cp's OEIS Frontend

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.

%I A068914 #30 May 16 2020 08:55:19
%S A068914 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,
%T A068914 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,
%U A068914 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
%N 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.
%C A068914 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
%H A068914 Stefano Spezia, <a href="/A068914/b068914.txt">First 151 antidiagonals of the array, flattened</a>
%H A068914 Johann Cigler, <a href="http://arxiv.org/abs/1501.04750">Some remarks and conjectures related to lattice paths in strips along the x-axis</a>, arXiv:1501.04750 [math.CO], 2015. See formula 0.2, p. 2.
%H A068914 Nancy S. S. Gu, Helmut Prodinger, <a href="https://arxiv.org/abs/2004.00684">Combinatorics on lattice paths in strips</a>, arXiv:2004.00684 [math.CO], 2020. See p. 2.
%H A068914 R. Kemp, <a href="http://dx.doi.org/10.1016/0012-365X(81)90235-1">On the average depth of a prefix of the Dycklanguage D_1</a>, Discrete Math., 36, 1981, 155-170.
%F A068914 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
%F A068914 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
%e A068914 Rows start:
%e A068914 1,0,0,0,0,...;
%e A068914 1,1,1,1,1,...;
%e A068914 1,1,2,2,4,...;
%e A068914 1,1,2,3,5,...;
%e A068914 etc.
%p A068914 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
%p A068914 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
%t A068914 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_ *)
%o A068914 (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
%Y A068914 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.
%Y A068914 Reflected version of A094718.
%K A068914 nonn,tabl
%O A068914 0,13
%A A068914 _Henry Bottomley_, Mar 06 2002