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%I #28 Nov 24 2016 09:46:50
%S 1,1,4,6,4,1,1,4,10,16,19,16,10,4,1,1,4,10,20,31,40,44,40,31,20,10,4,
%T 1,1,4,10,20,35,52,68,80,85,80,68,52,35,20,10,4,1,1,4,10,20,35,56,80,
%U 104,125,140,146,140,125,104,80,56,35,20,10,4,1
%N Triangle read by rows, in which row n gives coefficients in expansion of ((x^n - 1)/(x - 1))^4.
%C Sum of n-th row is n^4. The n-th row contains 4n-3 entries. Largest coefficients of each row are listed in A005900.
%C The n-th row is the fourth row of the n-nomial triangle. For example, row 2 (1,4,6,4,1) is the fourth row in the binomial triangle.
%C T(n,k) gives the number of possible ways of randomly selecting k cards from n-1 sets, each with four different playing cards. It is also the number of lattice paths from (0,0) to (4,k) using steps (1,0), (1,1), (1,2), ..., (1,n-1).
%H Juan Pablo Herrera P., <a href="/A277949/b277949.txt">Rows n=1..60 of the triangle, flattened</a>
%F T(n,k) = Sum_{i=k-n+1..k} A109439(T(n,i)).
%F T(n,k) = A000292(k+1) = (k+3)!/(k!*6) if 0 =< k < n,
%F T(n,k) = ((k+3)*(k+2)*(k+1)-4*(k-n+3)*(k-n+2)*(k-n+1))/6 if n =< k < 2*n,
%F T(n,k) = ((4*n-1-k)*(4*n-2-k)*(4*n-3-k)-4*(3*n-1-k)*(3*n-2-k)*(3*n-3-k))/6 if 2*n-3 =< k < 3*n-3,
%F T(n,k) = A000292(4*n-3-k) = (4*n-1-k)!/((4*n-4-k)!*6) if 3*n-3 =< k < 4n-3.
%e Triangle starts:
%e 1;
%e 1, 4, 6, 4, 1;
%e 1, 4, 10, 16, 19, 16, 10, 4, 1;
%e 1, 4, 10, 20, 31, 40, 44, 40, 31, 20, 10, 4, 1;
%e 1, 4, 10, 20, 35, 52, 68, 80, 85, 80, 68, 52, 35, 20, 10, 4, 1;
%e 1, 4, 10, 20, 35, 56, 80, 104, 125, 140, 146, 140, 125, 104, 80, 56, 35, 20, 10, 4, 1.
%e ...
%e There are T(3,2) = 10 ways to select 2 cards from two sets of four playing cards ABCD, namely, {AA}, {AB}, {AC}, {AD}, {BB}, {BC}, {BD}, {CC}, {CD}, and {DD}.
%t Table[CoefficientList[Series[((x^n - 1)/(x - 1))^4, {x, 0, 4 n}], x], {n, 6}] // Flatten (* _Michael De Vlieger_, Nov 10 2016 *)
%o (PARI) row(n) = Vec(((1 - x^n)/(1 - x))^4);
%o tabf(nn) = for (n=1, nn, print(row(n)));
%Y Cf. A000292, A004737, A005900, A109439, A210440, A277950, A277951.
%Y Mentioned in: A273975
%K nonn,easy,tabf
%O 1,3
%A _Juan Pablo Herrera P._, Nov 05 2016
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