A188107 Triangle T(n,k) with the coefficient [x^k] of 1/(1 - x - 2*x^2 + x^3)^(n-k+1) in row n, column k.

1, 1, 1, 1, 2, 3, 1, 3, 7, 4, 1, 4, 12, 14, 9, 1, 5, 18, 31, 35, 14, 1, 6, 25, 56, 87, 70, 28, 1, 7, 33, 90, 175, 207, 154, 47, 1, 8, 42, 134, 310, 476, 504, 306, 89, 1, 9, 52, 189, 504, 941, 1274, 1137, 633, 155, 1, 10, 63, 256, 770, 1680, 2745, 3188, 2571

Offset: 0

Author

L. Edson Jeffery, Mar 20 2011

Keywords

• nonn
• easy
• tabl

Comments

Modified versions of the generating function for the diagonal, A006053, are related to rhombus substitution tilings (see A187065, A187066 and A187067).

Examples

			The triangle starts in row n=0 as
  1;
  1,   1;
  1,   2,   3;
  1,   3,   7,   4;
  1,   4,  12,  14,   9;
  1,   5,  18,  31,  35,  14;
  1,   6,  25,  56,  87,  70,  28;
  1,   7,  33,  90, 175, 207, 154,  47;
  1,   8,  42, 134, 310, 476, 504, 306,  89;
		

Links

• Nathaniel Johnston, Rows n = 0..100, flattened

Crossrefs

Cf. A001654, A187065, A187066, A187067.

Cf. Columns: A000012, A000027, A055998.

Programs

Maple

A188107 := proc(n,k) 1/(1-x-2*x^2+x^3)^(n-k+1) ; coeftayl(%,x=0,k) ; end proc:
seq(seq(A188107(n,k),k=0..n),n=0..10) ; # R. J. Mathar, Mar 22 2011

Formula

Sum_{k=0..n} T(n,k) = A001654(n+1).

T(n,k) = T(n-1,k) + T(n-1,k-1) + 2*T(n-2,k-2) - T(n-3,k-3). - Philippe Deléham, Feb 24 2012