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).

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

Table of n, a(n) for n=0..90.

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.

Columns: A000012, A001477, A000217, A000292, A145126, A051745.

Diagonals: A121262, A008621, A001972, A122046.

Sequence in context: A321918 A321754 A321752 * A247919 A127839 A017827

Adjacent sequences: A349836 A349837 A349838 * A349840 A349841 A349842

Keyword

easy,nonn,tabl

Author

Michael A. Allen, Dec 01 2021

Status

approved