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A134511 abs(A049310) * A128174 provided both arrays are read with offset (n,k) = (0,0). 8
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, 89, 0, 88, 0, 73, 0, 38, 0, 10, 0, 1, 0, 144, 0, 138, 0, 103, 0, 47, 0, 11, 0, 1, 233, 0, 232, 0, 211, 0, 141, 0, 57, 0, 12, 0, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

A112552(unsigned) = A128174 * A049310.

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) are:

   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

CROSSREFS

Cf. A038730, A038792, A049310, A128174, A134512.

A(4n,2n) gives: A038736.

Sequence in context: A112552 A048154 A320602 * A112554 A120616 A108044

Adjacent sequences:  A134508 A134509 A134510 * A134512 A134513 A134514

KEYWORD

nonn,tabl

AUTHOR

Gary W. Adamson, Oct 28 2007

EXTENSIONS

Edited by Robert Israel, Mar 02 2018

STATUS

approved

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Last modified July 16 13:35 EDT 2020. Contains 335788 sequences. (Running on oeis4.)