A277950 Triangle read by rows, in which row n gives coefficients in expansion of ((x^n - 1)/(x - 1))^5.

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, 155, 155, 135, 101, 65, 35, 15, 5, 1, 1, 5, 15, 35, 70, 121, 185, 255, 320, 365, 381, 365, 320, 255, 185, 121, 70, 35, 15, 5, 1

Offset: 1

Author

Juan Pablo Herrera P., Nov 05 2016

Keywords

• nonn
• easy
• tabf

Comments

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.

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.

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).

Examples

			Triangle starts:
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, 155, 155, 135, 101, 65, 35, 15, 5, 1;
1, 5, 15, 35, 70, 121, 185, 255, 320, 365, 381, 365, 320, 255, 185, 121, 70, 35, 15, 5, 1;
		

Links

• Juan Pablo Herrera P., Rows n=1..60 of the triangle, flattened

Crossrefs

Cf. A000332, A004737, A077044, A109439, A154286, A277949, A277951.

Mentioned in: A273975.

Programs

Mathematica

Table[CoefficientList[Series[((x^n - 1)/(x - 1))^5, {x, 0, 5 n}], x], {n, 10}] // Flatten

PARI

row(n) = Vec(((1 - x^n)/(1 - x))^5); tabf(nn) = for (n=1, nn, print(row(n)));

Formula

T(n,k) = Sum_{i=k-n+1..k} A277949(T(n,i)).

From Juan Pablo Herrera P., Dec 20 2016: (Start)

T(n,k) = A000332(k+4) = (k+4)!/(k!*24) if 0 =< k < n.

T(n,k) = ((k+4)!/k!-5*(k-n+4)!/(k-n)!)/24 if n =< k < 2*n.

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.

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.

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)