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).

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

Author

Clark Kimberling, May 02 2000

Keywords

• nonn
• tabl

Examples

			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]
		

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.

Crossrefs

Cf. A001519, A038792, A134511, A324242.

Programs

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

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 *)

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

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)