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

0,5

Comments

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

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.

Formula

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

Example

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;

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

Prog

(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
(Sage) f=factorial; flatten([[sum( sum(0 if n<k+j+m else f(n)/(f(k)*f(j)*f(n-k-j-m)*f(m)) for j in (0..m)) for m in (0..k)) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 29 2021

Crossrefs

Cf. A065109.

Sequence in context: A177984 A157013 A346876 * A208956 A271705 A320280

Adjacent sequences: A141721 A141722 A141723 * A141725 A141726 A141727

Keyword

nonn,tabl

Author

Roger L. Bagula, Sep 12 2008

Extensions

Edited by G. C. Greubel, Mar 29 2021

Status

approved