

A277949


Triangle read by rows, in which row n gives coefficients in expansion of ((x^n  1)/(x  1))^4.


4



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, 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, 104, 125, 140, 146, 140, 125, 104, 80, 56, 35, 20, 10, 4, 1
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OFFSET

1,3


COMMENTS

Sum of nth row is n^4. The nth row contains 4n3 entries. Largest coefficients of each row are listed in A005900.
The nth row is the fourth row of the nnomial triangle. For example, row 2 (1,4,6,4,1) is the fourth row in the binomial triangle.
T(n,k) gives the number of possible ways of randomly selecting k cards from n1 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,n1).


LINKS

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


FORMULA

T(n,k) = Sum_{i=kn+1..k} A109439(T(n,i)).
T(n,k) = A000292(k+1) = (k+3)!/(k!*6) if 0 =< k < n,
T(n,k) = ((k+3)*(k+2)*(k+1)4*(kn+3)*(kn+2)*(kn+1))/6 if n =< k < 2*n,
T(n,k) = ((4*n1k)*(4*n2k)*(4*n3k)4*(3*n1k)*(3*n2k)*(3*n3k))/6 if 2*n3 =< k < 3*n3,
T(n,k) = A000292(4*n3k) = (4*n1k)!/((4*n4k)!*6) if 3*n3 =< k < 4n3.


EXAMPLE

Triangle starts:
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, 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, 104, 125, 140, 146, 140, 125, 104, 80, 56, 35, 20, 10, 4, 1.
...
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}.


MATHEMATICA

Table[CoefficientList[Series[((x^n  1)/(x  1))^4, {x, 0, 4 n}], x], {n, 6}] // Flatten (* Michael De Vlieger, Nov 10 2016 *)


PROG

(PARI) row(n) = Vec(((1  x^n)/(1  x))^4);
tabf(nn) = for (n=1, nn, print(row(n)));


CROSSREFS

Cf. A000292, A004737, A005900, A109439, A210440, A277950, A277951.
Mentioned in: A273975
Sequence in context: A219234 A155675 A230207 * A244081 A279445 A217285
Adjacent sequences: A277946 A277947 A277948 * A277950 A277951 A277952


KEYWORD

nonn,easy,tabf


AUTHOR

Juan Pablo Herrera P., Nov 05 2016


STATUS

approved



