OFFSET
0,4
COMMENTS
Row sums = A134512: (1, 1, 3, 4, 10, 14, 32, 46, 99, 145, ...).
From Petros Hadjicostas, Sep 03 2019: (Start)
To prove Alois P. Heinz's claim (see the Formula section and his Maple program below) we note that, for n >= 0 and 0 <= k <= n, T(n, n-k) = Sum_{r = 0 .. infinity} abs(A049310(n,r)) * A128174(r,n-k) = Sum_{r = n-k..n} abs(A049310(n,r)) * A128174(r,n-k). But A049310(n,r) = 0 when n + r is odd and A128174(r,n-k) = 1 iff r + n - k is even. Thus, when k is odd, T(n, n-k) = 0.
Assume now k is even. Then T(n, n-k) = Sum_{r = n-k..n and n+r even} abs(A049310(n,r)) = Sum_{r = n-k..n and n+r even} binomial((n+r)/2, r). Letting m = n-r (which is even), we see that the summation ranges from m = 0 to k over even numbers. Thus, let s = m/2, and so T(n, n-k) = Sum_{s = 0 .. k/2} binomial(n-s, n-2*s) = Sum_{s = 0 .. k/2} binomial(n-s, s) = F(n+1, k/2), where F(.,.) is the incomplete Fibonacci number from the references (see also the Formula section below).
(End)
LINKS
Robert Israel, Table of n, a(n) for n = 0..10010 (rows 0 to 141, flattened)
H. Belbachir and A. Belkhir, Combinatorial Expressions Involving Fibonacci, Hyperfibonacci, and Incomplete Fibonacci Numbers, Journal of Integer Sequences, Vol. 17 (2014), Article 14.4.3.
A. Dil and I. Mezo, A symmetric algorithm for hyperharmonic and Fibonacci numbers, Appl. Math. Comp. 206 (2008), 942-951; in Eqs. (11), see the incomplete Fibonacci numbers.
Piero Filipponi, Incomplete Fibonacci and Lucas numbers, P. Rend. Circ. Mat. Palermo (Serie II) 45(1) (1996), 37-56; see Table 1 (p. 39) that contains the incomplete Fibonacci numbers.
A. Pintér and H.M. Srivastava, Generating functions of the incomplete Fibonacci and Lucas numbers, Rend. Circ. Mat. Palermo (Serie II) 48(3) (1999), 591-596.
FORMULA
abs(A049310) * A128174 as infinite lower triangular matrices assuming both of them have offset (n,k) = (0,0).
From Petros Hadjicostas, Sep 03 2019: (Start)
Let F(m,r) = Sum_{j = 0..r} binomial(m-1-j, j) be the incomplete Fibonacci numbers from the references (defined for m >= 1 and 0 <= r <= floor((m-1)/2)).
As Alois P. Heinz observed, for n >= 0 and 0 <= k <= n, T(n, n-k) = F(n+1, k/2) when k is even, and = 0 otherwise (see his Maple program below).
(End)
EXAMPLE
First few rows of the triangle T(n,k):
1;
0, 1;
2, 0, 1;
0, 3, 0, 1;
5, 0, 4, 0, 1;
0, 8, 0, 5, 0, 1;
13, 0, 12, 0, 6, 0, 1;
0, 21, 0, 17, 0, 7, 0, 1;
34, 0, 33, 0, 23, 0, 8, 0, 1;
0, 55, 0, 50, 0, 30, 0, 9, 0, 1;
...
MAPLE
N:= 20: # for the first N rows
T128174:= Matrix(N, N, (i, j) -> `if`(j<=i, (i-j+1) mod 2, 0)):
T049310:= Matrix(N, N):
for i from 1 to N do
P:= orthopoly[U](i-1, x/2);
for j from 1 to i do
T049310[i, j]:= abs(coeff(P, x, j-1))
od
od:
A:= T049310 . T128174:
for i from 1 to N do
convert(A[i, 1..i], list)
od; # Robert Israel, Mar 02 2018
# second Maple program:
T:= (n, k)-> `if`((n+k)::odd, 0, add(binomial(n-s, s), s=0..(n-k)/2)):
seq(seq(T(n, k), k=0..n), n=0..12); # Alois P. Heinz, Sep 02 2019
MATHEMATICA
T[n_, k_] := If[OddQ[n+k], 0, Sum[Binomial[n-s, s], {s, 0, (n-k)/2}]];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Dec 31 2021, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Gary W. Adamson, Oct 28 2007
EXTENSIONS
Edited by Robert Israel, Mar 02 2018
STATUS
approved