A357368 -id:A357368 - cp's OEIS frontend

A339067 Triangle read by rows: T(n,k) is the number of linear forests with n nodes and k rooted trees.

1, 1, 1, 2, 2, 1, 4, 5, 3, 1, 9, 12, 9, 4, 1, 20, 30, 25, 14, 5, 1, 48, 74, 69, 44, 20, 6, 1, 115, 188, 186, 133, 70, 27, 7, 1, 286, 478, 503, 388, 230, 104, 35, 8, 1, 719, 1235, 1353, 1116, 721, 369, 147, 44, 9, 1, 1842, 3214, 3651, 3168, 2200, 1236, 560, 200, 54, 10, 1

Offset: 1

Andrew Howroyd, Dec 03 2020

T(n,k) is the number of trees with n nodes rooted at two noninterchangeable nodes at a distance k-1 from each other.

Also the convolution triangle of A000081. - Peter Luschny, Oct 07 2022

			Triangle begins:
    1;
    1,    1;
    2,    2,    1;
    4,    5,    3,    1;
    9,   12,    9,    4,   1;
   20,   30,   25,   14,   5,   1;
   48,   74,   69,   44,  20,   6,   1;
  115,  188,  186,  133,  70,  27,   7,  1;
  286,  478,  503,  388, 230, 104,  35,  8, 1;
  719, 1235, 1353, 1116, 721, 369, 147, 44, 9, 1;
  ...

Alois P. Heinz, Rows n = 1..200, flattened

Columns 1..6 are A000081, A000106, A000242, A000300, A000343, A000395.
Row sums are A000107.
T(2n-1,n) gives A339440.
Cf. A033185, A038002, A217781, A339428.

b:= proc(n) option remember; `if`(n<2, n, (add(add(d*b(d),
      d=numtheory[divisors](j))*b(n-j), j=1..n-1))/(n-1))
    end:
T:= proc(n, k) option remember; `if`(k=1, b(n), (t->
      add(T(j, t)*T(n-j, k-t), j=1..n-1))(iquo(k, 2)))
    end:
seq(seq(T(n, k), k=1..n), n=1..12);  # Alois P. Heinz, Dec 04 2020
# Using function PMatrix from A357368. Adds row and column for n, k = 0.
PMatrix(10, A000081); # Peter Luschny, Oct 07 2022

b[n_] := b[n] = If[n < 2, n, (Sum[Sum[d*b[d], {d, Divisors[j]}]*b[n - j], {j, 1, n - 1}])/(n - 1)];
T[n_, k_] := T[n, k] = If[k == 1, b[n], With[{t = Quotient[k, 2]}, Sum[T[j, t]*T[n - j, k - t], {j, 1, n - 1}]]];
Table[Table[T[n, k], {k, 1, n}], {n, 1, 12}] // Flatten (* Jean-François Alcover, Jan 03 2021, after Alois P. Heinz *)

\\ TreeGf is A000081.
TreeGf(N) = {my(A=vector(N, j, 1)); for (n=1, N-1, A[n+1] = 1/n * sum(k=1, n, sumdiv(k, d, d*A[d]) * A[n-k+1] ) ); x*Ser(A)}
ColSeq(n,k)={my(t=TreeGf(max(0,n+1-k))); Vec(t^k, -n)}
M(n, m=n)=Mat(vector(m, k, ColSeq(n,k)~))
{ my(T=M(12)); for(n=1, #T~, print(T[n,1..n])) }

G.f. of k-th column: t(x)^k where t(x) is the g.f. of A000081.

Sum_{k=1..n} k * T(n,k) = A038002(n). - Alois P. Heinz, Dec 04 2020

A105422 Triangle read by rows: T(n,k) is the number of compositions of n having exactly k parts equal to 1.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 1, 2, 0, 1, 2, 2, 3, 0, 1, 3, 5, 3, 4, 0, 1, 5, 8, 9, 4, 5, 0, 1, 8, 15, 15, 14, 5, 6, 0, 1, 13, 26, 31, 24, 20, 6, 7, 0, 1, 21, 46, 57, 54, 35, 27, 7, 8, 0, 1, 34, 80, 108, 104, 85, 48, 35, 8, 9, 0, 1, 55, 139, 199, 209, 170, 125, 63, 44, 9, 10, 0, 1, 89, 240, 366, 404, 360

Offset: 0

Emeric Deutsch, Apr 07 2005

T(n,k) is also the number of length n bit strings beginning with 0 having k singletons. Example: T(4,2)=3 because we have 0010, 0100 and 0110. - Emeric Deutsch, Sep 21 2008
The cyclic version of this array is A320341(n,k), which counts the (unmarked) cyclic compositions of n with exactly k parts equal to 1, with a minor exception for k=0. The sequence (A320341(n, k=0) - 1: n >= 1) counts the (unmarked) cyclic compositions of n with no parts equal to 1. - Petros Hadjicostas, Jan 08 2019
Also the convolution triangle of Fibonacci(n-2). # Peter Luschny, Oct 07 2022
T(n,k) is the number of length n+1 bit strings beginning and ending with 0 having k length 2 substrings 00. This is equivalent to the compositions interpretation because each m part corresponds to a length m+1 bit string beginning with 0 and ending with the next 0 bit. Thus a substring 00 corresponds to a 1 part. Example: T(4,2)=3 because we have 00010 for 112, 00100 for 121 and 01000 for 211. - Michael Somos, Sep 24 2024
In the Baccherini et al. 2008 link on page 81: "Bloom[3] studies the number of singles in all the 2^n n-length bit strings, where a single is any isolated 1 or 0, i.e., any run of length 1. Let R_{n,k} be the number of n-length bit strings beginning with 0 and having k singles." Here T(n,k) = R_{n,k}. This combinatorial interpretation is equivalent to my previous comment since a part of size k corresponds to run of k identical bits and also to a length k+1 bit string with 0s only at the beginning and end. - Michael Somos, Sep 25 2024

			T(6,2) = 9 because we have (1,1,4), (1,4,1), (4,1,1), (1,1,2,2), (1,2,1,2), (1,2,2,1), (2,1,1,2), (2,1,2,1) and (2,2,1,1).
Triangle begins:
00:    1;
01:    0,   1;
02:    1,   0,   1;
03:    1,   2,   0,   1;
04:    2,   2,   3,   0,   1;
05:    3,   5,   3,   4,   0,   1;
06:    5,   8,   9,   4,   5,   0,   1;
07:    8,  15,  15,  14,   5,   6,   0,   1;
08:   13,  26,  31,  24,  20,   6,   7,   0,  1;
09:   21,  46,  57,  54,  35,  27,   7,   8,  0,  1;
10:   34,  80, 108, 104,  85,  48,  35,   8,  9,  0,  1;
11:   55, 139, 199, 209, 170, 125,  63,  44,  9, 10,  0,  1;
12:   89, 240, 366, 404, 360, 258, 175,  80, 54, 10, 11,  0, 1;
13:  144, 413, 666, 780, 725, 573, 371, 236, 99, 65, 11, 12, 0, 1;
...

Alois P. Heinz, Rows n = 0..140, flattened
D. Baccherini, D. Merlini and R. Sprugnoli, Level generating trees and proper Riordan arrays, Applicable Analysis and Discrete Mathematics, 2, 2008, 69-91 (see p. 83). [_Emeric Deutsch_, Sep 21 2008]
Jia Huang, Partially Palindromic Compositions, J. Int. Seq. (2023) Vol. 26, Art. 23.4.1. See pp. 4, 10-11.
Milan Janjić, Words and Linear Recurrences, J. Int. Seq. 21 (2018), #18.1.4.

Column 0 yields A000045 (the Fibonacci numbers). Column 1 yields A006367. Column 2 yields A105423. Row sums yield A011782. Cyclic version is A320341.

T(2n,n) gives A222763.

G:=(1-z)/(1-z-z^2-t*z+t*z^2): Gser:=simplify(series(G,z=0,15)): P[0]:=1: for n from 1 to 14 do P[n]:=coeff(Gser,z^n) od: for n from 0 to 13 do seq(coeff(t*P[n],t^k),k=1..n+1) od; # yields sequence in triangular form
# second Maple program:
T:= proc(n) option remember; local j; if n=0 then 1
      else []; for j to n do zip((x, y)-> x+y, %,
      [`if`(j=1, 0, [][]), T(n-j)], 0) od; %[] fi
    end:
seq(T(n), n=0..20);  # Alois P. Heinz, Nov 05 2012
# Uses function PMatrix from A357368. Adds a row above and a column to the left.
PMatrix(10, n -> combinat:-fibonacci(n-2)); # Peter Luschny, Oct 07 2022

nn = 10; a = x/(1 - x) - x + y x;
CoefficientList[CoefficientList[Series[1/(1 - a), {x, 0, nn}], x], y] // Flatten (* Geoffrey Critzer, Dec 23 2011 *)
T[ n_, k_] := Which[k<0 || k>n, 0, n<2, Boole[n==k], True, T[n, k] =  T[n-1, k] + T[n-1, k-1] + T[n-2, k] - T[n-2, k-1] ]; (* Michael Somos, Sep 24 2024 *)

{T(n, k) = if(k<0 || k>n, 0, n<2, n==k, T(n-1, k) + T(n-1, k-1) + T(n-2, k) - T(n-2, k-1) )}; /* Michael Somos, Sep 24 2024 */

G.f.: (1-z)/(1 - z - z^2 - tz + tz^2).
T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) - T(n-2,k-1), T(0,0)=1, T(1,0)=0. - Philippe Deléham, Nov 18 2009
If the triangle's columns are transposed into rows, then T(n,k) can be interpreted as the number of compositions of n+k having exactly k 1's. Then g.f.: ((1-x)/(1-x-x^2))^(k-1) and T(n,k) = T(n-2,k) + T(n-1,k) - T(n-1, k-1) + T(n, k-1). Also, Sum_{j=1..n} T(n-j, j) = 2^(n-1), the number of compositions of n. - Gregory L. Simay, Jun 28 2019

A319083 Coefficients of polynomials related to the D'Arcais polynomials and Dedekind's eta(q) function, triangle read by rows, T(n,k) for 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 3, 1, 0, 4, 6, 1, 0, 7, 17, 9, 1, 0, 6, 38, 39, 12, 1, 0, 12, 70, 120, 70, 15, 1, 0, 8, 116, 300, 280, 110, 18, 1, 0, 15, 185, 645, 885, 545, 159, 21, 1, 0, 13, 258, 1261, 2364, 2095, 942, 217, 24, 1, 0, 18, 384, 2262, 5586, 6713, 4281, 1498, 284, 27, 1

Offset: 0

Peter Luschny, Oct 03 2018

Column k is the k-fold self-convolution of sigma (A000203). - Alois P. Heinz, Feb 01 2021

For fixed k, Sum_{j=1..n} T(j,k) ~ Pi^(2*k) * n^(2*k) / (6^k * (2*k)!). - Vaclav Kotesovec, Sep 20 2024

			Triangle starts:
[0] 1;
[1] 0,  1;
[2] 0,  3,   1;
[3] 0,  4,   6,    1;
[4] 0,  7,  17,    9,    1;
[5] 0,  6,  38,   39,   12,    1;
[6] 0, 12,  70,  120,   70,   15,   1;
[7] 0,  8, 116,  300,  280,  110,  18,   1;
[8] 0, 15, 185,  645,  885,  545, 159,  21,  1;
[9] 0, 13, 258, 1261, 2364, 2095, 942, 217, 24, 1;

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

Columns k=0..6 give: A000007, A000203, A000385, A374951, A374977, A374978, A374979.
Row sums are A180305.
T(2n,n) gives A340993.
Cf. A008298, A078521, A283334, A319933.

P := proc(n, x) option remember; if n = 0 then 1 else
x*add(numtheory:-sigma(n-k)*P(k,x), k=0..n-1) fi end:
Trow := n -> seq(coeff(P(n, x), x, k), k=0..n):
seq(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[sigma](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..10);  # Alois P. Heinz, Feb 01 2021
# Uses function PMatrix from A357368.
PMatrix(10, NumberTheory:-sigma); # 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[1, 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, 10}] // 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} sigma(n-k)*p(k, x).
Sum_{k=0..n} (-1)^k * T(n,k) = A283334(n). - Alois P. Heinz, Feb 07 2025

A128899 Riordan array (1,(1-2x-sqrt(1-4x))/(2x)).

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 5, 4, 1, 0, 14, 14, 6, 1, 0, 42, 48, 27, 8, 1, 0, 132, 165, 110, 44, 10, 1, 0, 429, 572, 429, 208, 65, 12, 1, 0, 1430, 2002, 1638, 910, 350, 90, 14, 1, 0, 4862, 7072, 6188, 3808, 1700, 544, 119, 16, 1, 0, 16796, 25194, 23256, 15504, 7752, 2907, 798, 152, 18, 1

Offset: 0

Philippe Deléham, Apr 21 2007

Let the sequence A(n) = [0/1, 2/1, 1/2, 3/2, 2/3, 4/3, ...] defined by a(2n)=n/(n+1) and a(2n+1)=(n+2)/(n+1). T(n,k) is the triangle read by rows given by A(n) DELTA A000007 where DELTA is the operator defined in A084938.

T is the convolution triangle of the Catalan numbers (see A357368). - Peter Luschny, Oct 19 2022

			Triangle begins:
  1;
  0,     1;
  0,     2,     1;
  0,     5,     4,     1;
  0,    14,    14,     6,     1;
  0,    42,    48,    27,     8,    1;
  0,   132,   165,   110,    44,   10,    1;
  0,   429,   572,   429,   208,   65,   12,  1;
  0,  1430,  2002,  1638,   910,  350,   90,  14,   1;
  0,  4862,  7072,  6188,  3808, 1700,  544, 119,  16,  1;
  0, 16796, 25194, 23256, 15504, 7752, 2907, 798, 152, 18, 1;
  ...

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened)
Paul Barry, A Note on Riordan Arrays with Catalan Halves, arXiv:1912.01124 [math.CO], 2019.
Paul Barry, Chebyshev moments and Riordan involutions, arXiv:1912.11845 [math.CO], 2019.

Row sums give A088218.

Cf. A000108, A008313, A039598, A039599, A050166.

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

T[n_, n_] := 1; T[, 0] = 0; T[n, k_] /; 0 < k < n := T[n, k] = T[n - 1, k - 1] + 2 T[n - 1, k] + T[n - 1, k + 1]; T[, ] = 0;
Table[T[n, k], {n, 0, 10}, {k, 0, n}] (* Jean-François Alcover, Jun 14 2019 *)

T(n, k) = binomial(2*n-2, n-k)-binomial(2*n-2, n-k-2); \\ Seiichi Manyama, Mar 24 2025

@cached_function
def T(k,n):
    if k==n: return 1
    if k==0: return 0
    return sum(catalan_number(i)*T(k-1,n-i) for i in (1..n-k+1))
A128899 = lambda n,k: T(k,n)
for n in (0..10): print([A128899(n,k) for k in (0..n)]) # Peter Luschny, Mar 12 2016

T(n,k) = A039598(n-1,k-1) for n >= 1, k >= 1; T(n,0)=0^n.
T(n,k) = T(n-1,k-1) + 2*T(n-1,k) + T(n-1,k+1) for k >= 1, T(n,0)=0^n, T(n,k)=0 if k > n.
T(n,k) + T(n,k+1) = A039599(n,k). - Philippe Deléham, Sep 12 2007

Typo in data corrected by Jean-François Alcover, Jun 14 2019

A340991 Triangle T(n,k) whose k-th column is the k-fold self-convolution of the primes; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 2, 0, 3, 4, 0, 5, 12, 8, 0, 7, 29, 36, 16, 0, 11, 58, 114, 96, 32, 0, 13, 111, 291, 376, 240, 64, 0, 17, 188, 669, 1160, 1120, 576, 128, 0, 19, 305, 1386, 3121, 4040, 3120, 1344, 256, 0, 23, 462, 2678, 7532, 12450, 12864, 8288, 3072, 512, 0, 29, 679, 4851, 16754, 34123, 44652, 38416, 21248, 6912, 1024

Offset: 0

Alois P. Heinz, Feb 01 2021

			Triangle T(n,k) begins:
  1;
  0,  2;
  0,  3,   4;
  0,  5,  12,    8;
  0,  7,  29,   36,   16;
  0, 11,  58,  114,   96,    32;
  0, 13, 111,  291,  376,   240,    64;
  0, 17, 188,  669, 1160,  1120,   576,  128;
  0, 19, 305, 1386, 3121,  4040,  3120, 1344,  256;
  0, 23, 462, 2678, 7532, 12450, 12864, 8288, 3072, 512;
  ...

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

Columns k=0-4 give (offsets may differ): A000007, A000040, A014342, A014343, A014344.
Main diagonal gives A000079.
Row sums give A030017(n+1).
T(2n,n) gives A340990.
Cf. A030018, A030281.

T:= proc(n, k) option remember; `if`(k=0, `if`(n=0, 1, 0),
      `if`(k=1, `if`(n=0, 0, ithprime(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);
# Uses function PMatrix from A357368.
PMatrix(10, ithprime); # Peter Luschny, Oct 09 2022

T[n_, k_] := T[n, k] = If[k == 0, If[n == 0, 1, 0],
     If[k == 1, If[n == 0, 0, Prime[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 10 2021, after Alois P. Heinz *)

T(n,k) = [x^n] (Sum_{j>=1} prime(j)*x^j)^k.
Sum_{k=0..n} k * T(n,k) = A030281(n).
Sum_{k=0..n} (-1)^k * T(n,k) = A030018(n).
Conjecture: row polynomials are x*R(n,1) for n > 0 where R(n,k) = R(n-1,k+1) + x*R(n-1,1)*R(1,k) for n > 1, k > 0 with R(1,k) = prime(k) for k > 0. The same recursion seems to work for self-convolution of any other sequence. - Mikhail Kurkov, Apr 05 2025

A130123 Infinite lower triangular matrix with 2^k in the right diagonal and the rest zeros. Triangle, T(n,k), n zeros followed by the term 2^k. Triangle by columns, (2^k, 0, 0, 0, ...).

Original entry on oeis.org

1, 0, 2, 0, 0, 4, 0, 0, 0, 8, 0, 0, 0, 0, 16, 0, 0, 0, 0, 0, 32, 0, 0, 0, 0, 0, 0, 64, 0, 0, 0, 0, 0, 0, 0, 128, 0, 0, 0, 0, 0, 0, 0, 0, 256, 0, 0, 0, 0, 0, 0, 0, 0, 0, 512, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1024, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2048, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4096

Offset: 0

Gary W. Adamson, May 11 2007

A 2^n transform matrix.
A130123 * A007318 = A038208. A007318 * A130123 = A013609. A130124 = A130123 * A002260. A130125 = A128174 * A130123.
Triangle T(n,k), 0 <= k <= n, given by [0,0,0,0,0,0,...] DELTA [2,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, May 26 2007
Also the Bell transform of A000038. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 27 2016
T is the convolution triangle of the characteristic function of 2 (see A357368). - Peter Luschny, Oct 19 2022

			First few terms of the triangle:
  1;
  0, 2;
  0, 0, 4;
  0, 0, 0, 8;
  0, 0, 0, 0, 16;
  0, 0, 0, 0,  0, 32; ...

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Cf. A130124, A130125.

[[k eq n select 2^n else 0: k in [0..n]]: n in [0..14]]; // G. C. Greubel, Jun 05 2019

# The function BellMatrix is defined in A264428.
BellMatrix(n -> `if`(n=0,2,0), 9); # Peter Luschny, Jan 27 2016
# Uses function PMatrix from A357368.
PMatrix(10, n -> ifelse(n=1, 2, 0)); # Peter Luschny, Oct 19 2022

BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
rows = 12;
M = BellMatrix[If[# == 0, 2, 0]&, rows];
Table[M[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 23 2018, after Peter Luschny *)
Table[If[k==n, 2^n, 0], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Jun 05 2019 *)

{T(n,k) = if(k==n, 2^n, 0)}; \\ G. C. Greubel, Jun 05 2019

def T(n, k):
    if (k==n): return 2^n
    else: return 0
[[T(n, k) for k in (0..n)] for n in (0..14)] # G. C. Greubel, Jun 05 2019

G.f.: 1/(1-2*x*y). - R. J. Mathar, Aug 11 2015

A155112 Triangle T(n,k), 0<=k<=n, read by rows given by [0,2,-1/2,-1/2,0,0,0,0,0,0,0,0,...] DELTA [1,0,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 3, 4, 1, 0, 5, 10, 6, 1, 0, 8, 22, 21, 8, 1, 0, 13, 45, 59, 36, 10, 1, 0, 21, 88, 147, 124, 55, 12, 1, 0, 34, 167, 339, 366, 225, 78, 14, 1, 0, 55, 310, 741, 976, 770, 370, 105, 16, 1, 0, 89, 566, 1557, 2422, 2337, 1443, 567, 136, 18, 1, 0, 144, 1020, 3174, 5696, 6505, 4920, 2485, 824, 171, 20, 1

Offset: 0

Philippe Deléham, Jan 20 2009

A Fibonacci convolution triangle; Riordan array (1, x*(1+x)/(1-x-x^2)).

			Triangle begins:
  1;
  0,  1;
  0,  2,  1;
  0,  3,  4,  1;
  0,  5, 10,  6,  1;
  0,  8, 22, 21,  8,  1;
  0, 13, 45, 59, 36, 10, 1;
  ...

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

Cf. A000007, A155020, A155116, A155117, A155119, A155127, A155130, A155132, A155144, A155157.
Cf. A000012, A155020, A154964, A154968, A154996, A154997, A154999, A155000, A155001, A155017.
Cf. A000045, A154929.
T(2n,n) gives A262910.
Cf. A291385.

T:= func< n,k | n eq 0 select 1 else (&+[ Binomial(n-j,j)*Binomial(n-j,k)*k/(n-j): j in [0..Floor(n/2)]]) >;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 26 2021

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

T[n_, k_]:= If[n==0, 1, Sum[Binomial[n-j, j]*Binomial[n-j, k]*k/(n-j), {j, 0, Floor[n/2]}]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 26 2021 *)

def T(n,k): return 1 if n==0 else sum( binomial(n-j,j)*binomial(n-j,k)*k/(n-j) for j in (0..n//2) )
flatten([[T(n,k) for k in [0..n]] for n in [0..12]]) # G. C. Greubel, Mar 26 2021

Recurrence: T(n+2,k+1) = T(n+1,k+1) + T(n+1,k) + T(n,k+1) + T(n,k).
Explicit formula: T(n,k) = Sum_{i=0..floor(n/2)} binomial(n-i, i)*binomial(n-i, k)*k/(n-i), for n > 0.
G.f.: (1-x-x^2)/(1-(1+y)*x-(1+y)*x^2). - Philippe Deléham, Feb 21 2012
Sum_{k=0..n} T(n,k)*x^(n-k) = A000012(n), A155020(n), A154964(n), A154968(n), A154996(n), A154997(n), A154999(n), A155000(n), A155001(n), A155017(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, respectively.
Sum_{k=0..n} T(n,k)*x^k = A000007(n), A155020(n), A155116(n), A155117(n), A155119(n), A155127(n), A155130(n), A155132(n), A155144(n), A155157(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, respectively. - Philippe Deléham, Feb 21 2012
Sum_{k=0..n} T(n, k)*(m-1)^k = (1/m)*[n=0] - (m-1)*(i*sqrt(m))^(n-2)*ChebyshevU(n, -i*sqrt(m)/2). - G. C. Greubel, Mar 26 2021
Sum_{k=0..n} k * T(n,k) = A291385(n-1) for n>=1. - Alois P. Heinz, Sep 29 2022

Typos in two terms corrected by Alois P. Heinz, Aug 08 2015

A038243 Triangle whose (i,j)-th entry is 5^(i-j)*binomial(i,j).

Original entry on oeis.org

1, 5, 1, 25, 10, 1, 125, 75, 15, 1, 625, 500, 150, 20, 1, 3125, 3125, 1250, 250, 25, 1, 15625, 18750, 9375, 2500, 375, 30, 1, 78125, 109375, 65625, 21875, 4375, 525, 35, 1, 390625, 625000, 437500, 175000, 43750, 7000, 700, 40, 1, 1953125, 3515625, 2812500, 1312500, 393750, 78750, 10500, 900, 45, 1

Offset: 0

N. J. A. Sloane

Mirror image of A013612. - Zerinvary Lajos, Nov 25 2007
T(i,j) is the number of i-permutations of 6 objects a,b,c,d,e,f, with repetition allowed, containing j a's. - Zerinvary Lajos, Dec 21 2007
Triangle of coefficients in expansion of (5+x)^n - N-E. Fahssi, Apr 13 2008
Also the convolution triangle of A000351. - Peter Luschny, Oct 09 2022

			Triangle begins as:
       1;
       5,      1;
      25,     10,      1;
     125,     75,     15,      1;
     625,    500,    150,     20,     1;
    3125,   3125,   1250,    250,    25,    1;
   15625,  18750,   9375,   2500,   375,   30,   1;
   78125, 109375,  65625,  21875,  4375,  525,  35,  1;
  390625, 625000, 437500, 175000, 43750, 7000, 700, 40, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
B. N. Cyvin, J. Brunvoll, and S. J. Cyvin, Isomer enumeration of unbranched catacondensed polygonal systems with pentagons and heptagons, Match, No. 34 (Oct 1996), 109-121.

Cf. A000351, A000400 (row sums), A036071, A050982, A053464, A081135, A081143.

Sequences of the form q^(n-k)*binomial(n, k): A007318 (q=1), A038207 (q=2), A027465 (q=3), A038231 (q=4), this sequence (q=5), A038255 (q=6), A027466 (q=7), A038279 (q=8), A038291 (q=9), A038303 (q=10), A038315 (q=11), A038327 (q=12), A133371 (q=13), A147716 (q=14), A027467 (q=15).

[5^(n-k)*Binomial(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, May 12 2021

for i from 0 to 8 do seq(binomial(i, j)*5^(i-j), j = 0 .. i) od; # Zerinvary Lajos, Dec 21 2007
# Uses function PMatrix from A357368. Adds column 1, 0, 0, ... to the left.
PMatrix(10, n -> 5^(n-1)); # Peter Luschny, Oct 09 2022

With[{q=5}, Table[q^(n-k)*Binomial[n,k], {n,0,12}, {k,0,n}]//Flatten] (* G. C. Greubel, May 12 2021 *)

flatten([[5^(n-k)*binomial(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 12 2021

See A038207 and A027465 and replace 2 and 3 in analogous formulas with 5. - Tom Copeland, Oct 26 2012

A110441 Triangular array formed by the Mersenne numbers.

Original entry on oeis.org

1, 3, 1, 7, 6, 1, 15, 23, 9, 1, 31, 72, 48, 12, 1, 63, 201, 198, 82, 15, 1, 127, 522, 699, 420, 125, 18, 1, 255, 1291, 2223, 1795, 765, 177, 21, 1, 511, 3084, 6562, 6768, 3840, 1260, 238, 24, 1, 1023, 7181, 18324, 23276, 16758, 7266, 1932, 308, 27, 1

Offset: 0

Asamoah Nkwanta (nkwanta(AT)jewel.morgan.edu), Aug 08 2005

This sequence factors A038255 into a product of Riordan arrays.
Subtriangle of the triangle given by (0, 3, -2/3, 2/3, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 19 2012
From Peter Bala, Jul 22 2014: (Start)
Let M denote the lower unit triangular array A130330 and for k = 0,1,2,... define M(k) to be the lower unit triangular block array
/I_k 0\
\ 0  M/
having the k x k identity matrix I_k as the upper left block; in particular, M(0) = M. Then the present triangle equals the infinite matrix product M(0)*M(1)*M(2)*... (which is clearly well-defined). See the Example section. (End)
For 1<=k<=n, T(n,k) equals the number of (n-1)-length ternary words containing k-1 letters equal 2 and avoiding 01 and 02. - Milan Janjic, Dec 20 2016
The convolution triangle of the Mersenne numbers. - Peter Luschny, Oct 09 2022

			Triangle starts:
   1;
   3,  1;
   7,  6,  1;
  15, 23,  9,  1;
  31, 72, 48, 12,  1;
(0, 3, -2/3, 2/3, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, ...) begins:
  1
  0,  1
  0,  3,  1
  0,  7,  6,  1
  0, 15, 23,  9,  1
  0, 31, 72, 48, 12, 1. - _Philippe Deléham_, Mar 19 2012
With the arrays M(k) as defined in the Comments section, the infinite product M(0*)M(1)*M(2)*... begins
/ 1          \/1         \/1        \      / 1       \
| 3  1       ||0  1      ||0 1      |      | 3  1    |
| 7  3 1     ||0  3 1    ||0 0 1    |... = | 7  6 1  |
|15  7 3 1   ||0  7 3 1  ||0 0 3 1  |      |15 23 9 1|
|31 15 7 3 1 ||0 15 7 3 1||0 0 7 3 1|      |...      |
|...         ||...       ||...      |      |...      | - _Peter Bala_, Jul 22 2014

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150)
Naiomi T. Cameron and Asamoah Nkwanta, On Some (Pseudo) Involutions in the Riordan Group, J. Int. Seq. 8 (2005), #05.3.7.
Milan Janjić, Words and Linear Recurrences, J. Int. Seq. 21 (2018), #18.1.4.

Cf. A000225, A130330, A206306, A116414.

# Uses function PMatrix from A357368. Adds column 1, 0, 0, ... to the left.
PMatrix(10, n -> 2^n - 1); # Peter Luschny, Oct 09 2022

With[{n = 9}, DeleteCases[#, 0] & /@ CoefficientList[Series[1/(1 - (3 + y) x + 2 x^2), {x, 0, n}, {y, 0, n}], {x, y}]] // Flatten (* Michael De Vlieger, Apr 25 2018 *)

Riordan array M(n, k): (1/(1-3z+2z^2), z/(1-3z+2z^2)). Leftmost column M(n, 0) is the Mersenne numbers A000225, first column is A045618, second column is A055582, row sum is A007070 and diagonal sum is even-indexed Fibonacci numbers A001906.
T(n,k) = Sum_{j=0..n} C(j+k,k)C(n-j,k)2^(n-j-k). - Paul Barry, Feb 13 2006
From Philippe Deléham, Mar 19 2012: (Start)
G.f.: 1/(1-(3+y)*x+2*x^2).
T(n,k) = 3*T(n-1,k) + T(n-1,k-1) -2*T(n-2,k), T(0,0) = 1, T(n,k) = 0 if k<0 or if k>n.
Sum_{k, 0<=k<=n} T(n,k)*x^k = A000225(n+1), A007070(n), A107839(n), A154244(n), A186446(n), A190975(n+1), A190979(n+1), A190869(n+1) for x = 0, 1, 2, 3, 4, 5, 6, 7 respectively. (End)
Recurrence: T(n+1,k+1) = Sum_{i=0..n-k} (2^(i+1) - 1)*T(n-i,k). - Peter Bala, Jul 22 2014
From Peter Bala, Oct 07 2019: (Start)
Recurrence for row polynomials: R(n,x) = (3 + x)*R(n-1,x) - 2*R(n-2,x) with R(0,x) = 1 and R(1,x) = 3 + x.
The row reverse polynomial x^n*R(n,1/x) is equal to the numerator polynomial of the finite continued fraction 1 + x/(1 + 2*x/(1 + ... + x/(1 + 2*x/(1)))) (with 2*n partial numerators). Cf. A116414. (End)

A122896 Riordan array (1, (1 - x - sqrt(1 - 2x - 3x^2)) / (2*x)), a Riordan array for directed animals. Triangle read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 4, 5, 3, 1, 0, 9, 12, 9, 4, 1, 0, 21, 30, 25, 14, 5, 1, 0, 51, 76, 69, 44, 20, 6, 1, 0, 127, 196, 189, 133, 70, 27, 7, 1, 0, 323, 512, 518, 392, 230, 104, 35, 8, 1, 0, 835, 1353, 1422, 1140, 726, 369, 147, 44, 9, 1

Offset: 0

Paul Barry, Sep 18 2006

Also the convolution triangle of the Motzkin numbers A001006. - Peter Luschny, Oct 08 2022

			Triangle begins:
[0] 1;
[1] 0,   1;
[2] 0,   1,   1;
[3] 0,   2,   2,   1;
[4] 0,   4,   5,   3,   1;
[5] 0,   9,  12,   9,   4,   1;
[6] 0,  21,  30,  25,  14,   5,   1;
[7] 0,  51,  76,  69,  44,  20,   6,  1;
[8] 0, 127, 196, 189, 133,  70,  27,  7, 1;
[9] 0, 323, 512, 518, 392, 230, 104, 35, 8, 1.

Row sums are A005773, number of directed animals of size n.
Product of A007318 and this sequence is A122897.
Cf. A001006, A007318, A064189.

T := proc(n,k) option remember;
if k=0 then return 0^n fi; if k>n then return 0 fi;
T(n-1,k-1) + T(n-1,k) + T(n-1,k+1) end:
for n from 0 to 9 do seq(T(n,k), k=0..n) od; # Peter Luschny, Aug 17 2016
# Uses function PMatrix from A357368.
PMatrix(10, n -> simplify(hypergeom([1 -n/2, -n/2+1/2], [2], 4))); # Peter Luschny, Oct 08 2022

T[n_, n_] = 1; T[, 0] = 0; T[n, k_] /; 0, ] = 0;
Table[T[n, k], {n, 0, 10}, {k, 0, n}] (* Jean-François Alcover, Jun 13 2019 *)

# uses[riordan_array from A256893]
riordan_array(1, (1-x-sqrt(1-2*x-3*x^2))/(2*x), 11) # Peter Luschny, Aug 17 2016

Inverse of Riordan array (1, x / (1 + x + x^2)).
T(n+1, k+1) = A064189(n, k). - Philippe Deléham, Apr 21 2007
Riordan array (1, x*m(x)) where m(x) is the g.f. of Motzkin numbers (A001006). - Philippe Deléham, Nov 04 2009

cp's OEIS Frontend

A339067 Triangle read by rows: T(n,k) is the number of linear forests with n nodes and k rooted trees.

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A105422 Triangle read by rows: T(n,k) is the number of compositions of n having exactly k parts equal to 1.

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A319083 Coefficients of polynomials related to the D'Arcais polynomials and Dedekind's eta(q) function, triangle read by rows, T(n,k) for 0 <= k <= n.

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A128899 Riordan array (1,(1-2x-sqrt(1-4x))/(2x)).

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A340991 Triangle T(n,k) whose k-th column is the k-fold self-convolution of the primes; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A130123 Infinite lower triangular matrix with 2^k in the right diagonal and the rest zeros. Triangle, T(n,k), n zeros followed by the term 2^k. Triangle by columns, (2^k, 0, 0, 0, ...).

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A122896 Riordan array (1, (1 - x - sqrt(1 - 2x - 3x^2)) / (2*x)), a Riordan array for directed animals. Triangle read by rows.