A201730 Triangle T(n,k), read by rows, given by (2,1/2,3/2,0,0,0,0,0,0,0,...) DELTA (0,1/2,-1/2,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.

1, 2, 0, 5, 1, 0, 14, 6, 0, 0, 41, 26, 1, 0, 0, 122, 100, 10, 0, 0, 0, 365, 363, 63, 1, 0, 0, 0, 1094, 1274, 322, 14, 0, 0, 0, 0, 3281, 4372, 1462, 116, 1, 0, 0, 0, 0, 9842, 14760, 6156, 744, 18, 0, 0, 0, 0, 0

Offset

0,2

Comments

Riordan array ((1-2x)/(1-4x+3x^2),x^2/(1-4x+3x^2)).

A007318*A201701 as lower triangular matrices.

Links

Table of n, a(n) for n=0..54.

Formula

G.f.: (1-2x)/(1-4x+(3-y)*x^2).

Sum_{k, 0<=k<=n} T(n,k)*x^k = A139011(n), A000079(n), A007051(n), A006012(n), A001075(n), A081294(n), A001077(n), A084059(n), A108851(n), A084128(n), A081340(n), A084132(n) for x = -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 respectively.

Sum_{k, k>+0} T(n+k,k) = A081704(n) .

T(n,k) = 3*T(n-1,k)+ Sum_{j>0} T(n-1-j,k-1).

T(n,k) = 4*T(n-1,k)+ T(n-2,k-1) - 3*T(n-2,k) with T(0,0)=1, T(1,0)= 2, T(1,1) = 0 and T(n,k) = 0 if k<0 or if n<k.

Example

Triangle begins:
1
2, 0
5, 1, 0
14, 6, 0, 0
41, 26, 1, 0, 0
122, 100, 10, 0, 0, 0
365, 363, 63, 1, 0, 0, 0

Maple

A201730 := proc(n, k)
    (1-2*x)/(1-4*x+(3-y)*x^2) ;
    coeftayl(%, y=0, k) ;
    coeftayl(%, x=0, n) ;
end proc:
seq(seq(A201730(n, k), k=0..n), n=0..12) ; # R. J. Mathar, Dec 06 2011

Mathematica

m = 13;
(* DELTA is defined in A084938 *)
DELTA[Join[{2, 1/2, 3/2}, Table[0, {m}]], Join[{0, 1/2, -1/2}, Table[0, {m}]], m] // Flatten (* Jean-François Alcover, Feb 19 2020 *)

Crossrefs

Cf. A007051 (1st column), A261064 (2nd column).

Sequence in context: A117780 A155759 A202209 * A188449 A177267 A188445

Adjacent sequences: A201727 A201728 A201729 * A201731 A201732 A201733

Keyword

nonn,tabl

Author

Philippe Deléham, Dec 04 2011

Status

approved