A349841 - OEIS

%I #9 Jul 31 2022 15:56:56

%S 1,1,0,1,1,0,1,2,1,0,1,3,3,1,0,1,4,6,4,1,1,1,5,10,10,5,2,0,1,6,15,20,

%T 15,7,2,0,1,7,21,35,35,22,9,2,0,1,8,28,56,70,57,31,11,2,0,1,9,36,84,

%U 126,127,88,42,13,2,1

%N Triangle T(n,k) built by placing all ones on the left edge, [1,0,0,0,0] repeated on the right edge, and filling the body using the Pascal recurrence T(n,k) = T(n-1,k) + T(n-1,k-1).

%C This is the m=5 member in the sequence of triangles A007318, A059259, A118923, A349839, A349841 which have all ones on the left side, ones separated by m-1 zeros on the other side, and whose interiors obey Pascal's recurrence.

%C T(n,k) is the (n,n-k)-th entry of the (1/(1-x^5),x/(1-x)) Riordan array.

%C For n>0, T(n,n-1) = A002266(n+4).

%C For n>1, T(n,n-2) = A008732(n-2).

%C For n>2, T(n,n-3) = A122047(n-1).

%C Sums of rows give A349842.

%C Sums of antidiagonals give A349843.

%H Michael A. Allen and Kenneth Edwards, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL25/Allen/allen3.html">On Two Families of Generalizations of Pascal's Triangle</a>, J. Int. Seq. 25 (2022) Article 22.7.1.

%F G.f.: (1-x*y)/((1-(x*y)^5)(1-x-x*y)) in the sense that T(n,k) is the coefficient of x^n*y^k in the series expansion of the g.f.

%F T(n,0) = 1.

%F T(n,n) = delta(n mod 5,0).

%F T(n,1) = n-1 for n>0.

%F T(n,2) = (n-1)*(n-2)/2 for n>1.

%F T(n,3) = (n-1)*(n-2)*(n-3)/6 for n>2.

%F T(n,4) = (n-1)*(n-2)*(n-3)*(n-4)/24 for n>3.

%F T(n,5) = C(n-1,5) + 1 for n>4.

%F T(n,6) = C(n-1,6) + n - 6 for n>5.

%F For 0 <= k < n, T(n,k) = (n-k)*Sum_{j=0..floor(k/5)} binomial(n-5*j,n-k)/(n-5*j).

%F The g.f. of the n-th subdiagonal is 1/((1-x^5)(1-x)^n).

%e Triangle begins:

%e   1;

%e   1,   0;

%e   1,   1,   0;

%e   1,   2,   1,   0;

%e   1,   3,   3,   1,   0;

%e   1,   4,   6,   4,   1,   1;

%e   1,   5,  10,  10,   5,   2,   0;

%e   1,   6,  15,  20,  15,   7,   2,   0;

%e   1,   7,  21,  35,  35,  22,   9,   2,   0;

%e   1,   8,  28,  56,  70,  57,  31,  11,   2,   0;

%e   1,   9,  36,  84, 126, 127,  88,  42,  13,   2,   1;

%t Flatten[Table[CoefficientList[Series[(1 - x*y)/((1 - (x*y)^5)(1 - x - x*y)), {x, 0, 20}, {y, 0, 10}], {x, y}][[n+1,k+1]],{n,0,10},{k,0,n}]]

%Y Other members of sequence of triangles: A007318, A059259, A118923, A349839.

%Y Columns: A000012, A001477, A000217, A000292, A000332, A323228.

%Y Diagonals: A079998, A002266, A008732, A122047.

%K easy,nonn,tabl

%O 0,8

%A _Michael A. Allen_, Dec 13 2021