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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: A029394 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 July 3 17:50 EDT 2022. Contains 355055 sequences. (Running on oeis4.)