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%I #19 Nov 19 2021 15:48:17
%S 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,
%T 1094,1274,322,14,0,0,0,0,3281,4372,1462,116,1,0,0,0,0,9842,14760,
%U 6156,744,18,0,0,0,0,0
%N 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.
%C Riordan array ((1-2x)/(1-4x+3x^2),x^2/(1-4x+3x^2)).
%C A007318*A201701 as lower triangular matrices.
%F G.f.: (1-2x)/(1-4x+(3-y)*x^2).
%F 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.
%F Sum_{k, k>+0} T(n+k,k) = A081704(n) .
%F T(n,k) = 3*T(n-1,k)+ Sum_{j>0} T(n-1-j,k-1).
%F 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.
%e Triangle begins:
%e 1
%e 2, 0
%e 5, 1, 0
%e 14, 6, 0, 0
%e 41, 26, 1, 0, 0
%e 122, 100, 10, 0, 0, 0
%e 365, 363, 63, 1, 0, 0, 0
%p A201730 := proc(n,k)
%p (1-2*x)/(1-4*x+(3-y)*x^2) ;
%p coeftayl(%,y=0,k) ;
%p coeftayl(%,x=0,n) ;
%p end proc:
%p seq(seq(A201730(n,k),k=0..n),n=0..12) ; # _R. J. Mathar_, Dec 06 2011
%t m = 13;
%t (* DELTA is defined in A084938 *)
%t 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 *)
%Y Cf. A007051 (1st column), A261064 (2nd column).
%K nonn,tabl
%O 0,2
%A _Philippe Deléham_, Dec 04 2011
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