A059317 Pascal's "rhombus" (actually a triangle T(n,k), n >= 0, 0<=k<=2n) read by rows: each entry is sum of 3 terms above it in previous row and one term above it two rows back.

1, 1, 1, 1, 1, 2, 4, 2, 1, 1, 3, 8, 9, 8, 3, 1, 1, 4, 13, 22, 29, 22, 13, 4, 1, 1, 5, 19, 42, 72, 82, 72, 42, 19, 5, 1, 1, 6, 26, 70, 146, 218, 255, 218, 146, 70, 26, 6, 1, 1, 7, 34, 107, 261, 476, 691, 773, 691, 476, 261, 107, 34, 7, 1, 1, 8, 43, 154, 428, 914, 1574, 2158

Offset: 0

Author

N. J. A. Sloane, Jan 26 2001

Keywords

• tabf
• easy
• nice
• nonn

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

Examples

			Triangle begins:
  1;
  1, 1, 1;
  1, 2, 4, 2, 1;
  1, 3, 8, 9, 8, 3, 1;
  ...
		

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.
• Steven 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.
• Index entries for triangles and arrays related to Pascal's triangle

Crossrefs

Cf. A059318, A007318. Row sums give A006190. Central column is A059345.

Other columns: A106050, A106053, A034856, A106058, A106113, A106150, A106173, A267192.

Cf. also A006190, A140750.

Programs

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

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 *)

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

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), Jan 30 2001