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

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, 120, 216, 336, 456, 546, 580, 546, 456, 336, 216, 120, 56, 21, 6, 1, 1, 6, 21, 56, 126, 246, 426, 666, 951, 1246, 1506, 1686, 1751, 1686, 1506, 1246, 951, 666, 426, 246, 126, 56, 21, 6, 1

Offset: 1

Author

Juan Pablo Herrera P., Nov 18 2016

Keywords

• nonn
• easy
• tabf

Comments

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.

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

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).

Examples

			Triangle starts:
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, 120, 216, 336, 456, 546, 580, 546, 456, 336, 216, 120, 56, 21, 6, 1.
		

Links

• Juan Pablo Herrera P., Rows n=1..50 of the triangle, flattened

Crossrefs

Cf. A000332, A004737, A071816, A109439, A154286, A277949, A277950.

Mentioned in: A273975

Programs

Mathematica

Table[CoefficientList[Series[((x^n - 1)/(x - 1))^6, {x, 0, 6 n}], x], {n, 10}] // Flatten

PARI

row(n) = Vec(((1 - x^n)/(1 - x))^6);
tabf(nn) = for (n=1, nn, print(row(n)));

Formula

T(n,k) = Sum_{i=k-n+1..k} A277950(T(n,i))