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A277951 Triangle read by rows, in which row n gives coefficients in expansion of ((x^n - 1)/(x - 1))^6.

%I A277951 #9 Jun 02 2025 12:20:00
%S A277951 1,1,6,15,20,15,6,1,1,6,21,50,90,126,141,126,90,50,21,6,1,1,6,21,56,
%T A277951 120,216,336,456,546,580,546,456,336,216,120,56,21,6,1,1,6,21,56,126,
%U A277951 246,426,666,951,1246,1506,1686,1751,1686,1506,1246,951,666,426,246,126,56,21,6,1
%N A277951 Triangle read by rows, in which row n gives coefficients in expansion of ((x^n - 1)/(x - 1))^6.
%C A277951 Sum of n-th row is n^6. The n-th row contains 6n-5 entries. Largest coefficients of each row are listed in A071816.
%C A277951 The n-th row is the sixth row of the n-nomial triangle.  For example, row 2 (1,6,15,20,15,6,1) is the sixth row in the binomial triangle
%C A277951 T(n,k) gives the number of possible ways of randomly selecting k cards from n-1 sets, each with six different playing cards. It is also the number of lattice paths from (0,0) to (6,k) using steps (1,0), (1,1), (1,2), ..., (1,n-1).
%H A277951 Juan Pablo Herrera P., <a href="/A277951/b277951.txt">Rows n=1..50 of the triangle, flattened</a>
%F A277951 T(n,k) = Sum_{i=k-n+1..k} A277950(T(n,i))
%e A277951 Triangle starts:
%e A277951 1;
%e A277951 1, 6, 15, 20, 15, 6, 1;
%e A277951 1, 6, 21, 50, 90, 126, 141, 126, 90, 50, 21, 6, 1;
%e A277951 1, 6, 21, 56, 120, 216, 336, 456, 546, 580, 546, 456, 336, 216, 120, 56, 21, 6, 1.
%t A277951 Table[CoefficientList[Series[((x^n - 1)/(x - 1))^6, {x, 0, 6 n}], x], {n, 10}] // Flatten
%o A277951 (PARI) row(n) = Vec(((1 - x^n)/(1 - x))^6);
%o A277951 tabf(nn) = for (n=1, nn, print(row(n)));
%Y A277951 Cf. A000332, A004737, A071816, A109439, A154286, A277949, A277950.
%Y A277951 Mentioned in: A273975
%K A277951 nonn,easy,tabf
%O A277951 1,3
%A A277951 _Juan Pablo Herrera P._, Nov 18 2016