A003056 -id:A003056 - cp's OEIS frontend

A116942 Permutation of the natural numbers in conjunction with A116939 and A003056.

0, 1, 3, 2, 4, 6, 5, 7, 10, 8, 11, 9, 12, 15, 13, 16, 14, 17, 21, 18, 22, 19, 23, 20, 24, 28, 25, 29, 26, 30, 27, 31, 36, 32, 37, 33, 38, 34, 39, 35, 40, 45, 41, 46, 42, 47, 43, 48, 44, 49, 55, 50, 56, 51, 57, 52, 58, 53, 59, 54, 60, 66, 61, 67, 62, 68, 63, 69, 64, 70, 65, 71, 78

Offset: 0

Reinhard Zumkeller, Feb 27 2006

nonn

Inverse: A116941;

A116939(n) = A003056(a(n)).

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for sequences that are permutations of the natural numbers

import Data.List (elemIndex)
import Data.Maybe (fromJust)
a116942 = fromJust . (`elemIndex` a116941_list)
-- Reinhard Zumkeller, Jun 28 2013

A262048 Irregular triangle read by rows: T(n, k) = A262045(n, k) * A237593(n, k), n >= 1, 1 <= k <= 2 * A003056(n).

Original entry on oeis.org

1, 1, 2, 2, 2, 0, 0, 2, 3, 1, 1, 3, 3, 0, 0, 3, 4, 1, 2, 2, 1, 4, 4, 0, 0, 0, 0, 4, 5, 2, 1, 1, 2, 5, 5, 0, 2, 2, 0, 5, 6, 2, 1, 0, 0, 1, 2, 6, 6, 0, 0, 0, 0, 0, 0, 6, 7, 2, 4, 2, 2, 4, 2, 7, 7, 0, 0, 0, 0, 0, 0, 7, 8, 3, 1, 0, 0, 1, 3, 8, 8, 0, 2, 1, 2, 2, 1, 2, 0, 8, 9, 3, 2, 1, 1, 1, 1, 2, 3, 9

Offset: 1

Hartmut F. W. Hoft, Sep 09 2015

This sequence lists the areas of rectangular pieces computed from the lengths of the legs in the Dyck path for n given in the n-th row of A237593 and the widths between the legs of the (n-1)-st and n-th Dyck paths given in the n-th row of A262045. The sum of the areas of all pieces appears to equal sigma(n), adjusting for a double count of the width at the center (that matters when the two symmetric Dyck paths enclose an odd number of regions).

T(n,k) is the sum of the widths of the symmetric representation of sigma(n) that are associated with the k-th line segment of the n-th Dyck path of the original diagram of the symmetric representation of sigma in the first quadrant after the diagram has been partitioned into two octants whose vertices are (for example) at (0, 1) and (1, 0). Note that the new diagram contains two main diagonals that are parallel between them, see example. - Omar E. Pol, Sep 30 2015

			For the 9th row: [5 2 2 2 2 5] in A237593 and [1 0 1 1 0 1] in A262045 result in [5 0 2 2 0 5] in this sequence which defines three regions between the Dyck paths for 8 and 9. The areas of the three regions are 5, 3 and 5; note that the width at the center of the middle region is 1 and is counted twice when adding up contiguous subsequences of nonzero numbers. The sum of the three areas is sigma(9) = 13.
For the 19th row: [10 4 2 2 1 1 2 2 4 10] in A237593 and [1 0 0 0 0 0 0 0 0 1] in A262045 result in [10 0 0 0 0 0 0 0 0 10] in this sequence which defines two disjoint regions of area 10 each between the Dyck paths for 18 and 19; their sum is sigma(19) = 20.
The first 20 rows of the triangle:
  -------------------------------------
  n\k   1  2  3  4  5  6  7  8  9  10
  -------------------------------------
  1:    1, 1;
  2:    2, 2;
  3:    2, 0, 0, 2;
  4:    3, 1, 1, 3;
  5:    3, 0, 0, 3;
  6:    4, 1, 2, 2, 1, 4;
  7:    4, 0, 0, 0, 0, 4;
  8:    5, 2, 1, 1, 2, 5;
  9:    5, 0, 2, 2, 0, 5;
  10:   6, 2, 1, 0, 0, 1, 2, 6;
  11:   6, 0, 0, 0, 0, 0, 0, 6;
  12:   7, 2, 4, 2, 2, 4, 2, 7;
  13:   7, 0, 0, 0, 0, 0, 0, 7;
  14:   8, 3, 1, 0, 0, 1, 3, 8;
  15:   8, 0, 2, 1, 2, 2, 1, 2, 0,  8;
  16:   9, 3, 2, 1, 1, 1, 1, 2, 3,  9;
  17:   9, 0, 0, 0, 0, 0, 0, 0, 0,  9;
  18:  10, 3, 4, 2, 1, 1, 2, 4, 3, 10;
  19:  10, 0, 0, 0, 0, 0, 0, 0, 0, 10;
  20:  11, 4, 2, 1, 4, 4, 1, 2, 4, 11;
  ...
A diagram of the first six rows shows the regions with an "x" marking the cells on the diagonal that are double counted.
.                                        _ _ _ _
.                                       |_|_|_|_|_
.                                             |_|x|_
.                                             |x|_|_|
.                  [4 1 2 2 1 4] & sigma(6)=12    |_|
.                                                 |_|
.                             _ _ _               |_|
.                            |_|_|_|
.                                  |_ _
.           [3 0 0 3] & sigma(5)=6   |_|
.                                    |_|
.                    _ _ _           |_|
.                   |_|_|_|_
.                       |x|_|
.                         |_| [3 1 1 3] & sigma(4)=7
.             _ _         |_|
.            |_|_|_
.                |_|    [2 0 0 2] & sigma(3)=4
.                |_|
.      _ _
.     |_ x|    [2 2] & sigma(2)=3
.       |_|
.  _
. |x|    [1 1] & sigma(1)=1
.
From _Omar E. Pol_, Sep 30 2015: (Start)
Illustration of the 12th row of the triangle:
.
.           7
.     _ _ _ _ _ _ _  2
.     / / / / / / /|   4
.                 /|_ _  2
.                 / / /|
.                 / / /      2
.                   /        _ 4
.                          / /|  2
.                        / / /|_ _
.                          / / / /|
.                                /|
.                                /|
.                                /| 7
.                                /|
.                                /|
.                                /|
.
The original diagram of the symmetric representation of sigma(12) in the first quadrant has been partitioned into two octants. Note that now there are two main diagonals in the diagram. The sums of the widths associated to the successive line segments of the partitioned zig-zag path give [7, 2, 4, 2, 2, 4, 2, 7], the same as the 12th row of the triangle.
(End)

Cf. A000203, A003056, A196020, A236104, A237048, A237270, A237271, A237591, A237593, A262045.

(* auxiliary functions are defined in A237048, A237593 and A262045 respectively *)
s[n_,k_]:=Ceiling[(n+1)/k-(k+1)/2] - Ceiling[(n+1)/(k+1)-(k+2)/2]
t[n_,k_]:=If[k<=row[n], s[n, k], s[n, 2*row[n]+1-k]]
a262048[n_]:=Map[t[n, #]&, Range[2*row[n]]] a262045[n]
Flatten[Map[a262048, Range[16]]] (* data *)

A265208 Total number T(n,k) of lambda-parking functions induced by all partitions of n into exactly k distinct parts; triangle T(n,k), n>=0, 0<=k<=A003056(n), read by rows.

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 3, 3, 0, 4, 5, 0, 5, 10, 0, 6, 14, 16, 0, 7, 21, 25, 0, 8, 27, 43, 0, 9, 36, 74, 0, 10, 44, 107, 125, 0, 11, 55, 146, 189, 0, 12, 65, 207, 307, 0, 13, 78, 267, 471, 0, 14, 90, 342, 786, 0, 15, 105, 436, 1058, 1296, 0, 16, 119, 538, 1490, 1921

Offset: 0

Alois P. Heinz, Dec 04 2015

Differs from A265020 first at T(5,2). See example.

			T(5,2) = 10: There are two partitions of 5 into 2 distinct parts: [2,3], [1,4]. Together they have 10 lambda-parking functions: [1,1], [1,2], [1,3], [1,4], [2,1], [2,2], [2,3], [3,1], [3,2], [4,1]. Here [1,1], [1,2], [1,3], [2,1], [3,1] are induced by both partitions. But they are counted only once.
T(6,1) = 6: [1], [2], [3], [4], [5], [6].
T(6,2) = 14: [1,1], [1,2], [1,3], [1,4], [1,5], [2,1], [2,2], [2,3], [2,4], [3,1], [3,2], [4,1], [4,2], [5,1].
T(6,3) = 16: [1,1,1], [1,1,2], [1,1,3], [1,2,1], [1,2,2], [1,2,3], [1,3,1], [1,3,2], [2,1,1], [2,1,2], [2,1,3], [2,2,1], [2,3,1], [3,1,1], [3,1,2], [3,2,1].
Triangle T(n,k) begins:
00 :  1;
01 :  0,  1;
02 :  0,  2;
03 :  0,  3,   3;
04 :  0,  4,   5;
05 :  0,  5,  10;
06 :  0,  6,  14,  16;
07 :  0,  7,  21,  25;
08 :  0,  8,  27,  43;
09 :  0,  9,  36,  74;
10 :  0, 10,  44, 107,  125;
11 :  0, 11,  55, 146,  189;
12 :  0, 12,  65, 207,  307;
13 :  0, 13,  78, 267,  471;
14 :  0, 14,  90, 342,  786;
15 :  0, 15, 105, 436, 1058, 1296;
16 :  0, 16, 119, 538, 1490, 1921;

Alois P. Heinz, Rows n = 0..400, flattened
R. Stanley, Parking Functions, 2011

Columns k=0-2 give: A000007, A000027, A176222(n+1).
Row sums give A265202.
Cf. A000217, A000272, A003056, A206735 (the same for general partitions), A265020, A265145.

b:= proc(p, g, n, i, t) option remember; `if`(g=0, 0, p!/g!*x^p)+
      `if`(n (p-> seq(coeff(p, x, i), i=0..degree(p)))(
        `if`(n=0, 1, b(0$2, n, 1$2))):
seq(T(n), n=0..25);

b[p_, g_, n_, i_, t_] := b[p, g, n, i, t] = If[g==0, 0, p!/g!*x^p] + If[nJean-François Alcover, Feb 02 2017, translated from Maple *)

T(A000217(n),n) = A000272(n+1).

A292047 Triangle read by rows: T(n,k) = (-1)^k * T(n-k,k-1) + T(n-k,k) with T(0,0) = 1 for 0 <= k <= A003056(n).

Original entry on oeis.org

1, 0, -1, 0, -1, 0, -1, -1, 0, -1, -1, 0, -1, -2, 0, -1, -2, 1, 0, -1, -3, 1, 0, -1, -3, 2, 0, -1, -4, 3, 0, -1, -4, 4, 1, 0, -1, -5, 5, 1, 0, -1, -5, 7, 2, 0, -1, -6, 8, 3, 0, -1, -6, 10, 5, 0, -1, -7, 12, 6, -1, 0, -1, -7, 14, 9, -1, 0, -1, -8, 16, 11, -2, 0, -1

Offset: 0

Seiichi Manyama, Sep 08 2017

			First few rows are:
  1;
  0, -1;
  0, -1;
  0, -1, -1;
  0, -1, -1;
  0, -1, -2;
  0, -1, -2, 1;
  0, -1, -3, 1;
  0, -1, -3, 2;
  0, -1, -4, 3;
  0, -1, -4, 4, 1.

Seiichi Manyama, Rows n = 0..481, flattened

Row sums give A278399.
Columns 0-1 give A000007, (-1)*A000012.
Cf. A292049.

A292049 Triangle read by rows: T(n,k) = (-1)^(k-1) * T(n-k,k-1) + T(n-k,k) with T(0,0) = 1 for 0 <= k <= A003056(n).

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 1, -1, 0, 1, -1, 0, 1, -2, 0, 1, -2, -1, 0, 1, -3, -1, 0, 1, -3, -2, 0, 1, -4, -3, 0, 1, -4, -4, 1, 0, 1, -5, -5, 1, 0, 1, -5, -7, 2, 0, 1, -6, -8, 3, 0, 1, -6, -10, 5, 0, 1, -7, -12, 6, 1, 0, 1, -7, -14, 9, 1, 0, 1, -8, -16, 11, 2, 0, 1, -8, -19

Offset: 0

Seiichi Manyama, Sep 08 2017

			First few rows are:
  1;
  0, 1;
  0, 1;
  0, 1, -1;
  0, 1, -1;
  0, 1, -2;
  0, 1, -2, -1;
  0, 1, -3, -1;
  0, 1, -3, -2;
  0, 1, -4, -3;
  0, 1, -4, -4, 1.

Seiichi Manyama, Rows n = 0..481, flattened

Row sums give (-1)*A278400.
Columns 0-1 give A000007, A000012.
Cf. A292047.

A330759 Number T(n,k) of set partitions into k blocks of strict integer partitions of n; triangle T(n,k), n>=0, 0<=k<=A003056(n), read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 2, 1, 0, 2, 1, 0, 3, 2, 0, 4, 5, 1, 0, 5, 6, 1, 0, 6, 9, 2, 0, 8, 13, 3, 0, 10, 23, 10, 1, 0, 12, 27, 11, 1, 0, 15, 40, 19, 2, 0, 18, 51, 26, 3, 0, 22, 71, 40, 5, 0, 27, 100, 73, 16, 1, 0, 32, 127, 93, 19, 1, 0, 38, 163, 132, 31, 2, 0, 46, 215, 184, 45, 3

Offset: 0

Alois P. Heinz, Dec 29 2019

			T(10,1) = 10: (10), 1234, 127, 136, 145, 19, 235, 28, 37, 46.
T(10,2) = 23: 123|4, 124|3, 12|34, 12|7, 134|2, 13|24, 13|6, 14|23, 14|5, 15|4, 16|3, 17|2, 1|234, 1|27, 1|36, 1|45, 1|9, 23|5, 25|3, 2|35, 2|8, 3|7, 4|6.
T(10,3) = 10: 12|3|4, 13|2|4, 14|2|3, 1|23|4, 1|24|3, 1|2|34, 1|2|7, 1|3|6, 1|4|5, 2|3|5.
T(10,4) = 1: 1|2|3|4.
Triangle T(n,k) begins:
  1;
  0,  1;
  0,  1;
  0,  2,   1;
  0,  2,   1;
  0,  3,   2;
  0,  4,   5,  1;
  0,  5,   6,  1;
  0,  6,   9,  2;
  0,  8,  13,  3;
  0, 10,  23, 10,  1;
  0, 12,  27, 11,  1;
  0, 15,  40, 19,  2;
  0, 18,  51, 26,  3;
  0, 22,  71, 40,  5;
  0, 27, 100, 73, 16, 1;
  ...

Alois P. Heinz, Rows n = 0..1000, flattened
Wikipedia, Partition of a set

Columns k=0-1 give: A000007, A000009 (for n>0).
Row sums give A294617.
Cf. A000041, A000096, A000217, A003056, A072706, A330460 (another version), A330765.

b:= proc(n, i, k) option remember; `if`(i*(i+1)/2 b(n-i, t, k)*k
        +b(n-i, t, k+1))(min(n-i, i-1))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2, 0)):
seq(T(n), n=0..20);

b[n_, i_, k_] := b[n, i, k] = If[i(i+1)/2 < n, 0,
     If[n == 0, x^k, b[n, i-1, k] + With[{t = Min[n-i, i-1]},
     b[n-i, t, k]*k + b[n-i, t, k+1]]]];
T[n_] := CoefficientList[b[n, n, 0], x];
T /@ Range[0, 20] // Flatten (* Jean-François Alcover, Mar 12 2021, after Alois P. Heinz *)

Sum_{k=0..2} T(n,k) = A072706(n).
Sum_{k=1..A003056(n)} k * T(n,k) = A330765(n).
T(A000217(n),n) = 1.
T(A000096(n),n) = A000041(n).
T(n*(n+1)/2+j,n) = A000041(j) for 0 <= j <= n.

A265020 Total sum T(n,k) of number of lambda-parking functions of partitions lambda of n into exactly k distinct parts; triangle T(n,k), n>=0, 0<=k<=A003056(n), read by rows.

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 3, 3, 0, 4, 5, 0, 5, 15, 0, 6, 21, 16, 0, 7, 42, 25, 0, 8, 54, 68, 0, 9, 90, 142, 0, 10, 110, 248, 125, 0, 11, 165, 409, 189, 0, 12, 195, 710, 496, 0, 13, 273, 1033, 967, 0, 14, 315, 1562, 2096, 0, 15, 420, 2291, 3265, 1296, 0, 16, 476, 3180

Offset: 0

Alois P. Heinz, Nov 30 2015

Differs from A265208 first at T(5,2). See example.

			T(5,2) = 15 because there are two partitions of 5 into 2 distinct parts: [2,3] and [1,4]. And [2,3] has 8 lambda-parking functions: [1,1], [1,2], [1,3], [2,1], [2,2], [2,3], [3,1], [3,2] and [1,4] has 7: [1,1], [1,2], [1,3], [1,4], [2,1], [3,1], [4,1]. So [1,1], [1,2], [1,3], [2,1], [3,1] are counted twice.
Triangle T(n,k) begins:
00 :  1;
01 :  0,  1;
02 :  0,  2;
03 :  0,  3,   3;
04 :  0,  4,   5;
05 :  0,  5,  15;
06 :  0,  6,  21,   16;
07 :  0,  7,  42,   25;
08 :  0,  8,  54,   68;
09 :  0,  9,  90,  142;
10 :  0, 10, 110,  248,  125;
11 :  0, 11, 165,  409,  189;
12 :  0, 12, 195,  710,  496;
13 :  0, 13, 273, 1033,  967;
14 :  0, 14, 315, 1562, 2096;
15 :  0, 15, 420, 2291, 3265, 1296;
16 :  0, 16, 476, 3180, 6057, 1921;

Alois P. Heinz, Rows n = 0..100, flattened
R. Stanley, Parking Functions, 2011

Row sums give A265016.
Columns k=0-1 give: A000007, A000027.
Cf. A000217, A000272, A003056, A265208.

p:= l-> (n-> n!*LinearAlgebra[Determinant](Matrix(n, (i, j)
         -> (t->`if`(t<0, 0, l[i]^t/t!))(j-i+1))))(nops(l)):
g:= (n, i, l)->  `if`(i*(i+1)/2n, 0, g(n-i, i-1, [i, l[]])))):
T:= n-> (f-> seq(coeff(f, x, i), i=0..degree(f)))(g(n$2, [])):
seq(T(n), n=0..20);

p[l_] := With[{n = Length[l]}, n!*Det[Table[With[{t = j - i + 1}, l[[i]]^t/t!], {i, 1, n}, {j, 1, n}]]];
g[n_, i_, l_] := If[i*(i + 1)/2 < n, 0, If[n == 0, p[l]*x^Length[l], g[n, i - 1, l] + If[i > n, 0, g[n - i, i - 1, Join[{i}, l]]]]];
T[n_] := If[n == 0, {1}, CoefficientList[g[n, n, {}], x]];
Table[T[n], {n, 0, 20}] // Flatten (* Jean-François Alcover, Jul 29 2024, after Alois P. Heinz *)

T(A000217(n),n) = A000272(n+1).

A288267 Triangle read by rows: T(n,k) = T(n,k+1) + T(n-k,k-1) with T(0,0) = 1 and T(n,k) = 0 if k<0 or k > A003056(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 1, 3, 3, 1, 5, 5, 2, 9, 9, 4, 1, 15, 15, 6, 1, 26, 26, 11, 2, 45, 45, 19, 4, 78, 78, 33, 7, 1, 135, 135, 57, 12, 1, 234, 234, 99, 21, 2, 406, 406, 172, 37, 4, 704, 704, 298, 64, 7, 1222, 1222, 518, 112, 13, 1, 2120, 2120, 898, 194, 22, 1, 3679

Offset: 0

Seiichi Manyama, Sep 01 2017

			First few rows are:
   1;
   1,  1;
   1,  1;
   2,  2, 1;
   3,  3, 1;
   5,  5, 2;
   9,  9, 4, 1;
  15, 15, 6, 1.

Seiichi Manyama, Rows n = 0..481, flattened

Columns 0+1,2 give A005169, A289080 (for n>0).

Cf. A003056.

T:= proc(n,k) option remember; `if`(k<0 or k*(k+1)/2>n, 0,
      `if`(n=0, 1, T(n, k+1)+T(n-k, k-1)))
    end:
seq(seq(T(n, k), k=0..floor((sqrt(1+8*n)-1)/2)), n=0..20); # Alois P. Heinz, Sep 01 2017

T[n_, k_] := T[n, k] = If[k < 0 || k(k+1)/2 > n, 0, If[n == 0, 1, T[n, k+1] + T[n-k, k-1]]];
Table[T[n, k], {n, 0, 20}, {k, 0, Floor[(Sqrt[8n+1]-1)/2]}] // Flatten (* Jean-François Alcover, Nov 14 2020, after Alois P. Heinz *)

A291940 Triangle read by rows: T(n,k) = T(n-k,k-1) - 2*T(n-k,k) + T(n-k,k+1) with T(0,0) = 1 for 0 <= k <= A003056(n).

Original entry on oeis.org

1, 0, 1, 0, -2, 0, 4, 1, 0, -7, -2, 0, 12, 2, 0, -22, -3, 1, 0, 41, 8, -2, 0, -74, -15, 2, 0, 133, 23, -5, 0, -243, -42, 12, 1, 0, 444, 82, -19, -2, 0, -806, -147, 33, 2, 0, 1465, 261, -65, -5, 0, -2669, -479, 118, 10, 0, 4859, 878, -211, -15, 1, 0, -8840, -1593, 386

Offset: 0

Seiichi Manyama, Sep 06 2017

			First few rows are:
  1;
  0,    1;
  0,   -2;
  0,    4,   1;
  0,   -7,  -2;
  0,   12,   2;
  0,  -22,  -3,  1;
  0,   41,   8, -2;
  0,  -74, -15,  2;
  0,  133,  23, -5;
  0, -243, -42, 12, 1.

Seiichi Manyama, Rows n = 0..481, flattened

Row sums give A291942.
Columns 0-1 give A000007, (-1)*A275762 (for n>0).
Cf. A008289, A291895, A291929.

A291970 Triangle read by rows: T(n,k) = 2 * T(n-k,k-1) + T(n-k,k) with T(0,0) = 1 for 0 <= k <= A003056(n).

Original entry on oeis.org

1, 0, 2, 0, 2, 0, 2, 4, 0, 2, 4, 0, 2, 8, 0, 2, 8, 8, 0, 2, 12, 8, 0, 2, 12, 16, 0, 2, 16, 24, 0, 2, 16, 32, 16, 0, 2, 20, 40, 16, 0, 2, 20, 56, 32, 0, 2, 24, 64, 48, 0, 2, 24, 80, 80, 0, 2, 28, 96, 96, 32, 0, 2, 28, 112, 144, 32, 0, 2, 32, 128, 176, 64, 0, 2, 32

Offset: 0

Seiichi Manyama, Sep 07 2017

			First few rows are:
  1;
  0, 2;
  0, 2;
  0, 2,  4;
  0, 2,  4;
  0, 2,  8;
  0, 2,  8,  8;
  0, 2, 12,  8;
  0, 2, 12, 16;
  0, 2, 16, 24;
  0, 2, 16, 32, 16.

Seiichi Manyama, Rows n = 0..481, flattened

Row sums give A032302.
Columns 0-1 give A000007, A007395.
Cf. A008289 (m=1), this sequence (m=2), A291971 (m=3).

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A116942 Permutation of the natural numbers in conjunction with A116939 and A003056.

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A262048 Irregular triangle read by rows: T(n, k) = A262045(n, k) * A237593(n, k), n >= 1, 1 <= k <= 2 * A003056(n).

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A265208 Total number T(n,k) of lambda-parking functions induced by all partitions of n into exactly k distinct parts; triangle T(n,k), n>=0, 0<=k<=A003056(n), read by rows.

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A292047 Triangle read by rows: T(n,k) = (-1)^k * T(n-k,k-1) + T(n-k,k) with T(0,0) = 1 for 0 <= k <= A003056(n).

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A292049 Triangle read by rows: T(n,k) = (-1)^(k-1) * T(n-k,k-1) + T(n-k,k) with T(0,0) = 1 for 0 <= k <= A003056(n).

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A330759 Number T(n,k) of set partitions into k blocks of strict integer partitions of n; triangle T(n,k), n>=0, 0<=k<=A003056(n), read by rows.

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A265020 Total sum T(n,k) of number of lambda-parking functions of partitions lambda of n into exactly k distinct parts; triangle T(n,k), n>=0, 0<=k<=A003056(n), read by rows.

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A288267 Triangle read by rows: T(n,k) = T(n,k+1) + T(n-k,k-1) with T(0,0) = 1 and T(n,k) = 0 if k<0 or k > A003056(n).

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Maple

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A291940 Triangle read by rows: T(n,k) = T(n-k,k-1) - 2*T(n-k,k) + T(n-k,k+1) with T(0,0) = 1 for 0 <= k <= A003056(n).

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