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A059317
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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.
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15
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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; internal format)
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OFFSET
| 0,6
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COMMENTS
| The rows have lengths 1, 3, 5, 7, ...
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 (deutsch(AT)duke.poly.edu), Sep 03 2007
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REFERENCES
| J. Goldwasser et al., The density of ones in Pascal's rhombus, Discrete Math., 204 (1999), 231-236.
Y. Moshe, The density of 0's in recurrence double sequences, J. Number Theory, 103 (2003), 109-121.
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LINKS
| S. R. Finch, P. Sebah and Z.-Q. Bai, Odd Entries in Pascal's Trinomial Triangle (arXiv:0802.2654)
W. F. Klostermeyer, M. E. Mays, L. Soltes and G. Trapp, A Pascal rhombus, Fibonacci Quarterly, 35 (1997), 318-328.
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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..inf, 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).
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 (deutsch(AT)duke.poly.edu), Sep 03 2007
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EXAMPLE
| 1; 1,1,1; 1,2,4,2,1; 1,3,8,9,8,3,1; ...
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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); (from 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 (deutsch(AT)duke.poly.edu), Sep 03 2007
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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}]] (* From Jean-François Alcover, Feb 01 2012 *)
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CROSSREFS
| Cf. A059318, A007318. Row sums give A006190. Central column is A059345.
Other columns: A106050, A106053, A034856, A106058, A106113, A106150, A106173.
Cf. A006190.
Sequence in context: A052285 A046858 A132823 * A087266 A160801 A177002
Adjacent sequences: A059314 A059315 A059316 * A059318 A059319 A059320
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KEYWORD
| tabf,easy,nice,nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Jan 26 2001
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EXTENSIONS
| More terms from Larry Reeves (larryr(AT)acm.org), Jan 30 2001
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