A296449 Triangle I(m,n) read by rows: number of perfect lattice paths on the m*n board.

1, 2, 4, 3, 7, 17, 4, 10, 26, 68, 5, 13, 35, 95, 259, 6, 16, 44, 122, 340, 950, 7, 19, 53, 149, 421, 1193, 3387, 8, 22, 62, 176, 502, 1436, 4116, 11814, 9, 25, 71, 203, 583, 1679, 4845, 14001, 40503, 10, 28, 80, 230, 664, 1922, 5574, 16188, 47064, 136946, 11, 31, 89, 257, 745, 2165, 6303, 18375, 53625, 156629, 457795

Offset: 1

Author

R. J. Mathar, Dec 13 2017

Keywords

• nonn
• tabl
• easy

Examples

			Triangle begins:
   1;
   2,  4;
   3,  7, 17;
   4, 10, 26,  68;
   5, 13, 35,  95, 259;
   6, 16, 44, 122, 340,  950;
   7, 19, 53, 149, 421, 1193, 3387;
   8, 22, 62, 176, 502, 1436, 4116, 11814;
   9, 25, 71, 203, 583, 1679, 4845, 14001, 40503;
  10, 28, 80, 230, 664, 1922, 5574, 16188, 47064, 136946;
		

Links

• D. Yaqubi, M. Farrokhi D. G., and H. Ghasemian Zoeram, Lattice paths inside a table, I, arXiv:1612.08697 [math.CO], 2016-2017, array I(m,n).

Crossrefs

Cf. A081113 (diagonal), A000079 (2nd row), A001333 (3rd row), A126358, A057960, A126360, A002714, A126362, A188866.

Programs

Maple

Inm := proc(n,m)
    if m >= n then
        (n+2)*3^(n-2)+(m-n)*add(A005773(i)*A005773(n-i),i=0..n-1)
            +2*add((n-k-2)*3^(n-k-3)*A001006(k),k=0..n-3) ;
    else
        0 ;
    end if;
end proc:
for m from 1 to 13 do
for n from 1 to m do
    printf("%a,",Inm(n,m)) ;
end do:
printf("\n") ;
end do:
# Second program:
A296449row := proc(n) local gf, ser;
gf := n -> 1 + (x*(n - (3*n + 2)*x) + (2*x^2)*(1 +
ChebyshevU(n - 1, (1 - x)/(2*x))) / ChebyshevU(n, (1 - x)/(2*x)))/(1 - 3*x)^2;
ser := n -> series(expand(gf(n)), x, n + 1);
seq(coeff(ser(n), x, k), k = 1..n) end:
for n from 0 to 11 do A296449row(n) od; # Peter Luschny, Sep 07 2021

Mathematica

(* b = A005773 *) b[0] = 1; b[n_] := Sum[k/n*Sum[Binomial[n, j] * Binomial[j, 2*j - n - k], {j, 0, n}], {k, 1, n}];
(* c = A001006 *) c[0] = 1; c[n_] := c[n] = c[n-1] + Sum[c[k] * c[n-2-k], {k, 0, n-2}];
Inm[n_, m_] := If[m >= n, (n + 2)*3^(n - 2) + (m - n)*Sum[b[i]*b[n - i], {i, 0, n - 1}] + 2*Sum[(n - k - 2)*3^(n - k - 3)*c[k], {k, 0, n-3}], 0];
Table[Inm[n, m], {m, 1, 13}, {n, 1, m}] // Flatten (* Jean-François Alcover, Jan 23 2018, adapted from first Maple program. *)

Formula

I(m,n) = (n+2)*3^(n-2) + (m-n)*Sum_{i=0..n-1} A005773(i)*A005773(n-i) + 2*Sum_{k=0..n-3} (n-k-2)*3^(n-k-3)*A001006(k). [Yaqubi Corr. 2.10]

I(m,n) = A188866(m-1,n) for m > 1. - Pontus von Brömssen, Sep 06 2021