A357368 -id:A357368 - cp's OEIS frontend

A073266 Triangle read by rows: T(n,k) is the number of compositions of n as the sum of k integral powers of 2.

1, 1, 1, 0, 2, 1, 1, 1, 3, 1, 0, 2, 3, 4, 1, 0, 2, 4, 6, 5, 1, 0, 0, 6, 8, 10, 6, 1, 1, 1, 3, 13, 15, 15, 7, 1, 0, 2, 3, 12, 25, 26, 21, 8, 1, 0, 2, 6, 10, 31, 45, 42, 28, 9, 1, 0, 0, 6, 16, 30, 66, 77, 64, 36, 10, 1, 0, 2, 4, 18, 40, 76, 126, 126, 93, 45, 11, 1, 0, 0, 6, 16, 50, 96, 168, 224, 198, 130, 55, 12, 1

Offset: 1

Antti Karttunen, Jun 25 2002

Upper triangular region of the table A073265 read by rows. - Emeric Deutsch, Feb 04 2005

Also the convolution triangle of A209229. - Peter Luschny, Oct 07 2022

			T(6,3) = 4 because there are four ordered partitions of 6 into 3 powers of 2, namely: 4+1+1, 1+4+1, 1+1+4 and 2+2+2.
Triangle begins:
  1;
  1, 1;
  0, 2, 1;
  1, 1, 3, 1;
  0, 2, 3, 4, 1;
  0, 2, 4, 6, 5, 1;

G. C. Greubel, Rows n = 1..100 of triangle, flattened
S. Lehr, J. Shallit and J. Tromp, On the vector space of the automatic reals, Theoret. Comput. Sci. 163 (1996), no. 1-2, 193-210.

Cf. A048298, A073265, A023359 (row sums), A089052 (partitions of n).

T(2n,n) gives A333047.

b:= proc(n) option remember; expand(`if`(n=0, 1,
       add(b(n-2^j)*x, j=0..ilog2(n))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n)):
seq(T(n), n=1..14);  # Alois P. Heinz, Mar 06 2020
# Uses function PMatrix from A357368. Adds a row above and a column to the left.
PMatrix(10, n -> if n = 2^ilog2(n) then 1 else 0 fi); # Peter Luschny, Oct 07 2022

m:= 10; T[n_, k_]:= T[n, k]= Coefficient[(Sum[x^(2^j), {j,0,m+1}])^k, x, n]; Table[T[n, k], {n,10}, {k,n}]//Flatten (* G. C. Greubel, Mar 06 2020 *)

T(n, k) = coefficient of x^n in the formal power series (x + x^2 + x^4 + x^8 + x^16 + ...)^k. - Emeric Deutsch, Feb 04 2005
T(0, k) = T(n, 0) = 0, T(n, k) = 0 if k > n, T(n, 1) = 1 if n = 2^m, 0 otherwise and in other cases T(n, k) = Sum_{i=0..floor(log_2(n-1))} T(n-(2^i), k-1). - Emeric Deutsch, Feb 04 2005
Sum_{k=0..n} T(n,k) = A023359(n). - Philippe Deléham, Nov 04 2006

A085771 Triangle read by rows. T(n, k) = A059438(n, k) for 1 <= k <= n, and T(n, 0) = n^0.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 3, 2, 1, 0, 13, 7, 3, 1, 0, 71, 32, 12, 4, 1, 0, 461, 177, 58, 18, 5, 1, 0, 3447, 1142, 327, 92, 25, 6, 1, 0, 29093, 8411, 2109, 531, 135, 33, 7, 1, 0, 273343, 69692, 15366, 3440, 800, 188, 42, 8, 1, 0, 2829325, 642581, 125316, 24892, 5226, 1146, 252, 52, 9, 1

Offset: 0

Philippe Deléham, Jul 22 2003

The convolution triangle of A003319, the number of irreducible permutations. - Peter Luschny, Oct 09 2022

			Triangle starts:
[0] [1]
[1] [0,      1]
[2] [0,      1,     1]
[3] [0,      3,     2,     1]
[4] [0,     13,     7,     3,    1]
[5] [0,     71,    32,    12,    4,   1]
[6] [0,    461,   177,    58,   18,   5,   1]
[7] [0,   3447,  1142,   327,   92,  25,   6,  1]
[8] [0,  29093,  8411,  2109,  531, 135,  33,  7, 1]
[9] [0, 273343, 69692, 15366, 3440, 800, 188, 42, 8, 1]

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 262 (#14).

FindStat - Combinatorial Statistic Finder, The decomposition number of a permutation.

T(2*n, n) = A308650(n).
Variants: A059439, A263484 (row reversed).
Cf. A003319, A059440, A055998, A059438, A000007, A084938.

# Uses function PMatrix from A357368.
PMatrix(10, A003319); # Peter Luschny, Oct 09 2022

# Using function delehamdelta from A084938.
def A085771_triangle(n) :
    a = [0, 1] + [(i + 3) // 2 for i in range(1, n-1)]
    b = [0^i for i in range(n)]
    return delehamdelta(a, b)
A085771_triangle(9) # Peter Luschny, Sep 10 2022

Let f(x) = Sum_{n>=0} n!*x^n, g(x) = 1 - 1/f(x). Then g(x) is the g.f. of the second column, A003319.
Triangle T(n, k) read by rows, given by [0, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, ...] DELTA A000007, where DELTA is Deléham's operator defined in A084938.
G.f.: 1/(1 - xy/(1 - x/(1 - 2x/(1 - 2x/(1 - 3x/(1 - 3x/(1 - 4x/(1-.... (continued fraction). - Paul Barry, Jan 29 2009

A205574 Triangle T(n,k), 0<=k<=n, given by (0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 5, 5, 3, 1, 0, 15, 14, 9, 4, 1, 0, 52, 44, 28, 14, 5, 1, 0, 203, 154, 93, 48, 20, 6, 1, 0, 877, 595, 333, 169, 75, 27, 7, 1, 0, 4140, 2518, 1289, 624, 280, 110, 35, 8, 1, 0, 21147, 11591, 5394, 2442, 1071, 435, 154, 44, 9, 1

Offset: 0

Philippe Deléham, Jan 29 2012

Bell convolution triangle ; g.f. for column k : (x*B(x))^k with  B(x) g.f. for A000110 (Bell numbers).
Riordan array (1, x*B(x)), when B(x) the g.f. of A000110.
Row sums are in A137551.

			Triangle begins:
  1;
  0,   1;
  0,   1,   1;
  0,   2,   2,  1;
  0,   5,   5,  3,  1;
  0,  15,  14,  9,  4,  1;
  0,  52,  44, 28, 14,  5, 1;
  0, 203, 154, 93, 48, 20, 6, 1;
  ...

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

Cf. Columns : A000007, A000110, A014322, A014323, A014325 ; Diagonals : A000012, A001477, A000096, A005586.
Another version: A292870.
T(2n,n) gives: A292871.

# Uses function PMatrix from A357368.
PMatrix(10, n -> combinat:-bell(n-1)); # Peter Luschny, Oct 19 2022

Sum_{k=0..n} T(n,k) = A137551(n), n>0.

A105475 Triangle read by rows: T(n,k) is number of compositions of n into k parts when each even part can be of two kinds.

Original entry on oeis.org

1, 2, 1, 1, 4, 1, 2, 6, 6, 1, 1, 8, 15, 8, 1, 2, 11, 26, 28, 10, 1, 1, 12, 42, 64, 45, 12, 1, 2, 16, 60, 122, 130, 66, 14, 1, 1, 16, 82, 208, 295, 232, 91, 16, 1, 2, 21, 108, 324, 582, 621, 378, 120, 18, 1, 1, 20, 135, 480, 1035, 1404, 1176, 576, 153, 20, 1, 2, 26, 170, 675

Offset: 1

Emeric Deutsch, Apr 09 2005

Riordan array ((1+2x)/(1-x^2),x(1+2x)/(1-x^2)). Factorizes as ((1+2x)/(1-x^2),x)*(1,x(1+2x)/(1-x^2)). Row sums A105476 form an eigensequence for ((1+2x)/(1-x^2),x). - Paul Barry, Feb 10 2011
Triangle T(n,k), 1<=k<=n, given by (0, 2, -3/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Jan 18 2012
Also the convolution triangle of A000034. - Peter Luschny, Oct 08 2022

			T(4,2) = 6 because we have (1,3), (3,1), (2,2), (2,2'), (2',2) and (2',2').
Triangle begins:
1;
2, 1;
1, 4,  1;
2, 6,  6, 1;
1, 8, 15, 8, 1;
Triangle (0, 2, -3/2, -1/2, 0, 0, 0...) DELTA (1, 0, 0, 0, 0, ...) begins:
1
0, 1
0, 2, 1
0, 1, 4, 1
0, 2, 6, 6, 1
0, 1, 8, 15, 8, 1
0, 2, 11, 26, 28, 10, 1
0, 1, 12, 42, 64, 45, 12, 1

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

Row sums yield A105476.

Cf. Diagonals: A000007, A000034, A000012, A005843, A000384, A100504.

G:=t*z*(1+2*z)/(1-t*z-z^2-2*t*z^2): Gser:=simplify(series(G,z=0,14)): for n from 1 to 12 do P[n]:=sort(coeff(Gser,z^n)) od: for n from 1 to 12 do seq(coeff(P[n],t^k),k=1..n) od; # yields sequence in triangular form
# second Maple program:
b:= proc(n) option remember; `if`(n=0, 1,
      expand(add((2-irem(i, 2))*b(n-i)*x, i=1..n)))
    end:
T:= n-> (p-> seq(coeff(p, x, k), k=1..n))(b(n)):
seq(T(n), n=1..14);  # Alois P. Heinz, Oct 16 2013
# Uses function PMatrix from A357368. Adds a row above and a column to the left.
PMatrix(10, n -> [1, 2][irem(n-1, 2) + 1]); # Peter Luschny, Oct 08 2022

max = 14; g = t*z*(1 + 2*z)/(1 - t*z - z^2 - 2*t*z^2); gser = Series[g, {z, 0, max}]; coes = CoefficientList[gser, {z, t}]; Table[ Table[ coes[[n, k]], {k, 2, n}], {n, 2, max}] // Flatten (* Jean-François Alcover, Oct 02 2013, after Maple *)

A111106 Riordan array (1, x*g(x)) where g(x) is g.f. of double factorials A001147.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 3, 2, 1, 0, 15, 7, 3, 1, 0, 105, 36, 12, 4, 1, 0, 945, 249, 64, 18, 5, 1, 0, 10395, 2190, 441, 100, 25, 6, 1, 0, 135135, 23535, 3807, 691, 145, 33, 7, 1, 0, 2027025, 299880, 40032, 5880, 1010, 200, 42, 8, 1

Offset: 0

Philippe Deléham, Oct 13 2005, Dec 20 2008

Triangle T(n,k), 0 <= k <= n, given by [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.

			Rows begin:
  1;
  0,       1;
  0,       1,      1;
  0,       3,      2,     1;
  0,      15,      7,     3,    1;
  0,     105,     36,    12,    4,    1;
  0,     945,    249,    64,   18,    5,   1;
  0,   10395,   2190,   441,  100,   25,   6,  1:
  0,  135135,  23535,  3807,  691,  145,  33,  7, 1;
  0, 2027025, 299880, 40032, 5880, 1010, 200, 42, 8, 1;

Columns: A000007, A001147, A034430; diagonals: A000012, A001477, A055998.

Cf. A084938, A111088, A112934, A168441.

# Uses function PMatrix from A357368.
PMatrix(10, n -> doublefactorial(2*n-3)); # Peter Luschny, Oct 19 2022

T(n, k) = Sum_{j=0..n-k} T(n-1, k-1+j)*A111088(j).
Sum_{k=0..n} T(n, k) = A112934(n).
G.f.: 1/(1-xy/(1-x/(1-2x/(1-3x/(1-4x/(1-... (continued fraction). - Paul Barry, Jan 29 2009
Sum_{k=0..n} T(n,k)*2^(n-k) = A168441(n). - Philippe Deléham, Nov 28 2009

A111960 Renewal array for central trinomial numbers A002426.

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 7, 7, 3, 1, 19, 20, 12, 4, 1, 51, 61, 40, 18, 5, 1, 141, 182, 135, 68, 25, 6, 1, 393, 547, 441, 251, 105, 33, 7, 1, 1107, 1640, 1428, 888, 420, 152, 42, 8, 1, 3139, 4921, 4572, 3076, 1596, 654, 210, 52, 9, 1, 8953, 14762, 14535, 10456, 5880, 2652, 966, 280, 63, 10, 1

Offset: 0

Paul Barry, Aug 23 2005

Also the convolution triangle of A002426. - Peter Luschny, Oct 06 2022

			Triangle T(n,k) begins:
   1;
   1,  1;
   3,  2,  1;
   7,  7,  3,  1;
  19, 20, 12,  4, 1;
  51, 61, 40, 18, 5, 1;
  ...
From _Paul Barry_, May 12 2009: (Start)
Production matrix is
  1, 1,
  2, 1, 1,
  0, 2, 1, 1,
  -2, 0, 2, 1, 1,
  0, -2, 0, 2, 1, 1,
  4, 0, -2, 0, 2, 1, 1. (End)

Row sums are A111961.
Diagonal sums are A111962.
Inverse is A111963.
Factors as A007318*A111959.
Column k=0 gives A002426.
Cf. A026325.

# Uses function PMatrix from A357368. Adds a row and column above and to the left.
PMatrix(10, n -> A002426(n - 1)); # Peter Luschny, Oct 06 2022

Factors as (1/(1-x), x/(1-x))*(1/sqrt(1-4x^2), x/sqrt(1-4x^2)).
From Paul Barry, May 12 2009: (Start)
Equals ((1-x^2)/(1+x+x^2),x/(1+x+x^2))^{-1}*(1,x/(1-x^2))=A094531*(1,x/(1-x^2)).
Riordan array (1/sqrt(1-2x-3x^2), x/sqrt(1-2x-3x^2));
T(n,k) = Sum_{j=0..n} C(n,j)*C((j-1)/2,(j-k)/2)*2^(j-k)*(1+(-1)^(j-k))/2.
G.f.: 1/(1-xy-x-2x^2/(1-x-x^2/(1-x-x^2/(1-x-x^2/(1-... (continued fraction). (End)

A341418 Triangle read by rows: T(n, m) gives the sum of the weights of weighted compositions of n with m parts from generalized pentagonal numbers {A001318(k)}_{k>=1}.

Original entry on oeis.org

1, 1, 1, 0, 2, 1, 0, 1, 3, 1, -1, 0, 3, 4, 1, 0, -2, 1, 6, 5, 1, -1, -2, -3, 4, 10, 6, 1, 0, -2, -6, -3, 10, 15, 7, 1, 0, -2, -6, -12, 0, 20, 21, 8, 1, 0, 1, -6, -16, -19, 9, 35, 28, 9, 1, 0, 0, 0, -16, -35, -24, 28, 56, 36, 10, 1, 1, 2, 3, -6, -40, -65, -21, 62, 84, 45, 11, 1

Offset: 1

Wolfdieter Lang, Feb 15 2021

The sums of row n are given in A000041(n), for n >= 1 (number of partitions).
A differently signed triangle is A047265.
One could add a column m = 0 starting at n = 0 with T(0, 0) = 1 and T(n, 0) = 0 otherwise, by including the empty partition with no parts.
For the weights w of positive integer numbers n see a comment in A339885. It is w(n) = -A010815(n), for n >= 0. Also w(n) = A257628(n), for n >= 1.
The weight of a composition is the one of the respective partition, obtained by the product of the weights of the parts.
That the row sums give the number of partitions follows from the pentagonal number theorem. See also the Apr 04 2013 conjecture in A000041 by Gary W. Adamson, and the hint for the proof by Joerg Arndt. The INVERT map of A = {1, 1, 0, 0, -5, -7, ...}, with offset 1, gives the A000041(n) numbers, for n >= 0.
If the above mentioned column for m = 0, starting at n = 0 is added this is an ordinary convolution triangle of the Riordan type R(1, f(x)), with f(x) = -(Product_{j>=1} (1 - x^j) - 1), generating {A257628(n)}{n>=0}. See the formulae below. - _Wolfdieter Lang, Feb 16 2021

			The triangle T(n, m) begins:
  n\m   1  2  3   4   5   6   7  8  9 10 11 12 ... A000041
  --------------------------------------------------------
  1:    1                                                1
  2:    1  1                                             2
  3:    0  2  1                                          3
  4:    0  1  3   1                                      5
  5:   -1  0  3   4   1                                  7
  6:    0 -2  1   6   5   1                             11
  7:   -1 -2 -3   4  10   6   1                         15
  8:    0 -2 -6  -3  10  15   7  1                      22
  9:    0 -2 -6 -12   0  20  21  8  1                   30
  10:   0  1 -6 -16 -19   9  35 28  9  1                42
  11:   0  0  0 -16 -35 -24  28 56 36 10  1             56
  12:   1  2  3  -6 -40 -65 -21 62 84 45 11  1          77
  ...
For instance the case n = 6: The relevant weighted partitions with parts from the pentagonal numbers and number of compositions are: m = 2: 2*(1,-5) = -2*(1,5), m = 3: 1*(2^3), m = 4: 3*(1^2,2^2), m = 5: 1*(1^4,2), m = 6: 1*(1^6). The other partitions have weight 0.

Wikipedia, Pentagonal number theorem.

Cf. A000041, A008284, A010815, A047265, A257628, -A307059 (alternating row sums), A339885 (for partitions).

# Using function PMatrix from A357368. Adds a row and a column for n, m = 0.
PMatrix(14, proc(n) 24*n+1; if issqr(%) then sqrt(%); -(-1)^irem(iquo(%+irem(%,6),6),2) else 0 fi end); # Peter Luschny, Oct 06 2022

nmax = 12;
col[m_] := col[m] = (-(Product[(1-x^j), {j, 1, nmax}]-1))^m // CoefficientList[#, x]&;
T[n_, m_] := col[m][[n+1]];
Table[T[n, m], {n, 1, nmax}, {m, 1, n}] // Flatten (* Jean-François Alcover, Oct 23 2023 *)

T(n, m) = Sum_{j=1..p(n,m)} w(Part(n, m, j))*M0(n, m, j), where p(n, m) = A008284(n, m), M0(n, m, j) are the multinomials from A048996, i.e., m!/Prod_{k=1..m} e(n,m,j,k)! with the exponents of the parts, and the ternary weight of the j-th partition of n with m parts Part(n,m,j), in Abramowitz-Stegun order, is defined as the product of the weights of the parts, using w(n) = -A010815(n), for n >= 1, and m = 1, 2, ..., n.
From Wolfdieter Lang, Feb 16 2021: (Start)
G.f. column m: G(m, x) =  ( -(Product_{j>=1} (1 - x^j) - 1) )^m, for  m >= 1.
G.f. of row polynomials R(n, x) = Sum_{m=1..n}, that is g. f. of the triangle:
  GfT(z, x) = 1/(1 - x*G(1, z)) - 1. Riordan triangle (without m = 0 column). (End)

A104580 Tribonacci convolution triangle.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 4, 5, 3, 1, 7, 12, 9, 4, 1, 13, 26, 25, 14, 5, 1, 24, 56, 63, 44, 20, 6, 1, 44, 118, 153, 125, 70, 27, 7, 1, 81, 244, 359, 336, 220, 104, 35, 8, 1, 149, 499, 819, 864, 646, 357, 147, 44, 9, 1, 274, 1010, 1830, 2144, 1800, 1134, 546, 200, 54, 10, 1

Offset: 0

Paul Barry, Mar 16 2005

			Rows begin
  {1},
  {1,1},
  {2,2,1},
  {4,5,3,1},
  {7,12,9,4,1},
   ...

First column is A000073(n+2). Row sums are A077939. Diagonal sums are A002478.

# Uses function PMatrix from A357368. Adds column 1,0,0,0,... to the left.
PMatrix(10, n -> A000073(n+1)); # Peter Luschny, Oct 19 2022

trinomial(n,k):=coeff(expand((1+x+x^2)^n),x,k);
create_list(sum(binomial(i+k,k)*trinomial(i,n-k-i),i,0,n-k),n,0,8,k,0,n); /* Emanuele Munarini, Mar 15 2011 */

Riordan array (1/(1-x-x^2-x^3), x/(1-x-x^2-x^3)).
From Paul Barry, Jun 02 2009: (Start)
T(n,m) = T'(n-1,m-1) + T'(n-1,m) + T'(n-2,m) + T'(n-3,m), where T'(n,m) = T(n,m) for n >= 0 and 0 <= m <= n and T'(n,m) = 0 otherwise. (End)
T(n,k) = Sum_{i=0..n-k} binomial(i+k,k)*A027907(i,n-k-i). - Emanuele Munarini, Mar 15 2011

A205813 Triangle T(n,k), read by rows, given by (0, 2, 1, 1, 1, 1, 1, 1, 1, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 6, 4, 1, 0, 20, 16, 6, 1, 0, 70, 64, 30, 8, 1, 0, 252, 256, 140, 48, 10, 1, 0, 924, 1024, 630, 256, 70, 12, 1, 0, 3432, 4096, 2772, 1280, 420, 96, 14, 1, 0, 12870, 16384, 12012, 6144, 2310, 640, 126, 16, 1

Offset: 0

Philippe Deléham, Feb 01 2012

Riordan array (1, x/sqrt(1-4*x)). Inverse of Riordan array (1, x*exp(arcsinh(-2*x))).

T is the convolution triangle of the shifted central binomial coefficients binomial(2*(n-1), n-1). - Peter Luschny, Oct 19 2022

			Triangle begins:
  1;
  0,   1;
  0,   2,   1;
  0,   6,   4,   1;
  0,  20,  16,   6,   1;
  0,  70,  64,  30,   8,   1;
  0, 252, 256, 140,  48,  10,   1;

Cf. A054335 and columns listed there.

# Uses function PMatrix from A357368.
PMatrix(10, n -> binomial(2*(n-1), n-1)); # Peter Luschny, Oct 19 2022

T(n,n) = 1 = A000012(n); T(n+1,n) = 2*n = A005843(n); T(n+2,n) = 2*n*(n+2) = A054000(n+1).
Sum_{k=0..n} T(n,k)*x^k = -A081696(n-1), A000007(n), A026671(n-1), A084868(n) for x = -1, 0, 1, 2 respectively.
G.f.: sqrt(1-4*x)/(sqrt(1-4*x)-y*x).
Sum_{k=0..n} T(n,k)*A090192(k) = A000108(n), A000108 = Catalan numbers.

A320019 Coefficients of polynomials related to the number of divisors, triangle read by rows, T(n,k) for 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 2, 4, 1, 0, 3, 8, 6, 1, 0, 2, 14, 18, 8, 1, 0, 4, 20, 41, 32, 10, 1, 0, 2, 28, 78, 92, 50, 12, 1, 0, 4, 37, 132, 216, 175, 72, 14, 1, 0, 3, 44, 209, 440, 490, 298, 98, 16, 1, 0, 4, 58, 306, 814, 1172, 972, 469, 128, 18, 1

Offset: 0

Peter Luschny, Oct 03 2018

Column k is the k-fold self-convolution of tau (A000005). - Alois P. Heinz, Feb 01 2021

			Triangle starts:
[0] 1
[1] 0, 1
[2] 0, 2,  1
[3] 0, 2,  4,   1
[4] 0, 3,  8,   6,   1
[5] 0, 2, 14,  18,   8,   1
[6] 0, 4, 20,  41,  32,  10,   1
[7] 0, 2, 28,  78,  92,  50,  12,  1
[8] 0, 4, 37, 132, 216, 175,  72, 14,  1
[9] 0, 3, 44, 209, 440, 490, 298, 98, 16, 1

Alois P. Heinz, Rows n = 0..200, flattened

Columns k=0-4 give: A000007, A000005, A055507, A191829, A375002.
Row sums are A129921.
T(2n,n) gives A340992.
Cf. A319083.

P := proc(n, x) option remember; if n = 0 then 1 else
x*add(numtheory:-tau(n-k)*P(k,x), k=0..n-1) fi end:
Trow := n -> seq(coeff(P(n, x), x, k), k=0..n):
seq(lprint([n], Trow(n)), n=0..9);
# second Maple program:
T:= proc(n, k) option remember; `if`(k=0, `if`(n=0, 1, 0),
      `if`(k=1, `if`(n=0, 0, numtheory[tau](n)), (q->
       add(T(j, q)*T(n-j, k-q), j=0..n))(iquo(k, 2))))
    end:
seq(seq(T(n, k), k=0..n), n=0..12);  # Alois P. Heinz, Feb 01 2021
# Uses function PMatrix from A357368.
PMatrix(10, NumberTheory:-tau); # Peter Luschny, Oct 19 2022

T[n_, k_] := T[n, k] = If[k == 0, If[n == 0, 1, 0],
     If[k == 1, If[n == 0, 0, DivisorSigma[0, n]],
     With[{q = Quotient[k, 2]}, Sum[T[j, q]*T[n-j, k-q], {j, 0, n}]]]];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Feb 11 2021, after Alois P. Heinz *)

The polynomials are defined by recurrence: p(0,x) = 1 and for n > 0 by
p(n, x) = x*Sum_{k=0..n-1} tau(n-k)*p(k, x).
Sigma[k](n) computes the sum of the k-th power of positive divisors of n. The recurrence applied with k = 0 gives this triangle, with k = 1 gives A319083.
T(n,k) = [x^n] (Sum_{j>=1} tau(j)*x^j)^k. - Alois P. Heinz, Feb 14 2021

cp's OEIS Frontend

A073266 Triangle read by rows: T(n,k) is the number of compositions of n as the sum of k integral powers of 2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A085771 Triangle read by rows. T(n, k) = A059438(n, k) for 1 <= k <= n, and T(n, 0) = n^0.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

SageMath

Formula

A205574 Triangle T(n,k), 0<=k<=n, given by (0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula

A105475 Triangle read by rows: T(n,k) is number of compositions of n into k parts when each even part can be of two kinds.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A111106 Riordan array (1, x*g(x)) where g(x) is g.f. of double factorials A001147.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Formula

A111960 Renewal array for central trinomial numbers A002426.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Formula

A341418 Triangle read by rows: T(n, m) gives the sum of the weights of weighted compositions of n with m parts from generalized pentagonal numbers {A001318(k)}_{k>=1}.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula