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A349839
Triangle T(n,k) built by placing all ones on the left edge, [1,0,0,0] 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).
3
1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 3, 1, 1, 1, 4, 6, 4, 2, 0, 1, 5, 10, 10, 6, 2, 0, 1, 6, 15, 20, 16, 8, 2, 0, 1, 7, 21, 35, 36, 24, 10, 2, 1, 1, 8, 28, 56, 71, 60, 34, 12, 3, 0, 1, 9, 36, 84, 127, 131, 94, 46, 15, 3, 0, 1, 10, 45, 120, 211, 258, 225, 140, 61, 18, 3, 0, 1, 11, 55, 165, 331, 469, 483, 365, 201, 79, 21, 3, 1
OFFSET
0,8
COMMENTS
This is the m=4 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.
T(n,k) is the (n,n-k)-th entry of the (1/(1-x^4),x/(1-x)) Riordan array.
For n>0, T(n,n-1) = A008621(n-1).
For n>1, T(n,n-2) = A001972(n-2).
For n>2, T(n,n-3) = A122046(n).
Sums of rows give A115451.
Sums of antidiagonals give A349840.
LINKS
Michael A. Allen and Kenneth Edwards, On Two Families of Generalizations of Pascal's Triangle, J. Int. Seq. 25 (2022) Article 22.7.1.
FORMULA
G.f.: (1-x*y)/((1-(x*y)^4)(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.
T(n,0) = 1.
T(n,n) = delta(n mod 4,0).
T(n,1) = n-1 for n>0.
T(n,2) = (n-1)*(n-2)/2 for n>1.
T(n,3) = (n-1)*(n-2)*(n-3)/6 for n>2.
T(n,4) = C(n-1,4) + 1 for n>3.
T(n,5) = C(n-1,5) + n - 5 for n>4.
For 0 <= k < n, T(n,k) = (n-k)*Sum_{j=0..floor(k/4)} binomial(n-4*j,n-k)/(n-4*j).
The g.f. of the n-th subdiagonal is 1/((1-x^4)(1-x)^n).
EXAMPLE
Triangle begins:
1;
1, 0;
1, 1, 0;
1, 2, 1, 0;
1, 3, 3, 1, 1;
1, 4, 6, 4, 2, 0;
1, 5, 10, 10, 6, 2, 0;
1, 6, 15, 20, 16, 8, 2, 0;
1, 7, 21, 35, 36, 24, 10, 2, 1;
1, 8, 28, 56, 71, 60, 34, 12, 3, 0;
1, 9, 36, 84, 127, 131, 94, 46, 15, 3, 0;
1, 10, 45, 120, 211, 258, 225, 140, 61, 18, 3, 0;
1, 11, 55, 165, 331, 469, 483, 365, 201, 79, 21, 3, 1;
MATHEMATICA
Flatten[Table[CoefficientList[Series[(1-x*y)/((1-(x*y)^4)(1 - x - x*y)), {x, 0, 24}, {y, 0, 12}], {x, y}][[n+1, k+1]], {n, 0, 12}, {k, 0, n}]]
CROSSREFS
Other members of sequence of triangles: A007318, A059259, A118923, A349841.
Sequence in context: A321918 A321754 A321752 * A247919 A127839 A017827
KEYWORD
easy,nonn,tabl
AUTHOR
Michael A. Allen, Dec 01 2021
STATUS
approved