A144385 -id:A144385 - cp's OEIS frontend

A144399 Triangle in A144385 with rows left-adjusted.

1, 1, 1, 1, 1, 3, 7, 10, 10, 1, 6, 25, 75, 175, 280, 280, 1, 10, 65, 315, 1225, 3780, 9100, 15400, 15400, 1, 15, 140, 980, 5565, 26145, 102025, 323400, 800800, 1401400, 1401400, 1, 21, 266, 2520, 19425, 125895, 695695, 3273270, 12962950

Offset: 0

David Applegate and N. J. A. Sloane, Dec 07 2008

Row n has 2n+1 terms.

			Triangle begins:
1
1, 1, 1
1, 3, 7, 10, 10
1, 6, 25, 75, 175, 280, 280
1, 10, 65, 315, 1225, 3780, 9100, 15400, 15400

Alois P. Heinz, Rows n = 0..100, flattened
Moa Apagodu, David Applegate, N. J. A. Sloane, and Doron Zeilberger, Analysis of the Gift Exchange Problem, arXiv:1701.08394, 2017.
David Applegate and N. J. A. Sloane, The Gift Exchange Problem (arXiv:0907.0513, 2009)

Cf. A144385. Row sums give A144416.

b:= proc(n) option remember; expand(`if`(n=0, 1, add(
       b(n-j)*binomial(n-1, j-1), j=1..min(3, n))*x))
    end:
T:= (n, k)-> coeff(b(k), x, n):
seq(seq(T(n, k), k=n..3*n), n=0..6);  # Alois P. Heinz, May 31 2018

b[n_] := b[n] = Expand[If[n == 0, 1, Sum[b[n - j]*Binomial[n - 1, j - 1], {j, 1, Min[3, n]}]*x]];
T[n_, k_] := Coefficient[b[k], x, n];
Table[T[n, k], {n, 0, 6}, { k, n, 3n}] // Flatten (* Jean-François Alcover, Jul 10 2018, after Alois P. Heinz *)

A144402 Triangle in A144385 read downwards by columns.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 3, 1, 0, 0, 7, 6, 1, 0, 0, 10, 25, 10, 1, 0, 0, 10, 75, 65, 15, 1, 0, 0, 0, 175, 315, 140, 21, 1, 0, 0, 0, 280, 1225, 980, 266, 28, 1, 0, 0, 0, 280, 3780, 5565, 2520, 462, 36, 1, 0, 0, 0, 0, 9100, 26145, 19425, 5670, 750, 45, 1, 0, 0, 0, 0, 15400

Offset: 0

David Applegate and N. J. A. Sloane, Dec 07 2008

The Bell transform of the sequence "g(n) = 1 if n<3 else 0". For the definition of the Bell transform see A264428. - Peter Luschny, Jan 19 2016

Moa Apagodu, David Applegate, N. J. A. Sloane, and Doron Zeilberger, Analysis of the Gift Exchange Problem, arXiv:1701.08394 [math.CO], 2017.
David Applegate and N. J. A. Sloane, The Gift Exchange Problem, arXiv:0907.0513 [math.CO], 2009.

Cf. A111246.

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[#<3, 1, 0]&, rows];
Table[M[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jul 14 2018, after Peter Luschny *)

# uses[bell_matrix from A264428]
bell_matrix(lambda n: 1 if n<3 else 0, 12) # Peter Luschny, Jan 19 2016

A144417 Triangle in A144385 read upwards by columns.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 6, 7, 0, 0, 1, 10, 25, 10, 0, 0, 1, 15, 65, 75, 10, 0, 0, 1, 21, 140, 315, 175, 0, 0, 0, 1, 28, 266, 980, 1225, 280, 0, 0, 0, 1, 36, 462, 2520, 5565, 3780, 280, 0, 0, 0, 1, 45, 750, 5670, 19425, 26145, 9100, 0, 0, 0, 0, 1, 55

Offset: 0

David Applegate and N. J. A. Sloane, Dec 07 2008

			Triangle begins:
1
1,0
1,1,0
1,3,1,0
1,6,7,0,0
1,10,25,10,0,0
1,15,65,75,10,0,0
1,21,140,315,175,0,0,0

Column sums give A001680. Cf. A144385.

A144416 a(n) is the total number of partitions of [1, 2, ..., k] into exactly n blocks, each of size 1, 2 or 3, for 0 <= k <= 3n.

Original entry on oeis.org

1, 3, 31, 842, 45296, 4061871, 546809243, 103123135501, 25942945219747, 8394104851717686, 3395846808758759686, 1679398297627675722593, 996789456118195908366641, 699283226713639676370419067, 572385833490097906671186099971, 540635257271794961275858251107746, 583630397618757664934692641037584628

Offset: 0

David Applegate and N. J. A. Sloane, Dec 07 2008, Dec 17 2008

nonn

Also, number of scenarios in the Gift Exchange Game when a gift can be stolen at most twice. - N. J. A. Sloane, Jan 25 2017

			a(0) = 1;
a(1) = 3: {1} {12} {123}
a(2) = 31: {1,2} {1,23} {2,13} {3,12} {1,234} {2,134} {3,124} {4,123}
{12,34} {13,24} {14,23} {12,345} {13,245} {14,235} {15,234} {23,145} {24,135}
{25,134} {34,125} {35,124} {45,123} {123,456} {124,356} {125,346} {126,345}
{134,256} {135,246} {136,245} {145,236} {146,235} {156,234}.

David Applegate and N. J. A. Sloane, Table of n, a(n) for n = 0..100
Moa Apagodu, David Applegate, N. J. A. Sloane, and Doron Zeilberger, Analysis of the Gift Exchange Problem, arXiv:1701.08394 [math.CO], 2017.
Moa Apagodu, David Applegate, N. J. A. Sloane, and Doron Zeilberger, On-Line Appendix I to "Analysis of the gift exchange problem", giving Type D recurrences for G_1(n) through G_15(n) (see A001515, A144416, A144508, A144509, A149187, A281358-A281361)
Moa Apagodu, David Applegate, N. J. A. Sloane, and Doron Zeilberger, On-Line Appendix II to "Analysis of the gift exchange problem", giving Type C recurrences for G_1(n) through G_15(n) (see A001515, A144416, A144508, A144509, A149187, A281358-A281361)
David Applegate and N. J. A. Sloane, The Gift Exchange Problem, arXiv:0907.0513 [math.CO], 2009.

Row sums of A144385. Slice sums of A144626.

The gift scenarios sequences when a gift can be stolen at most s times, for s = 1..9, are A001515, A144416, A144508, A144509, A149187, A281358, A281359, A281360, A281361.

t[n_, n_] = 1; t[n_ /; n >= 0, k_] /; 0 <= k <= 3*n := t[n, k] = t[n-1, k-1] + (k-1)*t[n-1, k-2] + (1/2)*(k-1)*(k-2)*t[n-1, k-3]; t[, ] = 0; a[n_] := Sum[t[n, k], {k, 0, 3*n}]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Feb 18 2017 *)

{a(n) = sum(i=n, 3*n, i!*polcoef(sum(j=1, 3, x^j/j!)^n, i))/n!} \\ Seiichi Manyama, May 22 2019

a(n) = Sum_{ b,c >= 0, b+c <= n } (n+b+2c)!/ ((n-b-c)! b! c! 2^b 6^c).

The sum is dominated by the b=0, c=n term, so a(n) ~ constant*(3*n)!/(n!*6^n).

A144299 Triangle of Bessel numbers read by rows. Row n gives T(n,n), T(n,n-1), T(n,n-2), ..., T(n,0) for n >= 0.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 3, 0, 0, 1, 6, 3, 0, 0, 1, 10, 15, 0, 0, 0, 1, 15, 45, 15, 0, 0, 0, 1, 21, 105, 105, 0, 0, 0, 0, 1, 28, 210, 420, 105, 0, 0, 0, 0, 1, 36, 378, 1260, 945, 0, 0, 0, 0, 0, 1, 45, 630, 3150, 4725, 945, 0, 0, 0, 0, 0, 1, 55, 990, 6930, 17325, 10395, 0, 0, 0, 0, 0, 0

Offset: 0

David Applegate and N. J. A. Sloane, Dec 06 2008

T(n,k) is the number of partitions of an n-set into k nonempty subsets, each of size at most 2.
The Grosswald and Choi-Smith references give many further properties and formulas.
Considered as an infinite lower triangular matrix T, lim_{n->infinity} T^n = A118930: (1, 1, 2, 4, 13, 41, 166, 652, ...) as a vector. - Gary W. Adamson, Dec 08 2008

			Triangle begins:
  n:
  0: 1
  1: 1  0
  2: 1  1   0
  3: 1  3   0    0
  4: 1  6   3    0   0
  5: 1 10  15    0   0  0
  6: 1 15  45   15   0  0  0
  7: 1 21 105  105   0  0  0  0
  8: 1 28 210  420 105  0  0  0  0
  9: 1 36 378 1260 945  0  0  0  0  0
  ...
The row sums give A000085.
For some purposes it is convenient to rotate the triangle by 45 degrees:
  1 0 0 0 0  0  0   0   0    0    0     0 ...
    1 1 0 0  0  0   0   0    0    0     0 ...
      1 3 3  0  0   0   0    0    0     0 ...
        1 6 15 15   0   0    0    0     0 ...
          1 10 45 105 105    0    0     0 ...
             1 15 105 420  945  945     0 ...
                1  21 210 1260 4725 10395 ...
                    1  28  378 3150 17325 ...
                        1   36  630  6930 ...
                             1   45   990 ...
  ...
The latter triangle is important enough that it has its own entry, A144331. Here the column sums give A000085 and the rows sums give A001515.
If the entries in the rotated triangle are denoted by b1(n,k), n >= 0, k <= 2n, then we have the recurrence b1(n, k) = b1(n - 1, k - 1) + (k - 1)*b1(n - 1, k - 2).
Then b1(n,k) is the number of partitions of [1, 2, ..., k] into exactly n blocks, each of size 1 or 2.

E. Grosswald, Bessel Polynomials, Lecture Notes Math., Vol. 698, 1978.

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
David Applegate and N. J. A. Sloane, The Gift Exchange Problem, arXiv:0907.0513 [math.CO], 2009.
J. Y. Choi and J. D. H. Smith, On the unimodality and combinatorics of Bessel numbers, Discrete Math., 264 (2003), 45-53.
Tom Copeland, Infinitesimal Generators, the Pascal Pyramid, and the Witt and Virasoro Algebras
Toufik Mansour, Matthias Schork, and Mark Shattuck, The Generalized Stirling and Bell Numbers Revisited, Journal of Integer Sequences, Vol. 15 (2012), #12.8.3.
T. Mansour and M. Shattuck, Partial matchings and pattern avoidance, Appl. Anal. Discrete Math. 7 (2013) 25-50.

Other versions of this same triangle are given in A111924 (which omits the first row), A001498 (which left-adjusts the rows in the bottom view), A001497 and A100861. Row sums give A000085.

Cf. A000085, A000217, A001464, A004526, A118930, A144385, A144643..

a144299 n k = a144299_tabl !! n !! k
a144299_row n = a144299_tabl !! n
a144299_tabl = [1] : [1, 0] : f 1 [1] [1, 0] where
   f i us vs = ws : f (i + 1) vs ws where
               ws = (zipWith (+) (0 : map (i *) us) vs) ++ [0]
-- Reinhard Zumkeller, Jan 01 2014

A144299:= func< n,k | k le Floor(n/2) select Factorial(n)/(Factorial(n-2*k)*Factorial(k)*2^k) else 0 >;
[A144299(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Sep 29 2023

Maple code producing the rotated version:
b1 := proc(n, k)
option remember;
if n = k then 1;
elif k < n then 0;
elif n < 1 then 0;
else b1(n - 1, k - 1) + (k - 1)*b1(n - 1, k - 2);
end if;
end proc;
for n from 0 to 12 do lprint([seq(b1(n,k),k=0..2*n)]); od:

T[n_,0]=0; T[1,1]=1; T[2,1]=1; T[n_, k_]:= T[n-1,k-1] + (n-1)T[n-2,k-1];
Table[T[n,k], {n,12}, {k,n,1,-1}]//Flatten (* Robert G. Wilson v *)
Table[If[k<=Floor[n/2],n!/((n-2 k)! k! 2^k),0], {n, 0, 12},{k,0,n}]//Flatten (* Stefano Spezia, Jun 15 2023 *)

def A144299(n,k): return factorial(n)/(factorial(n-2*k)*factorial(k)*2^k) if k <= (n//2) else 0
flatten([[A144299(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Sep 29 2023

T(n, k) = T(n-1, k-1) + (n-1)*T(n-2, k-1).
E.g.f.: Sum_{k >= 0} Sum_{n = 0..2k} T(n,k) y^k x^n/n! = exp(y(x+x^2/2)). (The coefficient of y^k is the e.g.f. for the k-th row of the rotated triangle shown below.)
T(n, k) = n!/((n - 2*k)!*k!*2^k) for 0 <= k <= floor(n/2) and 0 otherwise. - Stefano Spezia, Jun 15 2023
From G. C. Greubel, Sep 29 2023: (Start)
T(n, 1) = A000217(n-1).
Sum_{k=0..n} T(n,k) = A000085(n).
Sum_{k=0..n} (-1)^k*T(n,k) = A001464(n). (End)

Offset fixed by Reinhard Zumkeller, Jan 01 2014

A144633 Triangle of 3-restricted Stirling numbers of the first kind (T(n,k), 0 <= k <= n), read by rows.

Original entry on oeis.org

1, 0, 1, 0, -1, 1, 0, 2, -3, 1, 0, -5, 11, -6, 1, 0, 10, -45, 35, -10, 1, 0, 35, 175, -210, 85, -15, 1, 0, -910, -315, 1225, -700, 175, -21, 1, 0, 11935, -6265, -5670, 5565, -1890, 322, -28, 1, 0, -134750, 139755, -5005, -39270, 19425, -4410, 546, -36, 1

Offset: 0

N. J. A. Sloane, Jan 21 2009

Definition: take the triangle in A144385, write it as an (infinite) upper triangular square matrix, invert it and transpose it.

The Bell transform of A144636(n+1). Also the inverse Bell transform of the sequence "g(n) = 1 if n<3 else 0". For the definition of the Bell transform see A264428. - Peter Luschny, Jan 19 2016

			Triangle begins:
1;
0,    1;
0,   -1,    1;
0,    2,   -3,    1;
0,   -5,   11,   -6,    1;
0,   10,  -45,   35,  -10,   1;
0,   35,  175, -210,   85, -15,   1;
0, -910, -315, 1225, -700, 175, -21,  1;

J. Y. Choi and J. D. H. Smith, On the combinatorics of multi-restricted numbers, Ars. Com., 75(2005), pp. 44-63.

Alois P. Heinz, Rows n = 0..140, flattened
J. Y. Choi and J. D. H. Smith, The Tri-restricted Numbers and Powers of Permutation Representations, J. Comb. Math. Comb. Comp. 42 (2002), 113-125.
J. Y. Choi and J. D. H. Smith, On the Unimodality and Combinatorics of the Bessel Numbers, Discrete Math., 264 (2003), 45-53.
J. Y. Choi et al., Reciprocity for multirestricted Stirling numbers, J. Combin. Theory 113 A (2006), 1050-1060.

For another version of this triangle see A144634.
Columns give A144636-A144639.
Cf. A144402.

A:= proc(n,k) option remember; if n=k then 1 elif k A(i-1, j-1))^(-1) end:
T:= (n,k)-> M(n+1)[k+1, n+1]:
seq(seq(T(n,k), k=0..n), n=0..12); # Alois P. Heinz, Oct 23 2009

max = 10; t[n_, n_] = 1; t[n_ /; n >= 0, k_] /; (0 <= k <= 3*n) := t[n, k] = t[n-1, k-1] + (k-1)*t[n-1, k-2] + (1/2)*(k-1)*(k-2)*t[n-1, k-3]; t[, ] = 0; A144633 = Table[t[n, k], {n, 0, max}, {k, 0, max}] // Inverse // Transpose; Table[A144633[[n, k]], {n, 1, max}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jan 14 2014 *)

# uses[bell_matrix from A264428]
bell_matrix(lambda n: A144636(n+1), 10) # Peter Luschny, Jan 18 2016

Corrected and extended by Alois P. Heinz, Oct 23 2009

A111246 Triangle read by rows: a(n,k) = number of partitions of an n-set into exactly k nonempty subsets, each of size <= 3.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 0, 7, 6, 1, 0, 10, 25, 10, 1, 0, 10, 75, 65, 15, 1, 0, 0, 175, 315, 140, 21, 1, 0, 0, 280, 1225, 980, 266, 28, 1, 0, 0, 280, 3780, 5565, 2520, 462, 36, 1, 0, 0, 0, 9100, 26145, 19425, 5670, 750, 45, 1, 0, 0, 0, 15400, 102025, 125895, 56595, 11550, 1155

Offset: 1

Ji Young Choi, Oct 31 2005

a(n,k) = 0 if k > n; a(n,k) = 0 if n > 0 and k < 0; a(n,k) can be extended to negative n and k, just as the Stirling numbers or Pascal's triangle can be extended. The present triangle is called the tri-restricted Stirling numbers of the second kind.

Also the Bell transform of the sequence "a(n) = 1 if n<3 else 0". For the definition of the Bell transform see A264428. - Peter Luschny, Jan 27 2016

			a(1,1)=1;
a(2,1)=1; a(2,2)=1;
a(3,1)=1; a(3,2)=3; a(3,3)=1;
a(4,1)=0; a(4,2)=7; a(4,3)=6; a(4,4)=1;
a(5,1)=0; a(5,2)=10; a(5,3)=25; a(5,4)=10; a(5,5)=1;
a(6,1)=0; a(6,2)=10; a(6,3)=75; a(6,4)=65; a(6,5)=15; a(6,6)=1; ...

J. Y. Choi and J. D. H. Smith, On the combinatorics of multi-restricted numbers, Ars. Com., 75(2005), pp. 44-63.

J. Y. Choi and J. D. H. Smith, The Tri-restricted Numbers and Powers of Permutation Representations, J. Comb. Math. Comb. Comp. 42 (2002), 113-125.
J. Y. Choi and J. D. H. Smith, On the Unimodality and Combinatorics of the Bessel Numbers, Discrete Math., 264 (2003), 45-53.
J. Y. Choi et al., Reciprocity for multirestricted Stirling numbers, J. Combin. Theory 113 A (2006), 1050-1060.

A144385 and A144402 are other versions of this same triangle.

Cf. A001680, A008277 (Stirling numbers).

# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0,...) as column 0.
BellMatrix(n -> `if`(n<3,1,0), 10); # Peter Luschny, Jan 27 2016

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[# < 3, 1, 0]&, rows];
Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 24 2018, after Peter Luschny *)

row(n) = {x='x+O('x^(n+1)); polcoeff(serlaplace(exp(y*(x+x^2/2+x^3/6))), n, 'x); }
tabl(nn) = for(n=1, nn, print(Vecrev(row(n)/y))) \\ Jinyuan Wang, Dec 21 2019

a(n, k) = a(n-1, k-1) + k*a(n-1, k) - binomial(n-1, 3)*a(n-4, k-1).
G.f. = Sum_{k_1+k_2+k_3=k, k_1+ 2k_2+3k_3=n} frac{n!}{(1!)^{k_1}(2!)^{k_2}(3!)^{k_3}k_1!k_2!k_3!}.
E.g.f.: exp(y*(x+x^2/2+x^3/6)). - Vladeta Jovovic, Nov 01 2005

More terms from Vladeta Jovovic, Nov 01 2005

Recurrence, offset and example corrected by David Applegate, Jan 16 2009

A144331 Triangle b(n,k) for n >= 0, 0 <= k <= 2n, read by rows. See A144299 for definition and properties.

Original entry on oeis.org

1, 0, 1, 1, 0, 0, 1, 3, 3, 0, 0, 0, 1, 6, 15, 15, 0, 0, 0, 0, 1, 10, 45, 105, 105, 0, 0, 0, 0, 0, 1, 15, 105, 420, 945, 945, 0, 0, 0, 0, 0, 0, 1, 21, 210, 1260, 4725, 10395, 10395, 0, 0, 0, 0, 0, 0, 0, 1, 28, 378, 3150, 17325, 62370, 135135, 135135, 0, 0, 0, 0, 0, 0

Offset: 0

David Applegate and N. J. A. Sloane, Dec 07 2008

Although this entry is the last of the versions of the underlying triangle to be added to the OEIS, for some applications it is the most important.
Row n has 2n+1 entries.
A001498 has a b-file.

			Triangle begins:
  1
  0 1 1
  0 0 1 3 3
  0 0 0 1 6 15 15
  0 0 0 0 1 10 45 105 105
  0 0 0 0 0  1 15 105 420  945  945
  0 0 0 0 0  0  1  21 210 1260 4725 10395 10395
  ...

Reinhard Zumkeller, Rows n = 0..100 of triangle, flattened
Moa Apagodu, David Applegate, N. J. A. Sloane, and Doron Zeilberger, Analysis of the Gift Exchange Problem, arXiv:1701.08394 [math.CO], 2017.
David Applegate and N. J. A. Sloane, The Gift Exchange Problem (arXiv:0907.0513, 2009)

Cf. A144299, A278990.
Row sums give A001515, column sums A000085.
Other versions of this triangle are given in A001497, A001498, A111924 and A100861.
See A144385 for a generalization.

a144331 n k = a144331_tabf !! n !! k
a144331_row n = a144331_tabf !! n
a144331_tabf = iterate (\xs ->
  zipWith (+) ([0] ++ xs ++ [0]) $ zipWith (*) (0:[0..]) ([0,0] ++ xs)) [1]
-- Reinhard Zumkeller, Nov 24 2014

A144331:= func< n,k | k le n-1 select 0 else Factorial(k)/(2^(k-n)*Factorial(k-n)*Factorial(2*n-k)) >;
[A144331(n,k): k in [0..2*n], n in [0..12]]; // G. C. Greubel, Oct 04 2023

Flatten[Table[PadLeft[Table[(n+k)!/(2^k*k!*(n-k)!), {k,0,n}], 2*n+1, 0], {n,0,12}]] (* Jean-François Alcover, Oct 14 2011 *)

def A144331(n, k): return 0 if kA144331(n,k) for k in range(2*n+1)] for n in range(13)]) # G. C. Greubel, Oct 04 2023

E.g.f.: Sum_{n >= 0} Sum_{k = 0..2n} b(n,k) y^n * x^k/k! = exp(x*y*(1 + x/2)).
b(n, k) = 2^(n-k)*k!/((2*n-k)!*(k-n)!).
Sum_{k=0..2*n} b(n, k) = A001515(n).
Sum_{n >= 0} b(n, k) = A000085(k).
From G. C. Greubel, Oct 04 2023: (Start)
T(n, k) = 0 for 0 <= k <= n-1, otherwise T(n, k) = k!/(2^(k-n)*(k-n)!*(2*n-k)!) for n <= k <= 2*n.
Sum_{k=0..2*n} (-1)^k * T(n, k) = A278990(n). (End)

A144643 Triangle read by rows: T(n,k) = number of partitions of [1..k] into n nonempty clumps of sizes 1, 2, 3 or 4 (n >= 0, 0 <= k <= 4n).

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 0, 0, 1, 3, 7, 15, 25, 35, 35, 0, 0, 0, 1, 6, 25, 90, 280, 770, 1855, 3675, 5775, 5775, 0, 0, 0, 0, 1, 10, 65, 350, 1645, 6930, 26425, 90475, 275275, 725725, 1576575, 2627625, 2627625, 0, 0, 0, 0, 0, 1, 15, 140, 1050, 6825, 39795, 211750, 1033725, 4629625

Offset: 0

David Applegate and N. J. A. Sloane, Jan 25 2009

			Irregular triangle begins:
  1;
  0, 1, 1, 1, 1;
  0, 0, 1, 3, 7, 15, 25,  35,  35;
  0, 0, 0, 1, 6, 25, 90, 280, 770, 1855, 3675, 5775, 5775;
  ...

G. C. Greubel, Rows n = 0..25 of the irregular triangle, flattened
Moa Apagodu, David Applegate, N. J. A. Sloane, and Doron Zeilberger, Analysis of the Gift Exchange Problem, arXiv:1701.08394 [math.CO], 2017.
David Applegate and N. J. A. Sloane, The Gift Exchange Problem, arXiv:0907.0513 [math.CO], 2009.

Row sums give A144508.
See A144644 and A144645 for other versions.
Cf. A144299, A144385.

function t(n,k)
  if k eq n then return 1;
  elif k le n-1 or n le 0 then return 0;
  else return (&+[Binomial(k-1,j)*t(n-1,k-j-1): j in [0..3]]);
  end if;
end function;
A144643:= func< n,k | t(n,k) >;
[A144643(n,k): k in [0..4*n], n in [0..8]]; // G. C. Greubel, Oct 11 2023

T := proc(n, k) option remember;
if n = k then 1;
elif k < n then 0;
elif n < 1 then 0;
else T(n - 1, k - 1) + (k - 1)*T(n - 1, k - 2) + 1/2*(k - 1)*(k - 2)*T(n - 1, k - 3) + 1/6*(k - 1)*(k - 2)*(k - 3)*T(n - 1, k - 4);
end if;
end proc;

T[n_, k_]:= T[n, k]= Which[n==k, 1, kJean-François Alcover, Mar 20 2014, after Maple *)
Table[BellY[k, n, {1,1,1,1}], {n,0,12}, {k,0,4*n}]//Flatten (* G. C. Greubel, Oct 11 2023 *)

@CachedFunction
def t(n,k):
    if (k==n): return 1
    elif (kA144643(n,k): return t(n,k)
flatten([[A144643(n,k) for k in range(4*n+1)] for n in range(13)]) # G. C. Greubel, Oct 11 2023

T(n, k) = Sum_{j=0..3} binomial(k-1, j) * T(n-1, k-j-1), with T(n, n) = 1, T(n, k) = 0 if n < 1 or n > k.

Sum_{k=0..4*n} T(n, k) = A144508(n).

A151338 Triangle read by rows: T(n,k) = number of partitions of [1..k] into n nonempty clumps of sizes 1, 2, 3, 4 or 5 (n >= 0, 0 <= k <= 5n).

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 1, 0, 0, 1, 3, 7, 15, 31, 56, 91, 126, 126, 0, 0, 0, 1, 6, 25, 90, 301, 938, 2737, 7455, 18711, 41811, 81081, 126126, 126126, 0, 0, 0, 0, 1, 10, 65, 350, 1701, 7686, 32725, 132055, 505351, 1824823, 6173167, 19339320, 55096041

Offset: 0

N. J. A. Sloane, May 13 2009

Row n has 5n+1 entries.

			The triangle begins:
1
0, 1, 1, 1, 1, 1
0, 0, 1, 3, 7, 15, 31, 56, 91, 126, 126
0, 0, 0, 1, 6, 25, 90, 301, 938, 2737, 7455, 18711, 41811, 81081, 126126, 126126
0, 0, 0, 0, 1, 10, 65, 350, 1701, 7686, 32725, 132055, 505351, 1824823, 6173167, 19339320, 55096041, 138654516, 295891596, 488864376, 488864376
0, 0, 0, 0, 0, 1, 15, 140, 1050, 6951, 42315, 241780, 1310925, 6782776, 33549516, 158533375, 713733020, 3046944901, 12246267033, 45892143297, 158167994985, 491492022021, 1336310771796, 3030225834636, 5194672859376, 5194672859376
...

Moa Apagodu, David Applegate, N. J. A. Sloane, and Doron Zeilberger, Analysis of the Gift Exchange Problem, arXiv:1701.08394, 2017.
David Applegate and N. J. A. Sloane, The Gift Exchange Problem (arXiv:0907.0513, 2009)

This is one of a sequence of triangles: A144331, A144385, A144643, A151338, A151359, ...

See A151509, A151510 for other versions.

cp's OEIS Frontend

A144399 Triangle in A144385 with rows left-adjusted.

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A144402 Triangle in A144385 read downwards by columns.

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A144417 Triangle in A144385 read upwards by columns.

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A144416 a(n) is the total number of partitions of [1, 2, ..., k] into exactly n blocks, each of size 1, 2 or 3, for 0 <= k <= 3n.

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A144299 Triangle of Bessel numbers read by rows. Row n gives T(n,n), T(n,n-1), T(n,n-2), ..., T(n,0) for n >= 0.

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A144633 Triangle of 3-restricted Stirling numbers of the first kind (T(n,k), 0 <= k <= n), read by rows.

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A111246 Triangle read by rows: a(n,k) = number of partitions of an n-set into exactly k nonempty subsets, each of size <= 3.

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