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

0,5

Comments

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

Links

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

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

Example

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;

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

Crossrefs

Cf. A001654, A187065, A187066, A187067.

Cf. Columns: A000012, A000027, A055998.

Sequence in context: A271700 A136555 A343627 * A174014 A236376 A063967

Adjacent sequences: A188104 A188105 A188106 * A188108 A188109 A188110

Keyword

nonn,easy,tabl

Author

L. Edson Jeffery, Mar 20 2011

Status

approved