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A188106 Triangle T(n,k) with the coefficient [x^k] of 1/(1-2*x-x^2+x^3)^(n-k+1) in row n, column k. 3
1, 1, 2, 1, 4, 5, 1, 6, 14, 11, 1, 8, 27, 42, 25, 1, 10, 44, 101, 119, 56, 1, 12, 65, 196, 342, 322, 126, 1, 14, 90, 335, 770, 1080, 847, 283, 1, 16, 119, 526, 1495, 2772, 3248, 9414, 5521, 1429, 1, 18, 152, 777, 2625, 6032, 9366, 9414, 5521, 1429 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Modified versions of the generating function for D(0)={1,2,5,11,...}=A006054(m+2), m=0,1,2,..., are related to rhombus substitution tilings (see A187068, A187069 and A187070). The columns of the triangle have generating functions 1/(1-x), 2*x/(1-x)^2, x^2*(5-x)/(1-x)^3, x^3*(11-2*x-x^2)/(1-x)^4, x^4*(25-6*x-3*x^2)/(1-x)^5, ..., for which the sum of the signed coefficients in the n-th numerator equals 2^n. The diagonals {1,2,5,...}, {1,4,14,...}, ..., are generated by successive series expansion of F(n+1,x), n=0,1,..., where F(n,x)=1/(1-2*x-x^2+x^3)^n. For example, the second diagonal is {T{1,0},T{2,1},...}={1,4,14,...}=A189426, for which successive partial sums give A189427 (excluding the zero terms). Moreover, the diagonals correspond to successive convolutions of A006054 (= the first diagonal) with itself.

LINKS

Table of n, a(n) for n=0..55.

FORMULA

sum_{k=0..n} T(n,k)=A033505(n).

T(n,0) = 1. T(n,2) = A014106(n-1). T(n,3) = (n-2)*(4*n^2+2*n-9)/3. T(n,4) = (n-2)*(n-3)*(2*n+7)*(2*n-3)/6.

EXAMPLE

1;

1, 2;

1, 4, 5;

1, 6, 14, 11;

1, 8, 27, 42, 25;

1, 10, 44, 101, 119, 56;

1, 12, 65, 196, 342, 322, 126;

1, 14, 90, 335, 770, 1080, 847, 283;

1, 16, 119, 526, 1495 ...

MAPLE

A188106 := proc(n, k) 1/(1-2*x-x^2+x^3)^(n-k+1) ; coeftayl(%, x=0, k) ; end proc:

seq(seq(A188106(n, k), k=0..n), n=0..10) ; # R. J. Mathar, Mar 22 2011

CROSSREFS

Cf. A006054, A033505, A189426, A189427.

Sequence in context: A161135 A237274 A038730 * A050166 A124959 A081281

Adjacent sequences:  A188103 A188104 A188105 * A188107 A188108 A188109

KEYWORD

nonn,tabl

AUTHOR

L. Edson Jeffery, Mar 20 2011

STATUS

approved

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Last modified December 2 10:53 EST 2021. Contains 349440 sequences. (Running on oeis4.)