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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.

%I A059317 #80 Jan 05 2025 19:51:36
%S A059317 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,
%T A059317 82,72,42,19,5,1,1,6,26,70,146,218,255,218,146,70,26,6,1,1,7,34,107,
%U A059317 261,476,691,773,691,476,261,107,34,7,1,1,8,43,154,428,914,1574,2158
%N 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.
%C A059317 The rows have lengths 1, 3, 5, 7, ...; cf. A005408.
%C A059317 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
%C A059317 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
%D A059317 Lin Yang and S.-L. Yang, The parametric Pascal rhombus. Fib. Q., 57:4 (2019), 337-346.
%H A059317 Reinhard Zumkeller, <a href="/A059317/b059317.txt">Rows n = 0..100 of triangle, flattened</a>
%H A059317 Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL26/Barry/barry601.html">On Motzkin-Schröder Paths, Riordan Arrays, and Somos-4 Sequences</a>, J. Int. Seq. (2023) Vol. 26, Art. 23.4.7.
%H A059317 Steven R. Finch, P. Sebah and Z.-Q. Bai, <a href="http://arXiv.org/abs/0802.2654">Odd Entries in Pascal's Trinomial Triangle</a>, arXiv:0802.2654 [math.NT], 2008.
%H A059317 J. Goldwasser et al., <a href="http://dx.doi.org/10.1016/S0012-365X(98)00373-2">The density of ones in Pascal's rhombus</a>, Discrete Math., 204 (1999), 231-236.
%H A059317 W. F. Klostermeyer, M. E. Mays, L. Soltes and G. Trapp, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/35-4/klostermeyer.pdf">A Pascal rhombus</a>, Fibonacci Quarterly, 35 (1997), 318-328.
%H A059317 Y. Moshe, <a href="http://dx.doi.org/10.1016/S0022-314X(03)00103-3">The density of 0's in recurrence double sequences</a>, J. Number Theory, 103 (2003), 109-121.
%H A059317 José L. Ramírez, <a href="http://arxiv.org/abs/1511.04577">The Pascal Rhombus and the Generalized Grand Motzkin Paths</a>, arXiv:1511.04577 [math.CO], 2015.
%H A059317 Paul K. Stockmeyer, <a href="http://arxiv.org/abs/1504.04404">The Pascal Rhombus and the Stealth Configuration</a>, arXiv:1504.04404 [math.CO], 2015.
%H A059317 Sheng-Liang Yang and Yuan-Yuan Gao, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/56-4/yanggao1032018.pdf">The Pascal rhombus and Riordan arrays</a>, Fib. Q., 56:4 (2018), 337-347.
%H A059317 <a href="/index/Pas#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>
%F A059317 T(n+1, k) = T(n, k-1) + T(n, k) + T(n, k+1) + T(n-1, k).
%F A059317 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.
%F A059317 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).
%F A059317 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]
%F A059317 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
%F A059317 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
%F A059317 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
%e A059317 Triangle begins:
%e A059317   1;
%e A059317   1, 1, 1;
%e A059317   1, 2, 4, 2, 1;
%e A059317   1, 3, 8, 9, 8, 3, 1;
%e A059317   ...
%p A059317 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
%p A059317 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
%t A059317 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 *)
%o A059317 (Haskell)
%o A059317 -- import Data.List (zipWith4)
%o A059317 a059317 n k = a059317_tabf !! n !! k
%o A059317 a059317_row n = a059317_tabf !! n
%o A059317 a059317_tabf = [1] : [1,1,1] : f [1] [1,1,1] where
%o A059317    f ws vs = vs' : f vs vs' where
%o A059317      vs' = zipWith4 (\r s t x -> r + s + t + x)
%o A059317            (vs ++ [0,0]) ([0] ++ vs ++ [0]) ([0,0] ++ vs)
%o A059317            ([0,0] ++ ws ++ [0,0])
%o A059317 -- _Reinhard Zumkeller_, Jun 30 2012
%Y A059317 Cf. A059318, A007318. Row sums give A006190. Central column is A059345.
%Y A059317 Other columns: A106050, A106053, A034856, A106058, A106113, A106150, A106173, A267192.
%Y A059317 Cf. also A006190, A140750.
%K A059317 tabf,easy,nice,nonn
%O A059317 0,6
%A A059317 _N. J. A. Sloane_, Jan 26 2001
%E A059317 More terms from Larry Reeves (larryr(AT)acm.org), Jan 30 2001