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%I #36 Jun 25 2020 11:23:46
%S 1,1,0,1,1,0,2,0,1,1,1,3,2,3,1,1,1,2,5,6,8,6,5,2,1,1,3,8,13,19,20,19,
%T 13,8,3,1,1,4,12,24,40,52,58,52,40,24,12,4,1,1,5,17,40,76,116,150,162,
%U 150,116,76,40,17,5,1,1,6,23,62,133,232,342,428,462,428,342,232,133,62,23,6
%N Triangular array, read by rows: second differences in n,n direction of trinomial array A027907.
%C For n > 2, T(n,k) is the number of integer strings s(0), ..., s(n) such that s(n) = n - k, s(0) = 0, |s(i) - s(i-1)| = 1 for i = 1,2 and <= 1 for i >= 3.
%H G. C. Greubel, <a href="/A024996/b024996.txt">Table of n, a(n) for n = 0..1874</a> [a(676) ff. corrected by _Georg Fischer_, Jun 24 2020]
%F T(n, k) = T(n-1, k-2) + T(n-1, k-1) + T(n-1, k), starting with [1], [1, 0, 1], [1, 0, 2, 0, 1].
%F G.f.: y*z + (1-y*z)^2 / (1-z*(1+y+y^2)). - _Ralf Stephan_, Jan 09 2005 [corrected by _Peter Luschny_, Jun 25 2020]
%e 1
%e 1 0 1
%e 1 0 2 0 1
%e 1 1 3 2 3 1 1
%e 1 2 5 6 8 6 5 2 1
%e 1 3 8 13 19 20 19 13 8 3 1
%p A024996 := proc(n,k)
%p option remember;
%p if n < 0 or k < 0 or k > 2*n then
%p 0 ;
%p elif n <= 2 then
%p if k = 2*n or k = 0 then
%p 1;
%p elif k = 2*n-1 or k = 1 then
%p 0;
%p elif k =2 then
%p 2;
%p end if;
%p else
%p procname(n-1,k-1)+procname(n-1,k-2)+procname(n-1,k) ;
%p end if;
%p end proc: # _R. J. Mathar_, Jun 23 2013
%p seq(seq(A024996(n,k), k=0..2*n), n=0..11); # added by _Georg Fischer_, Jun 24 2020
%t nmax = 10; CoefficientList[CoefficientList[Series[y*x + (1 - y*x)^2/(1 - x*(1 + y + y^2)), {x, 0, nmax}, {y, 0, 2*nmax}], x], y] // Flatten (* _G. C. Greubel_, May 22 2017; amended by _Georg Fischer_, Jun 24 2020 *)
%o (PARI) T(n,k)=if(n<0||k<0||k>2*n,0,if(n==0,1,if(n==1,[1,0,1][k+1],if(n==2,[1,0,2,0,1][k+1],T(n-1,k-2)+T(n-1,k-1)+T(n-1,k))))) \\ _Ralf Stephan_, Jan 09 2004
%o nmax=8; for(n=0, nmax, for(k=0, 2*n, print1(T(n,k),","))) \\ added by __Georg Fischer_, Jun 24 2020
%o (Julia)
%o using Nemo
%o function A024996Expansion(prec)
%o R, t = PolynomialRing(ZZ, "t")
%o S, x = PowerSeriesRing(R, prec+1, "x")
%o ser = divexact(x^2*t^3 + x^2*t + x*t - 1, x*t^2 + x*t + x - 1)
%o L = zeros(ZZ, prec^2)
%o for k ∈ 0:prec-1, n ∈ 0:2*k
%o L[k^2+n+1] = coeff(coeff(ser, k), n)
%o end
%o L
%o end
%o A024996Expansion(8) |> println # _Peter Luschny_, Jun 25 2020
%Y First differences in n, n direction of array A025177.
%Y Central column is essentially A024997, other columns are A024998, A026069, A026070, A026071. Row sums are in A025579.
%Y Cf. A027907, A026552, A024072.
%K nonn,tabf,easy
%O 0,7
%A _Clark Kimberling_
%E Edited by _Ralf Stephan_, Jan 09 2004
%E Offset corrected by _R. J. Mathar_, Jun 23 2013
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