5, n) = 5-step Lucas numbers = A074048 Pentanacci numbers with initial conditions a(0)=5, a(1)=1, a(2)=3, a(3)=7, a(4)=15. L(6, n) = 6-step Lucas numbers = A074584 Esanacci ("6-anacci") numbers. L(7, n) = 7-step Lucas numbers = A104621 Heptanacci-Lucas numbers. L(8, n) = 8-step Lucas numbers = A105754. L(9, n) = 9-step Lucas numbers = A105755. See A000295, A125129 for comments on partial sums of diagonals.

Cf. A000012, A000032, A000204, A001644, A001648, A048887, A048888, A074048, A074584, A092921, A104621, A105754, A105755, A125129.

def L(k, n):
    if n < 0:
        return -1
    a = [-1]*(k-1) + [k] # [-1, -1, ..., -1, k]
    for i in range(1, n+1):
        a[:] = a[1:] + [sum(a)]
    return a[-1]
[L(k, n) for d in (1..12) for k, n in zip((d..1, step=-1), (1..d))] # Freddy Barrera, Jan 10 2019

L(k,n) = L(k,n-1) + L(k,n-2) + ... + L(k,n-k); L(k,n) = -1 for n < 0, and L(k,0) = k.

G.f. for row k: x*(dB(k,x)/dx)/(1-B(k,x)), where B(k,x) = x + x^2 + ... + x^k. - Petros Hadjicostas, Jan 24 2019

Corrected by Freddy Barrera, Jan 10 2019

A175331 Array A092921(n,k) without the first two rows, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 5, 4, 2, 1, 1, 8, 7, 4, 2, 1, 1, 13, 13, 8, 4, 2, 1, 1, 21, 24, 15, 8, 4, 2, 1, 1, 34, 44, 29, 16, 8, 4, 2, 1, 1, 55, 81, 56, 31, 16, 8, 4, 2, 1, 1, 89, 149, 108, 61, 32, 16, 8, 4, 2, 1, 1, 144, 274, 208, 120, 63, 32, 16, 8, 4, 2, 1, 1, 233, 504, 401, 236, 125, 64, 32, 16, 8, 4, 2, 1

Offset: 2

Roger L. Bagula, Dec 03 2010

Antidiagonal sums are A048888. This is a transposed version of A048887, so the bivariate generating function is obtained by swapping the two arguments.

Brlek et al. (2006) call this table "number of psp-polyominoes with flat bottom". - N. J. A. Sloane, Oct 30 2018

			The array starts in row n=2 with columns k >= 1 as:
  1   1   1   1   1   1   1   1   1   1
  1   2   2   2   2   2   2   2   2   2
  1   3   4   4   4   4   4   4   4   4
  1   5   7   8   8   8   8   8   8   8
  1   8  13  15  16  16  16  16  16  16
  1  13  24  29  31  32  32  32  32  32
  1  21  44  56  61  63  64  64  64  64
  1  34  81 108 120 125 127 128 128 128
  1  55 149 208 236 248 253 255 256 256

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, pp. 125, 155.

Srecko Brlek, Andrea Frosini, Simone Rinaldi, and Laurent Vuillon, Tilings by translation: enumeration by a rational language approach, The Electronic Journal of Combinatorics, vol.13, (2006). Table 1 is essentially this array. - _N. J. A. Sloane_, Jul 20 2014

Cf. A000045, A048887, A048004, A126198.

A092921 := proc(n,k) if k <= 0 or n <= 0 then 0; elif k = 1 or n = 1 then 1; else add( procname(n-i,k),i=1..k) ; end if; end proc:
A175331 := proc(n,k) A092921(n,k) ; end proc: # R. J. Mathar, Dec 17 2010

f[x_, n_] = (x - x^(m + 1))/(1 - 2*x + x^(m + 1))
a = Table[Table[SeriesCoefficient[
      Series[f[x, m], {x, 0, 10}], n], {n, 0, 10}], {m, 1, 10}];
Table[Table[a[[m, n - m + 1]], {m, 1, n - 1}], {n, 1, 10}];
Flatten[%]

T(n,k) = A092921(n,k), n >= 2.
T(n,2) = A000045(n).
T(n,3) = A000073(n+2).
T(n,4) = A000078(n+2).

A247506 Generalized Fibonacci numbers: square array A(n,k) read by ascending antidiagonals, A(n,k) = [x^k]((1-Sum_{j=1..n} x^j)^(-1)), (n>=0, k>=0).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 2, 3, 1, 0, 1, 1, 2, 4, 5, 1, 0, 1, 1, 2, 4, 7, 8, 1, 0, 1, 1, 2, 4, 8, 13, 13, 1, 0, 1, 1, 2, 4, 8, 15, 24, 21, 1, 0, 1, 1, 2, 4, 8, 16, 29, 44, 34, 1, 0, 1, 1, 2, 4, 8, 16, 31, 56, 81, 55, 1, 0

Offset: 0

Peter Luschny, Nov 02 2014

			[n\k] [0][1][2][3][4] [5] [6] [7]  [8]  [9] [10]  [11]  [12]
   [0] 1, 0, 0, 0, 0,  0,  0,  0,   0,   0,   0,    0,    0
   [1] 1, 1, 1, 1, 1,  1,  1,  1,   1,   1,   1,    1,    1
   [2] 1, 1, 2, 3, 5,  8, 13, 21,  34,  55,  89,  144,  233  [A000045]
   [3] 1, 1, 2, 4, 7, 13, 24, 44,  81, 149, 274,  504,  927  [A000073]
   [4] 1, 1, 2, 4, 8, 15, 29, 56, 108, 208, 401,  773, 1490  [A000078]
   [5] 1, 1, 2, 4, 8, 16, 31, 61, 120, 236, 464,  912, 1793  [A001591]
   [6] 1, 1, 2, 4, 8, 16, 32, 63, 125, 248, 492,  976, 1936  [A001592]
   [7] 1, 1, 2, 4, 8, 16, 32, 64, 127, 253, 504, 1004, 2000  [A066178]
   [8] 1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 509, 1016, 2028  [A079262]
   [.] .  .  .  .  .   .   .   .    .    .    .     .     .
  [oo] 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048  [A011782]
.
As a triangular array, starts:
  1,
  1, 0,
  1, 1, 0,
  1, 1, 1, 0,
  1, 1, 2, 1, 0,
  1, 1, 2, 3, 1, 0,
  1, 1, 2, 4, 5, 1, 0,
  1, 1, 2, 4, 7, 8, 1, 0,
  1, 1, 2, 4, 8, 13, 13, 1, 0,
  1, 1, 2, 4, 8, 15, 24, 21, 1, 0,
  ...

Stefano Spezia, First 140 antidiagonals of the array, flattened
Harold R. Parks and Dean C. Wills, Sum of k-bonacci Numbers, arXiv:2208.01224 [math.CO], 2022. See p. 5.

Cf. A247505, A011782, A000045, A000073, A000078, A001591, A001592, A066178, A079262.

Cf. A048887, A092921, A144406.

A := (n,k) -> coeff(series((1-add(x^j, j=1..n))^(-1),x,k+2),x,k):
seq(print(seq(A(n,k), k=0..12)), n=0..9);

A[n_, k_] := A[n, k] = If[k<0, 0, If[k==0, 1, Sum[A[n, j], {j, k-n, k-1}]]]; Table[A[n-k, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 08 2019 *)

A(n, k) = Sum_{j=0..floor(k/(n+1))} (-1)^j*((k - j*n) + j + delta(k,0))/(2*(k - j*n) + delta(k,0))*binomial(k - j*n, j)*2^(k-j*(n+1)), where delta denotes the Kronecker delta (see Corollary 3.2 in Parks and Wills). - Stefano Spezia, Aug 06 2022

A349802 Triangle read by rows: T(n,k) is the number of binary Lyndon words of length n that begin with exactly k 0's. 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 2, 2, 1, 1, 0, 0, 2, 3, 2, 1, 1, 0, 0, 4, 6, 4, 2, 1, 1, 0, 0, 5, 10, 7, 4, 2, 1, 1, 0, 0, 8, 18, 14, 8, 4, 2, 1, 1, 0, 0, 11, 31, 26, 15, 8, 4, 2, 1, 1, 0, 0, 18, 56, 50, 30, 16, 8, 4, 2, 1, 1, 0

Offset: 0

Peter Kagey, Nov 30 2021

Rows sum to A001037.

Conjecture: The Euler transform of column k=1 gives the Fibonacci numbers, the Euler transform of column k=2 gives the tribonacci numbers (A000073), and more generally, the Euler transform of column k >= 1 gives the (k+1)-bonacci numbers (A048887).

			For n = 6, the values correspond to the following Lyndon words:
T(6,1) = 2 via 010111 and 011111;
T(6,2) = 3 via 001011, 001101, and 001111;
T(6,3) = 2 via 000101 and 000111;
T(6,4) = 1 via 000011; and
T(6,5) = 1 via 000001.
Table begins:
n\k | 0  1   2   3  4  5  6  7  8  9
----+------------------------------------
  0 | 1
  1 | 1, 1
  2 | 0, 1,  0
  3 | 0, 1,  1,  0
  4 | 0, 1,  1,  1, 0
  5 | 0, 2,  2,  1, 1, 0
  6 | 0, 2,  3,  2, 1, 1, 0
  7 | 0, 4,  6,  4, 2, 1, 1, 0
  8 | 0, 5, 10,  7, 4, 2, 1, 1, 0
  9 | 0, 8, 18, 14, 8, 4, 2, 1, 1, 0
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows n=0..50)

Cf. A000045, A000073, A001037, A006206 (k=1), A048887.

B(k,n)=my(g=1/(1 - x*(1-x^k)/(1-x))); Vec(1 + sum(j=1, n, moebius(j)/j * log(subst(g + O(x*x^(n\j)), x, x^j))))
A(n,m)={my(M=Mat(vector(m, k, Col(B(k,n) - B(k-1,n))))); M[1,1]=M[2,2]=1; M}
{ my(M=A(10,10)); for(n=1, #M, print(M[n,1..n])) } \\ Andrew Howroyd, Dec 05 2021

T(n,n) = 0 for n >= 2.
T(n,n-1) = 1 for n >= 1.
T(n,n-m) = 2^(m-2) for n >= 2*m - 1 and m >= 2.

A125129 Partial sums of diagonals of array of k-step Lucas numbers as in A125127, read by antidiagonals.

Original entry on oeis.org

1, 1, 4, 1, 8, 11, 1, 12, 19, 26, 1, 19, 33, 45, 57, 1, 30, 58, 84, 102, 120, 1, 48, 101, 157, 197, 222, 247, 1, 77, 179, 292, 380, 436, 469, 502, 1, 124, 318, 546, 731, 855, 929, 971, 1013, 1, 200, 567, 1026, 1409, 1674, 1838, 1932, 1984, 2036

Offset: 1

Jonathan Vos Post, Nov 23 2006

Array of partial sums of diagonals of L(k,n) begins: 0.|.1...4..11...26...57..120..247..502.1013.2036.
|.1...8..19...45..102..222..469..971.1984.
|.1..12..33...84..197..436..929.1932.
|.1..19..58..157..380..855.1838.
|.1..30.101..292..731.1674.
|.1..48.179..546.1409.
|.1..77.318.1026.
|.1.124.567.
|.1.200.
|.1.

			Row 1 of the derived array is the partial sum of the diagonal above the main diagonal of array of k-step Lucas numbers as in A125127, hence the partial sums of: 1, 7, 11, 26, 57, 120, 247, 502, 103, ... are 1 = 1; 8 = 1 + 7; 19 = 1 + 7 + 11; 45 = 1 + 7 + 11 + 26; and so forth.

Cf. A000012, A000032, A000204, A001644, A001648, A048887, A048888, A074048, A074584, A092921, A104621, A105754, A105755, A125127, A000295.

Row 0 = SUM[i=1..n]L(i,i) = A127128 = partial sum of main diagonal of array of A125127. Row 1 = SUM[i=1..n]L(i,i+1) = partial sum of diagonal above main diagonal of array of A125127. Row 2 = SUM[i=1..n]L(i,i+2) = partial sum of diagonal 2 above main diagonal of array of A125127. .. Row m = SUM[i=1..n]L(i,i+m) = partial sum of diagonal 2 above main diagonal of array of A125127.

A144406 Rectangular array A read by upward antidiagonals: entry A(n,k) in row n and column k gives the number of compositions of k in which no part exceeds n, n>=1, k>=0.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 1, 2, 4, 5, 1, 1, 1, 2, 4, 7, 8, 1, 1, 1, 2, 4, 8, 13, 13, 1, 1, 1, 2, 4, 8, 15, 24, 21, 1, 1, 1, 2, 4, 8, 16, 29, 44, 34, 1, 1, 1, 2, 4, 8, 16, 31, 56, 81, 55, 1, 1, 1, 2, 4, 8, 16, 32, 61, 108, 149, 89, 1, 1, 1, 2, 4, 8, 16, 32, 63, 120, 208, 274, 144, 1

Offset: 1

Roger L. Bagula and Gary W. Adamson, Sep 29 2008

Polynomial expansion as antidiagonal of p(x,n) = (x-1)/(x^n*(-x+(2*x-1)/x^n)). Based on the Pisot general polynomial type q(x,n) = x^n - (x^n-1)/(x-1) (the original name of the sequence).
Row sums are 1, 2, 3, 5, 8, 14, ... (A079500).
Conjecture: Since the array row sequences successively tend to A000079, the absolute values of nonzero differences between two successive row sequences tend to A045623 = {1,2,5,12,28,64,144,320,704,1536,...}, as k -> infinity. - L. Edson Jeffery, Dec 26 2013

			Array A begins:
  {1, 1, 1, 1, 1,  1,  1,  1,   1,   1,   1, ...}
  {1, 1, 2, 3, 5,  8, 13, 21,  34,  55,  89, ...}
  {1, 1, 2, 4, 7, 13, 24, 44,  81, 149, 274, ...}
  {1, 1, 2, 4, 8, 15, 29, 56, 108, 208, 401, ...}
  {1, 1, 2, 4, 8, 16, 31, 61, 120, 236, 464, ...}
  {1, 1, 2, 4, 8, 16, 32, 63, 125, 248, 492, ...}
  {1, 1, 2, 4, 8, 16, 32, 64, 127, 253, 504, ...}
  {1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 509, ...}
  {1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, ...}
  ... - _L. Edson Jeffery_, Dec 26 2013
As a triangle:
  {1},
  {1, 1},
  {1, 1, 1},
  {1, 1, 2, 1},
  {1, 1, 2, 3, 1},
  {1, 1, 2, 4, 5, 1},
  {1, 1, 2, 4, 7, 8, 1},
  {1, 1, 2, 4, 8, 13, 13, 1},
  {1, 1, 2, 4, 8, 15, 24, 21, 1},
  {1, 1, 2, 4, 8, 16, 29, 44, 34, 1},
  {1, 1, 2, 4, 8, 16, 31, 56, 81, 55, 1},
  {1, 1, 2, 4, 8, 16, 32, 61, 108, 149, 89, 1},
  {1, 1, 2, 4, 8, 16, 32, 63, 120, 208, 274, 144, 1},
  {1, 1, 2, 4, 8, 16, 32, 64, 125, 236, 401, 504, 233, 1},
  {1, 1, 2, 4, 8, 16, 32, 64, 127, 248, 464, 773, 927, 377, 1}

Cf. A000079, A045623, A092921, A175331.
Same as A048887 but with a column of 1's added on the left (the number of compositions of 0 is defined to be equal to 1).
Array rows (with appropriate offsets) are A000012, A000045, A000073, A000078, A001591, A001592, etc.

g[x_, n_] = x^(n) - (x^n - 1)/(x - 1);
h[x_, n_] = FullSimplify[ExpandAll[x^(n)*g[1/x, n]]];
f[t_, n_] := 1/h[t, n];
a = Table[CoefficientList[Series[f[t, m], {t, 0, 30}], t], {m, 1, 31}];
b = Table[Table[a[[n - m + 1]][[m]], {m, 1, n }], {n, 1, 15}];
Flatten[b] (* Triangle version *)
Grid[Table[CoefficientList[Series[(1 - x)/(1 - 2 x + x^(n + 1)), {x, 0, 10}], x], {n, 1, 10}]] (* Array version - L. Edson Jeffery, Jul 18 2014 *)

t(n,m) = antidiagonal_expansion of p(x,n) where p(x,n) = (x-1)/(x^n*(-x+(2*x-1)/x^n)).

G.f. for array A: (1-x)/(1 - 2*x + x^(n+1)), n>=1. - L. Edson Jeffery, Dec 26 2013

Definition changed by L. Edson Jeffery, Jul 18 2014

A063686 Triangular array: T(n,k) is the number of binary necklaces (no turning over) of length n whose longest run of 1's has length k. Table begins at n=0, k=0.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 4, 4, 2, 1, 1, 1, 1, 4, 6, 4, 2, 1, 1, 1, 1, 7, 11, 8, 4, 2, 1, 1, 1, 1, 9, 19, 14, 8, 4, 2, 1, 1, 1, 1, 14, 33, 27, 16, 8, 4, 2, 1, 1, 1, 1, 18, 56, 50, 30, 16, 8, 4, 2, 1, 1, 1, 1, 30, 101, 96, 59, 32, 16, 8, 4, 2, 1, 1, 1

Offset: 0

Christopher Lenard (c.lenard(AT)bendigo.latrobe.edu.au), Aug 22 2001

Column k=1 appears to be A032190(n), n=2,3,...

			Triangle begins:
  1;
  1, 1;
  1, 1, 1;
  1, 1, 1, 1;
  1, 2, 1, 1, 1;
  1, 2, 2, 1, 1, 1;
  1, 4, 4, 2, 1, 1, 1;
  1, 4, 6, 4, 2, 1, 1, 1;
  1, 7, 11, 8, 4, 2, 1, 1, 1;
  1, 9, 19, 14, 8, 4, 2, 1, 1, 1;
  1, 14, 33, 27, 16, 8, 4, 2, 1, 1, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1274 (Rows n=0..49)

Cf. A032190, A048004, A048887.

Cf. A000358, A093305, A280218 (necklaces avoiding 00, 000, 0000).

\\ here R(n) is A048887 transposed
R(n)={Mat(vector(n, k, Col((1-x)/(1-2*x+x^(k+1)) - 1 + O(x*x^n))))}
S(M)={matrix(#M-1, #M-1, n, k, if(kAndrew Howroyd, Oct 15 2017

T(0,0)=1 from Andrew Howroyd, Oct 15 2017

cp's OEIS Frontend

A126198 Triangle read by rows: T(n,k) (1 <= k <= n) = number of compositions of n into parts of size <= k.

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A125127 Array L(k,n) read by antidiagonals: k-step Lucas numbers.

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Sage

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A175331 Array A092921(n,k) without the first two rows, read by antidiagonals.

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A247506 Generalized Fibonacci numbers: square array A(n,k) read by ascending antidiagonals, A(n,k) = [x^k]((1-Sum_{j=1..n} x^j)^(-1)), (n>=0, k>=0).

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A349802 Triangle read by rows: T(n,k) is the number of binary Lyndon words of length n that begin with exactly k 0's. 0 <= k <= n.

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A125129 Partial sums of diagonals of array of k-step Lucas numbers as in A125127, read by antidiagonals.

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A144406 Rectangular array A read by upward antidiagonals: entry A(n,k) in row n and column k gives the number of compositions of k in which no part exceeds n, n>=1, k>=0.

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