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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, 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
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
KEYWORD
nonn,tabl
AUTHOR
Roger L. Bagula, Sep 12 2008
EXTENSIONS
Edited by G. C. Greubel, Mar 29 2021
STATUS
approved