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A136664 Triangular vector sequence as weighted conversion between A137286 and A049310.

%I A136664 #8 Jan 07 2013 11:12:45
%S A136664 1,0,2,8,0,4,0,20,0,8,128,0,48,0,16,0,352,0,112,0,32,3072,0,928,0,256,
%T A136664 0,64,0,8928,0,2368,0,576,0,128,98304,0,24960,0,5888,0,1280,0,256,0,
%U A136664 296448,0,67584,0,14336,0,2816,0,512,3932160,0,863232,0,178176,0,34304
%N A136664 Triangular vector sequence as weighted conversion between A137286 and A049310.
%C A136664 Row sums:
%C A136664 {1, 2, 12, 28, 192, 496, 4320, 12000, 130688, 381696, 5015040};
%C A136664 Suppose that you have a Chebyshev-like recursion: (one type) P[x,n]=x*P[x,n-1]-P[x,n-2]
%C A136664 and an Hermite: Q[x,n]=x*Q[x,n-1]-n*Q[x,n-2]
%C A136664 You can define a set of Matrices on the Coefficient list vectors:
%C A136664 vp[n]=M[n].vq[n]
%C A136664 vq[n].vq[n]t=delta[i,j]
%C A136664 vp[n].vq[n]t=M[n]
%C A136664 where M[n] is a diagonal matrix (a vector)
%C A136664 Then a new set of polynomials is obtained.
%F A136664 T(n,m)=If[A137286(m)>0,A049310(n)/A137286(m),0] Out_vector=2^(n-1)*T(n,m)
%e A136664 {1},
%e A136664 {0, 2},
%e A136664 {8, 0, 4},
%e A136664 {0, 20, 0, 8},
%e A136664 {128, 0, 48, 0, 16},
%e A136664 {0, 352, 0, 112, 0, 32},
%e A136664 {3072, 0, 928, 0, 256, 0, 64},
%e A136664 {0, 8928, 0, 2368, 0, 576, 0, 128},
%e A136664 {98304, 0, 24960, 0, 5888, 0, 1280, 0, 256},
%e A136664 {0, 296448, 0, 67584, 0, 14336, 0, 2816, 0, 512},
%e A136664 {3932160, 0, 863232, 0, 178176, 0, 34304, 0, 6144, 0, 1024}
%t A136664 Clear[P, x, n, a] (*Hermite : A137286*) P[x, 0] = 1; P[x, 1] = x; P[x_, n_] := P[x, n] = x*P[x, n - 1] - n*P[x, n - 2]; a1 = Table[CoefficientList[P[x, n], x], {n, 0, 10}]; (* Chebyshev : other kind : A049310*) Clear[B, x, n] B[x, 0] = 1; B[x, 1] = x; B[x_, n_] := B[x, n] = x*B[x, n - 1] - B[x, n - 2]; a = Table[CoefficientList[B[x, n], x], {n, 0, 10}]; (* converter?*) b = Table[Table[If[a[[n]][[ i]] == 0, 0, 2^(n - 1)*a1[[n]][[i]]/a[[n]][[i]]], {i, 1, Length[a[[n]]]}], {n, 1, Length[a]}]; Flatten[b]
%Y A136664 Cf. A137286, A049310.
%K A136664 nonn,uned,tabl
%O A136664 1,3
%A A136664 _Roger L. Bagula_, Apr 01 2008