A140835 - OEIS

%I #2 Oct 12 2012 14:54:50

%S 0,1,1,1,2,1,2,6,6,2,3,12,18,12,3,5,25,25,25,25,5,8,48,120,160,120,48,

%T 8,13,91,91,91,91,91,91,13,21,168,588,168,1470,168,588,168,21,34,306,

%U 306,2856,306,306,2856,306,306,34,55,550,2475,550,2475,13860,2475,550,2475

%N A triangular sequence from a vector a(n) times a triangular tensor t(n,m): T(n,m)=a(n).t(n,m); a(n)=Fibonacci(n);A000045(n): t(n,m)=Binomial(n,GCD(n,m)).

%C Row sums are: {0, 2, 4, 16, 48, 110, 512, 572, 3360, 7616, 26070, 9968, 365184, 36814, 1532128, 4848280, 16897440, 437578, 228446272, 1438264, 1596986490, ...}

%C This tensor like approach is based on the operational ideas of Gary W. Adamson:

%C Thinking about triangular sequences as triangular tensors and Adamson's

%C operations on them as a new kind of "operator"calculus:

%C Operator.T[n,m]=T'[n,m]

%C The idea is that

%C since some of these triangular sequences are representations of

%C orthogonal / Hilbert space wave functions as polynomials

%C there should be a Hamiltonian:

%C H.T[n,m]=E[n].T[n,m]

%C where E[n] is an energy vector.

%C That approach opens up vector operators of the sort:

%C T[n,m].V[n]=T'[n,m]

%C The current sequence is a result of just such an operation.

%F T(n,m)=a(n).t(n,m); a(n)=Fibonacci(n): t(n,m)=Binomial(n,GCD(n,m)).

%e {0},

%e {1, 1},

%e {1, 2, 1},

%e {2, 6, 6, 2},

%e {3, 12, 18, 12, 3},

%e {5, 25, 25, 25, 25, 5},

%e {8, 48, 120, 160, 120, 48, 8},

%e {13, 91, 91, 91, 91, 91, 91, 13},

%e {21, 168, 588, 168, 1470, 168, 588, 168, 21},

%e {34, 306, 306, 2856, 306, 306, 2856, 306, 306, 34},

%e {55, 550, 2475, 550, 2475, 13860, 2475, 550, 2475, 550, 55}

%t Clear[t, a, n, m] t[n_, m_] = Binomial[n, GCD[n, m]]; a = Table[Table[Fibonacci[n]*t[n, m], {m, 0, n}], {n, 0, 10}]; Flatten[a]

%Y Cf. A000045.

%K nonn,tabl,uned

%O 1,5

%A _Roger L. Bagula_ and _Gary W. Adamson_, Jul 18 2008