OFFSET
1,5
COMMENTS
Row sums are: {0, 2, 4, 16, 48, 110, 512, 572, 3360, 7616, 26070, 9968, 365184, 36814, 1532128, 4848280, 16897440, 437578, 228446272, 1438264, 1596986490, ...}
This tensor like approach is based on the operational ideas of Gary W. Adamson:
Thinking about triangular sequences as triangular tensors and Adamson's
operations on them as a new kind of "operator"calculus:
Operator.T[n,m]=T'[n,m]
The idea is that
since some of these triangular sequences are representations of
orthogonal / Hilbert space wave functions as polynomials
there should be a Hamiltonian:
H.T[n,m]=E[n].T[n,m]
where E[n] is an energy vector.
That approach opens up vector operators of the sort:
T[n,m].V[n]=T'[n,m]
The current sequence is a result of just such an operation.
FORMULA
T(n,m)=a(n).t(n,m); a(n)=Fibonacci(n): t(n,m)=Binomial(n,GCD(n,m)).
EXAMPLE
{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, 8},
{13, 91, 91, 91, 91, 91, 91, 13},
{21, 168, 588, 168, 1470, 168, 588, 168, 21},
{34, 306, 306, 2856, 306, 306, 2856, 306, 306, 34},
{55, 550, 2475, 550, 2475, 13860, 2475, 550, 2475, 550, 55}
MATHEMATICA
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]
CROSSREFS
KEYWORD
AUTHOR
Roger L. Bagula and Gary W. Adamson, Jul 18 2008
STATUS
approved