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A296449 Triangle I(m,n) read by rows: number of perfect lattice paths on the m*n board. 1
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

Table of n, a(n) for n=1..63.

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).

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]

EXAMPLE

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;

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:

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-Fran├žois Alcover, Jan 23 2018, adapted from Maple *)

CROSSREFS

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

Sequence in context: A138612 A246680 A294244 * A253792 A058330 A124256

Adjacent sequences:  A296446 A296447 A296448 * A296450 A296451 A296452

KEYWORD

nonn,tabl,easy

AUTHOR

R. J. Mathar, Dec 13 2017

STATUS

approved

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Last modified May 25 22:17 EDT 2019. Contains 323576 sequences. (Running on oeis4.)