A141724 Triangle T(n, k) = Sum_{m=0..k} Sum_{j=0..m} Multinomial(n-k-m-j, j, m, k), read by rows.

1, 1, 1, 1, 4, 1, 1, 15, 6, 1, 1, 40, 36, 8, 1, 1, 85, 160, 60, 10, 1, 1, 156, 615, 340, 90, 12, 1, 1, 259, 2016, 1715, 595, 126, 14, 1, 1, 400, 5656, 7616, 3500, 952, 168, 16, 1, 1, 585, 13896, 30408, 18396, 6300, 1428, 216, 18, 1, 1, 820, 30645, 109320, 88620, 37044, 10500, 2040, 270, 20, 1

Offset: 0

Author

Roger L. Bagula, Sep 12 2008

Keywords

• nonn
• tabl

Comments

Row sums are: {1, 2, 6, 23, 86, 317, 1215, 4727, 18310, 71249, 279281, ...}.

Examples

			Triangle begins as:
  1;
  1,    1;
  1,    4,     1;
  1,   15,     6,      1;
  1,   40,    36,      8,      1;
  1,   85,   160,     60,     10,      1;
  1,  156,   615,    340,     90,     12,     1;
  1,  259,  2016,   1715,    595,    126,    14,     1;
  1,  400,  5656,   7616,   3500,    952,   168,    16,    1;
  1,  585, 13896,  30408,  18396,   6300,  1428,   216,   18,   1;
  1,  820, 30645, 109320,  88620,  37044, 10500,  2040,  270,  20,  1;
  1, 1111, 61930, 352605, 393030, 200508, 67914, 16500, 2805, 330, 22, 1;
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened
• Mohammad K. Azarian, A Double Sum, Problem 440, College Mathematics Journal, Vol. 21, No. 5, Nov. 1990, p. 424. Solution published in Vol. 22. No. 5, Nov. 1991, pp. 448-449.

Crossrefs

Cf. A065109.

Programs

Magma

F:= Factorial; [[ (&+[ (&+[ n lt k+j+m select 0 else F(n)/(F(k)*F(j)*F(n-k-j-m)*F(m)): j in [0..m]]) : m in [0..k]]): k in [0..n]]: n in [0..12]]; // G. C. Greubel, Mar 29 2021

Mathematica

Table[Sum[Sum[Multinomial[n-k-m-j,m,k,j], {j,0,m}], {m,0,k}], {n,0,12}, {k,0,n}]//Flatten

Sage

f=factorial; flatten([[sum( sum(0 if nG. C. Greubel, Mar 29 2021

Formula

T(n, k) = Sum_{m=0..k} Sum_{j=0..m} Multinomial(n-k-m-j, j, m, k).

Extensions

Edited by G. C. Greubel, Mar 29 2021