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A068914 Square array read by antidiagonals of number of k step walks (each step +-1 starting from 0) which are never more than n or less than 0. 7
1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 2, 2, 1, 1, 0, 1, 4, 3, 2, 1, 1, 0, 1, 4, 5, 3, 2, 1, 1, 0, 1, 8, 8, 6, 3, 2, 1, 1, 0, 1, 8, 13, 9, 6, 3, 2, 1, 1, 0, 1, 16, 21, 18, 10, 6, 3, 2, 1, 1, 0, 1, 16, 34, 27, 19, 10, 6, 3, 2, 1, 1, 0, 1, 32, 55, 54, 33, 20, 10, 6, 3, 2, 1, 1, 0, 1, 32, 89 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,13
COMMENTS
The (n,k)-entry of the square array is p(n,k) in the R. Kemp reference (see Table 1 on p. 160 and Theorem 2 on p. 159). - Emeric Deutsch, Jun 16 2011
LINKS
Johann Cigler, Some remarks and conjectures related to lattice paths in strips along the x-axis, arXiv:1501.04750 [math.CO], 2015. See formula 0.2, p. 2.
Nancy S. S. Gu, Helmut Prodinger, Combinatorics on lattice paths in strips, arXiv:2004.00684 [math.CO], 2020. See p. 2.
R. Kemp, On the average depth of a prefix of the Dycklanguage D_1, Discrete Math., 36, 1981, 155-170.
FORMULA
An explicit expression for the (n,k)-entry of the square array can be found in the R. Kemp reference (Theorem 2 on p. 159). - Emeric Deutsch, Jun 16 2011
The g.f. of column k is (1 + v^2)*(1 - v^(k+1))/((1 - v)*(1 + v^(k+2))), where v = (1 - sqrt(1-4*z^2))/(2*z) (see p. 159 of the R. Kemp reference). - Emeric Deutsch, Jun 16 2011
EXAMPLE
Rows start:
1,0,0,0,0,...;
1,1,1,1,1,...;
1,1,2,2,4,...;
1,1,2,3,5,...;
etc.
MAPLE
v := ((1-sqrt(1-4*z^2))*1/2)/z: G := proc (k) options operator, arrow: (1+v^2)*(1-v^(k+1))/((1-v)*(1+v^(k+2))) end proc: a := proc (n, k) options operator, arrow: coeff(series(G(k), z = 0, 80), z, n) end proc: for n from 0 to 15 do seq(a(n, k), k = 0 .. 15) end do; # yields the first 16 entries of the first 16 rows of the square array
v := ((1-sqrt(1-4*z^2))*1/2)/z: G := proc (k) options operator, arrow: (1+v^2)*(1-v^(k+1))/((1-v)*(1+v^(k+2))) end proc: a := proc (n, k) options operator, arrow: coeff(series(G(k), z = 0, 80), z, n) end proc: for n from 0 to 13 do seq(a(n-i, i), i = 0 .. n) end do; # yields the first 14 antidiagonals of the square array in triangular form
MATHEMATICA
v = (1-Sqrt[1-4z^2])/(2z); f[k_] = (1+v^2)*(1-v^(k+1))/((1-v)*(1+v^(k+2))) ; m = 14; a = Table[ PadRight[ CoefficientList[ Series[f[k], {z, 0, m}], z], m], {k, 0, m}]; Flatten[Table[a[[n+1-k, k]], {n, m}, {k, n, 1, -1}]][[;; 95]] (* Jean-François Alcover, Jul 13 2011, after Emeric Deutsch *)
PROG
(PARI) T(n, k) = sum(j=floor(-n/(k+2)), ceil(n/(k+2)), (-1)^j*binomial(n, floor((n+(k+2)*j)/2))); \\ Stefano Spezia, May 08 2020
CROSSREFS
Rows include effectively A000007, A000012, A016116, A000045, A038754, A028495, A030436, A061551. Central and lower diagonals are A001405. Cf. A068913 for starting in the middle rather than an edge.
Reflected version of A094718.
Sequence in context: A264391 A116598 A244925 * A090824 A264620 A302301
KEYWORD
nonn,tabl
AUTHOR
Henry Bottomley, Mar 06 2002
STATUS
approved

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Last modified April 19 13:40 EDT 2024. Contains 371792 sequences. (Running on oeis4.)