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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.

%I A141724 #11 Mar 29 2021 01:07:18
%S A141724 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,
%T A141724 12,1,1,259,2016,1715,595,126,14,1,1,400,5656,7616,3500,952,168,16,1,
%U A141724 1,585,13896,30408,18396,6300,1428,216,18,1,1,820,30645,109320,88620,37044,10500,2040,270,20,1
%N A141724 Triangle T(n, k) = Sum_{m=0..k} Sum_{j=0..m} Multinomial(n-k-m-j, j, m, k), read by rows.
%C A141724 Row sums are: {1, 2, 6, 23, 86, 317, 1215, 4727, 18310, 71249, 279281, ...}.
%H A141724 G. C. Greubel, <a href="/A141724/b141724.txt">Rows n = 0..50 of the triangle, flattened</a>
%H A141724 Mohammad K. Azarian, <a href="https://www.jstor.org/stable/2686918">A Double Sum, Problem 440</a>, College Mathematics Journal, Vol. 21, No. 5, Nov. 1990, p. 424. <a href="https://www.jstor.org/stable/2686609">Solution</a> published in Vol. 22. No. 5, Nov. 1991, pp. 448-449.
%F A141724 T(n, k) = Sum_{m=0..k} Sum_{j=0..m} Multinomial(n-k-m-j, j, m, k).
%e A141724 Triangle begins as:
%e A141724   1;
%e A141724   1,    1;
%e A141724   1,    4,     1;
%e A141724   1,   15,     6,      1;
%e A141724   1,   40,    36,      8,      1;
%e A141724   1,   85,   160,     60,     10,      1;
%e A141724   1,  156,   615,    340,     90,     12,     1;
%e A141724   1,  259,  2016,   1715,    595,    126,    14,     1;
%e A141724   1,  400,  5656,   7616,   3500,    952,   168,    16,    1;
%e A141724   1,  585, 13896,  30408,  18396,   6300,  1428,   216,   18,   1;
%e A141724   1,  820, 30645, 109320,  88620,  37044, 10500,  2040,  270,  20,  1;
%e A141724   1, 1111, 61930, 352605, 393030, 200508, 67914, 16500, 2805, 330, 22, 1;
%t A141724 Table[Sum[Sum[Multinomial[n-k-m-j,m,k,j], {j,0,m}], {m,0,k}], {n,0,12}, {k,0,n}]//Flatten
%o A141724 (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
%o A141724 (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
%Y A141724 Cf. A065109.
%K A141724 nonn,tabl
%O A141724 0,5
%A A141724 _Roger L. Bagula_, Sep 12 2008
%E A141724 Edited by _G. C. Greubel_, Mar 29 2021