OFFSET
0,6
COMMENTS
The rows have lengths 1, 3, 5, 7, ...; cf. A005408.
T(n,k) is the number of paths in the right half-plane from (0,0) to (n,k-n), consisting of steps U=(1,1), D=(1,-1), h=(1,0) and H=(2,0). Example: T(3,4)=8 because we have hhU, HU, hUh, Uhh, UH, DUU, UDU and UUD. Row sums yield A006190. - Emeric Deutsch, Sep 03 2007
Let p(n,x) denote the Fibonacci polynomial, defined by p(1,x) = 1, p(2,x) = x, p(n,x) = x*p(n-1,x) + p(n-2,x). The coefficients of the numerator polynomial of the rational function p(n, x + 1 + 1/x) form row n of the triangle A059317; the first three numerator polynomials are 1, 1 + x + x^2, 1 + 2*x + 4*x^2 + 2*x^3 + x^4. - Clark Kimberling, Nov 04 2013
REFERENCES
Lin Yang and S.-L. Yang, The parametric Pascal rhombus. Fib. Q., 57:4 (2019), 337-346.
LINKS
Reinhard Zumkeller, Rows n = 0..100 of triangle, flattened
Paul Barry, On Motzkin-Schröder Paths, Riordan Arrays, and Somos-4 Sequences, J. Int. Seq. (2023) Vol. 26, Art. 23.4.7.
S. R. Finch, P. Sebah and Z.-Q. Bai, Odd Entries in Pascal's Trinomial Triangle arXiv:0802.2654 [math.NT], 2008.
J. Goldwasser et al., The density of ones in Pascal's rhombus, Discrete Math., 204 (1999), 231-236.
W. F. Klostermeyer, M. E. Mays, L. Soltes and G. Trapp, A Pascal rhombus, Fibonacci Quarterly, 35 (1997), 318-328.
Y. Moshe, The density of 0's in recurrence double sequences, J. Number Theory, 103 (2003), 109-121.
José L. Ramírez, The Pascal Rhombus and the Generalized Grand Motzkin Paths, arXiv:1511.04577 [math.CO], 2015.
Paul K. Stockmeyer, The Pascal Rhombus and the Stealth Configuration, arXiv:1504.04404 [math.CO], 2015.
Sheng-Liang Yang and Yuan-Yuan Gao, The Pascal rhombus and Riordan arrays, Fib. Q., 56:4 (2018), 337-347.
FORMULA
T(n+1, k) = T(n, k-1) + T(n, k) + T(n, k+1) + T(n-1, k).
Another definition: T(i, j) is defined for i >= 0, -infinity <= j <= infinity; T(i, j) = T(i-1, j) + T(i-1, j-1) + T(i-1, j-2) + T(i-2, j-2) for i >= 2, all j; T(0, 0) = T(1, 1) = T(1, 1) = T(1, 2) = 1; T(0, j) = 0 for j != 0; T(1, j) = 0 for j != 0, 1, 2.
G.f.: Sum_{n>=0, k=0..2*n} T(n, k)*z^n*w^k = 1/(1-z-z*w-z*w^2-z^2*w^2).
There does not seem to be a simple expression for T(n, k). [That may have been true in 2001, but it is no longer true, as the following formulas show. - N. J. A. Sloane, Jan 22 2016]
If the rows of the sequence are displayed in the shape of an isosceles triangle, then, for k>=0, columns k and -k have g.f. z^k*g^k/sqrt((1+z-z^2)(1-3z-z^2)), where g=1+zg+z^2*g+z^2*g^2=[1-z-z^2-sqrt((1+z-z^2)(1-3z--z^2))]/(2z^2). - Emeric Deutsch, Sep 03 2007
T(i,j) = Sum_{m=0..i} Sum_{l=0..i-j-2*m} binomial(2*m+j,m)*binomial(l+j+2*m,l)*binomial(l,i-j-2*m-l) (see Ramirez link). - José Luis Ramírez Ramírez, Nov 18 2015
The e.g.f of the j-th column of the Pascal rhombus is L_j(x)=(F(x)^(j+1)*C(F(x)^2)^j)/(x*(1-2*F(x)^2*C(F(x)^2))), where F(x) and C(x) are the generating function of the Fibonacci numbers and Catalan numbers. - José Luis Ramírez Ramírez, Nov 18 2015
EXAMPLE
Triangle begins:
1;
1, 1, 1;
1, 2, 4, 2, 1;
1, 3, 8, 9, 8, 3, 1;
...
MAPLE
r:=proc(i, j) option remember; if i=0 then 0 elif i=1 and abs(j)>0 then 0 elif i=1 and j=0 then 1 elif i>=1 then r(i-1, j)+r(i-1, j-1)+r(i-1, j+1)+r(i-2, j) else 0 fi end: seq(seq(r(i, j), j=-i+1..i-1), i=0..9); # Emeric Deutsch, Jun 06 2004
g:=1/(1-z-z*w-z*w^2-z^2*w^2): gser:=simplify(series(g, z=0, 10)): for n from 0 to 8 do P[n]:=sort(coeff(gser, z, n)) end do: for n from 0 to 8 do seq(coeff(P[n], w, k), k=0..2*n) end do; # yields sequence in triangular form; Emeric Deutsch, Sep 03 2007
MATHEMATICA
t[0, 0] = t[1, 0] = t[1, 1] = t[1, 2] = 1; t[n_ /; n >= 0, k_ /; k >= 0] /; k <= 2n := t[n, k] = t[n-1, k] + t[n-1, k-1] + t[n-1, k-2] + t[n-2, k-2]; t[n_, k_] /; n < 0 || k < 0 || k > 2n = 0; Flatten[ Table[ t[n, k], {n, 0, 8}, {k, 0, 2n}]] (* Jean-François Alcover, Feb 01 2012 *)
PROG
(Haskell)
-- import Data.List (zipWith4)
a059317 n k = a059317_tabf !! n !! k
a059317_row n = a059317_tabf !! n
a059317_tabf = [1] : [1, 1, 1] : f [1] [1, 1, 1] where
f ws vs = vs' : f vs vs' where
vs' = zipWith4 (\r s t x -> r + s + t + x)
(vs ++ [0, 0]) ([0] ++ vs ++ [0]) ([0, 0] ++ vs)
([0, 0] ++ ws ++ [0, 0])
-- Reinhard Zumkeller, Jun 30 2012
CROSSREFS
KEYWORD
tabf,easy,nice,nonn
AUTHOR
N. J. A. Sloane, Jan 26 2001
EXTENSIONS
More terms from Larry Reeves (larryr(AT)acm.org), Jan 30 2001
STATUS
approved