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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) = 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 the first 50 rows, flattened

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.: (1-yz)^2 / [1-z(1+y+y^2)]. - Ralf Stephan, Jan 09 2005

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

MATHEMATICA

CoefficientList[CoefficientList[Series[(1 - y*x)^2/(1 - x*(1 + y + y^2)), {x, 0, 10}, {y, 0, 10}], x], y] // Flatten (* G. C. Greubel, May 22 2017 *)

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)))))

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. 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 February 21 02:57 EST 2019. Contains 320364 sequences. (Running on oeis4.)