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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. 21
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 (list; graph; refs; listen; history; text; internal format)
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

Sheng-Liang Yang et al., The Pascal rhombus and Riordan array, Fib. Q., 56:4 (2018), 337-347.

LINKS

Reinhard Zumkeller, Rows n = 0..100 of triangle, flattened

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.

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

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

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.

Sequence in context: A046858 A225812 A132823 * A322046 A247644 A220886

Adjacent sequences:  A059314 A059315 A059316 * A059318 A059319 A059320

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

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Last modified December 12 09:36 EST 2019. Contains 329953 sequences. (Running on oeis4.)