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 A024996 Triangular array, read by rows: second differences in n,n direction of trinomial array A027907. 18
 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, 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, 150, 116, 76, 40, 17, 5, 1, 1, 6, 23, 62, 133, 232, 342, 428, 462, 428, 342, 232, 133, 62, 23, 6 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,7 COMMENTS 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. LINKS G. C. Greubel, Table of n, a(n) for n = 0..1874 [a(676) ff. corrected by Georg Fischer, Jun 24 2020] FORMULA 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]. 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] EXAMPLE 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 13 8 3 1 MAPLE A024996 := proc(n, k) option remember; if n < 0 or k < 0 or k > 2*n then 0 ; elif n <= 2 then if k = 2*n or k = 0 then 1; elif k = 2*n-1 or k = 1 then 0; elif k =2 then 2; end if; else procname(n-1, k-1)+procname(n-1, k-2)+procname(n-1, k) ; end if; end proc: # R. J. Mathar, Jun 23 2013 seq(seq(A024996(n, k), k=0..2*n), n=0..11); # added by Georg Fischer, Jun 24 2020 MATHEMATICA 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 *) PROG (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 nmax=8; for(n=0, nmax, for(k=0, 2*n, print1(T(n, k), ", "))) \\ added by _Georg Fischer, Jun 24 2020 (Julia) using Nemo function A024996Expansion(prec) R, t = PolynomialRing(ZZ, "t") S, x = PowerSeriesRing(R, prec+1, "x") ser = divexact(x^2*t^3 + x^2*t + x*t - 1, x*t^2 + x*t + x - 1) L = zeros(ZZ, prec^2) for k ∈ 0:prec-1, n ∈ 0:2*k L[k^2+n+1] = coeff(coeff(ser, k), n) end L end A024996Expansion(8) |> println # Peter Luschny, Jun 25 2020 CROSSREFS First differences in n, n direction of array A025177. Central column is essentially A024997, other columns are A024998, A026069, A026070, A026071. Row sums are in A025579. Cf. A027907, A026552, A024072. Sequence in context: A357914 A035467 A254045 * A187596 A263863 A134655 Adjacent sequences: A024993 A024994 A024995 * A024997 A024998 A024999 KEYWORD nonn,tabf,easy AUTHOR Clark Kimberling EXTENSIONS Edited by Ralf Stephan, Jan 09 2004 Offset corrected by R. J. Mathar, Jun 23 2013 STATUS approved

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Last modified August 5 18:20 EDT 2024. Contains 374954 sequences. (Running on oeis4.)