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%I #24 Dec 22 2016 23:36:20
%S 1,1,5,10,10,5,1,1,5,15,30,45,51,45,30,15,5,1,1,5,15,35,65,101,135,
%T 155,155,135,101,65,35,15,5,1,1,5,15,35,70,121,185,255,320,365,381,
%U 365,320,255,185,121,70,35,15,5,1
%N Triangle read by rows, in which row n gives coefficients in expansion of ((x^n - 1)/(x - 1))^5.
%C Sum of n-th row is n^5. The n-th row contains 5n-4 entries. Largest coefficients of each row are listed in A077044.
%C The n-th row is the fifth row of the n-nomial triangle. For example, row 2 (1,5,10,10,5,1) is the fifth 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 five different playing cards. It is also the number of lattice paths from (0,0) to (5,k) using steps (1,0), (1,1), (1,2), ..., (1,n-1).
%H Juan Pablo Herrera P., <a href="/A277950/b277950.txt">Rows n=1..60 of the triangle, flattened</a>
%F T(n,k) = Sum_{i=k-n+1..k} A277949(T(n,i)).
%F From _Juan Pablo Herrera P._, Dec 20 2016: (Start)
%F T(n,k) = A000332(k+4) = (k+4)!/(k!*24) if 0 =< k < n.
%F T(n,k) = ((k+4)!/k!-5*(k-n+4)!/(k-n)!)/24 if n =< k < 2*n.
%F T(n,k) = ((k+4)!/k!-5*(k-n+4)!/(k-n)!+10*(k-2*n+4)!/(k-2*n)!)/24 if 2*n =< k < 3*n.
%F T(n,k) = ((5*n-k-1)!/(5*n-k-5)!-5*(4*n-k-1)!/(4*n-k-5)!)/24 if 3*n-4 =< k < 4*n-4.
%F T(n,k) = A000332(5*n-k-1) = (5*n-k-1)!/(5*n-k-5)!*24 4*n-4 =< k < 5*n-4. (End)
%e Triangle starts:
%e 1;
%e 1, 5, 10, 10, 5, 1;
%e 1, 5, 15, 30, 45, 51, 45, 30, 15, 5, 1;
%e 1, 5, 15, 35, 65, 101, 135, 155, 155, 135, 101, 65, 35, 15, 5, 1;
%e 1, 5, 15, 35, 70, 121, 185, 255, 320, 365, 381, 365, 320, 255, 185, 121, 70, 35, 15, 5, 1;
%t Table[CoefficientList[Series[((x^n - 1)/(x - 1))^5, {x, 0, 5 n}], x], {n, 10}] // Flatten
%o (PARI) row(n) = Vec(((1 - x^n)/(1 - x))^5); tabf(nn) = for (n=1, nn, print(row(n)));
%Y Cf. A000332, A004737, A077044, A109439, A154286, A277949, A277951.
%Y Mentioned in: A273975.
%K nonn,easy,tabf
%O 1,3
%A _Juan Pablo Herrera P._, Nov 05 2016
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