OFFSET
1,2
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).
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
EXAMPLE
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;
MAPLE
Inm := proc(n, m)
if m >= n then
+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. *)
CROSSREFS
KEYWORD
AUTHOR
R. J. Mathar, Dec 13 2017
STATUS
approved