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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.
2
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. Columns: A000012, A000027, A055998.
Sequence in context: A271700 A136555 A343627 * A174014 A236376 A063967
KEYWORD
nonn,easy,tabl
AUTHOR
L. Edson Jeffery, Mar 20 2011
STATUS
approved