A189427 -id:A189427 - cp's OEIS frontend

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.

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, 2180, 636, 1, 18, 152, 777, 2625, 6032, 9366, 9414, 5521, 1429, 1, 20, 189, 1096, 4284, 11718, 22590, 30148, 26517, 13804, 3211

Offset: 0

L. Edson Jeffery, Mar 20 2011

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.

			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 ...

Cf. A006054, A033505, A189426, A189427.

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

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.

a(43) and following corrected by Georg Fischer, Oct 14 2023

A189426 Expansion of (x^2)/(1-2*x-x^2+x^3)^2.

Original entry on oeis.org

0, 0, 1, 4, 14, 42, 119, 322, 847, 2180, 5521, 13804, 34160, 83818, 204204, 494494, 1191227, 2856666, 6823334, 16240714, 38534657, 91175154, 215179125, 506670394, 1190534467, 2792076392, 6536567296, 15278103876, 35656587624, 83101366684

Offset: 0

L. Edson Jeffery, Apr 22 2011

Convolution of A006054={0,0,1,2,5,11,25,56,126,...} with itself.

For n=0,1,2,..., partial sums are given by Sum_{k=0..n} a(k)=A189427(n), where A189427={0,0,1,5,19,61,180,...}.

Index entries for linear recurrences with constant coefficients, signature (4, -2, -6, 3, 2, -1).

Cf. A006054, A189427.

CoefficientList[Series[x^2/(1-2x-x^2+x^3)^2,{x,0,40}],x] (* or *) LinearRecurrence[{4,-2,-6,3,2,-1},{0,0,1,4,14,42},40] (* Harvey P. Dale, Feb 29 2012 *)

Vec((x^2)/(1-2*x-x^2+x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

G.f.: (x^2)/(1-2*x-x^2+x^3)^2.

a(n)=4*a(n-1)-2*a(n-2)-6*a(n-3)+3*a(n-4)+2*a(n-5)-a(n-6), n>=6.

cp's OEIS Frontend

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.

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