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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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

T(n,n) = A001519(n) for n >= 1 (odd-indexed Fibonacci numbers).

LINKS

Alois P. Heinz, Rows n = 1..200, 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 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

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

CROSSREFS

Cf. A038792, A134511, A324242.

Sequence in context: A194363 A161135 A237274 * A188106 A050166 A124959

Adjacent sequences:  A038727 A038728 A038729 * A038731 A038732 A038733

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, May 02 2000

STATUS

approved

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Last modified October 15 10:15 EDT 2019. Contains 328026 sequences. (Running on oeis4.)