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A038730
Path-counting triangular array T(i,j), read by rows, obtained from array t in A038792 by T(i,j) = t(2*i-j, j) (for i >= 1 and 1 <= j <= i).
11
1, 1, 2, 1, 4, 5, 1, 6, 12, 13, 1, 8, 23, 33, 34, 1, 10, 38, 73, 88, 89, 1, 12, 57, 141, 211, 232, 233, 1, 14, 80, 245, 455, 581, 609, 610, 1, 16, 107, 393, 888, 1350, 1560, 1596, 1597, 1, 18, 138, 593, 1594, 2881, 3805, 4135, 4180, 4181
OFFSET
1,3
LINKS
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 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
T(n, n) = A001519(n) for n >= 1 (odd-indexed Fibonacci numbers).
From Petros Hadjicostas, Sep 03 2019: (Start)
Following Dil and Mezo (2008, p. 944), define the incomplete Fibonacci numbers by F(n,k) = Sum_{s = 0..k} binomial(n-1-s, s) for n >= 1 and 0 <= k <= floor((n-1)/2). Then T(i, j) = F(2*i-1, j-1) for 1 <= j <= i.
G.f. for column j: Define g(t,j) = ((1+t)^j * (1+t-t^2) + (1-t)^j * (1-t-t^2))/2, which is a function of t^2. Then the g.f. for column j is Sum_{i >= j} T(i,j)*x^i = x^j * (Fibonacci(2*j-1) * (1-x)^(j+1) + Fibonacci(2*j-2) * x * (1-x)^j - x * g(sqrt(x), j)) / ((1-x)^j * (1-3*x+x^2)). This follows from the results in Pintér and Srivastava (1999).
(End)
EXAMPLE
Triangle T(i,j) begins as follows:
1;
1, 2;
1, 4, 5;
1, 6, 12, 13;
1, 8, 23, 33, 34;
1, 10, 38, 73, 88, 89;
1, 12, 57, 141, 211, 232, 233;
... [edited by Petros Hadjicostas, Sep 02 2019]
MAPLE
t:= proc(i, j) option remember; `if`(i=1 or j=1, 1,
max(t(i-1, j)+t(i-1, j-1), t(i-1, j-1)+t(i, j-1)))
end:
T:= (i, j)-> t(2*i-j, j):
seq(seq(T(i, j), j=1..i), i=1..10); # Alois P. Heinz, Sep 02 2019
MATHEMATICA
T[i_, j_]:= Sum[Binomial[2i-k-2, k], {k, 0, j-1}];
Table[T[i, j], {i, 1, 10}, {j, 1, i}] // Flatten (* Jean-François Alcover, Dec 06 2019 *)
PROG
(Magma) [(&+[Binomial(2*n-j-2, j): j in [0..k-1]]): k in [1..n], n in [1..12]]; // G. C. Greubel, Apr 05 2022
(SageMath)
def A038730(n, k): return sum( binomial(2*n-j-2, j) for j in (0..k-1))
flatten([[A038730(n, k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Apr 05 2022
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, May 02 2000
STATUS
approved