OFFSET
1,5
COMMENTS
Row sums are the Pell sequence A000129.
Right border = inverse binomial transform of the Pell sequence: (A016116).
This triangle = difference terms of columns from an array generated from binomial transforms of (1,0,0,0...); (1,1,0,0,0...); (1,1,2,2...); (1,1,2,2,4,...); where (1, 1, 2, 2, 4, 4,...) = A016116, the inverse binomial transform of the Pell sequence A000129.
Triangle read by rows, iterates of X * [1,0,0,0,...] where X = an infinite bidiagonal matrix with (1,1,1,...) in the main diagonal and (1,2,1,2,1,2,...) in the subdiagonal, with the rest zeros. - Gary W. Adamson, May 10 2008
This sequence is jointly generated with A135837 as a triangular array of coefficients of polynomials u(n,x): initially, u(1,x) = v(1,x) = 1; for n>1, u(n,x) = u(n-1,x) + x*v(n-1) and v(n,x) = 2*x*u(n-1,x) + v(n-1,x). See the Mathematica section. - Clark Kimberling, Feb 26 2012
LINKS
G. C. Greubel, Rows n = 1..50 of the triangle, flattened
FORMULA
From G. C. Greubel, Oct 23 2021:
T(n, k) = 2^floor((k-1)/2)*binomial(n-1, k-1).
Sum_{k=0..n} T(n, k) = A000129(n). (End)
EXAMPLE
First few rows of the generating array are:
1, 1, 1, 1, 1, ...
1, 2, 3, 4, 5, ...
1, 2, 5, 10, 17, ...
1, 2, 5, 12, 25, ...
1, 2, 5, 12, 29, ...
...
Taking difference terms of the columns, we get this triangle. First few rows are:
1;
1, 1;
1, 2, 2;
1, 3, 6, 2;
1, 4, 12, 8, 4;
1, 5, 20, 20, 20, 4;
1, 6, 30, 40, 60, 24, 8;
1, 7, 42, 70, 140, 84, 56, 8;
...
MATHEMATICA
(* First program *)
u[1, x_]:= 1; v[1, x_]:= 1; z = 13;
u[n_, x_]:= u[n-1, x] + x*v[n-1, x];
v[n_, x_]:= 2*x*u[n-1, x] + v[n-1, x];
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%] (* A117919 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A135837 *)
(* Second program *)
Table[2^Floor[(k-1)/2]*Binomial[n-1, k-1], {n, 15}, {k, n}]//Flatten (* G. C. Greubel, Oct 23 2021 *)
PROG
(Magma) [2^Floor((k-1)/2)*Binomial(n-1, k-1): k in [1..n], n in [1..15]]; // G. C. Greubel, Oct 23 2021
(Sage) flatten([[2^((k-1)//2)*binomial(n-1, k-1) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, Oct 23 2021
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Gary W. Adamson, Apr 02 2006
EXTENSIONS
Name changed and more terms added by G. C. Greubel, Oct 23 2021
STATUS
approved