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A115713 A divide-and-conquer related triangle. 4
1, -1, 1, -4, 0, 1, 0, 0, -1, 1, 0, -4, 0, 0, 1, 0, 0, 0, 0, -1, 1, 0, 0, -4, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, -4, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, -4, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, -4, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, -4, 0, 0, 0, 0, 0, 0, 0, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

LINKS

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

FORMULA

G.f.: (1-x+x*y)/(1-x^2*y^2) - 4*x^2/(1-x^2*y).

(1, x) - (x, x)/2 - (x, -x)/2 - 4*(x^2, x^2) expressed in the notation of stretched Riordan arrays.

Column k has g.f.: x^k - (x*(-x)^k + x^(k+1))/2 - 4*x^(2*k+2).

T(n, k) = if(n=k, 1, 0) OR if(n=2k+2, -4, 0) OR if(n=k+1, -(1+(-1)^k)/2, 0).

Sum_{k=0..n} T(n, k) = A115634(n).

Sum_{k=0..floor(n/2)} T(n-k, k) = A115714(n).

EXAMPLE

Triangle begins

   1;

  -1,  1;

  -4,  0,  1;

   0,  0, -1,  1;

   0, -4,  0,  0,  1;

   0,  0,  0,  0, -1,  1;

   0,  0, -4,  0,  0,  0,  1;

   0,  0,  0,  0,  0,  0, -1,  1;

   0,  0,  0, -4,  0,  0,  0,  0,  1;

   0,  0,  0,  0,  0,  0,  0,  0, -1,  1;

   0,  0,  0,  0, -4,  0,  0,  0,  0,  0,  1;

   0,  0,  0,  0,  0,  0,  0,  0,  0,  0, -1,  1;

   0,  0,  0,  0,  0, -4,  0,  0,  0,  0,  0,  0,  1;

   0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0, -1,  1;

   0,  0,  0,  0,  0,  0, -4,  0,  0,  0,  0,  0,  0,  0,  1;

MAPLE

A115713 := proc(n, k)

    coeftayl( (1-x+x*y)/(1-x^2*y^2)-4*x^2/(1-x^2*y), x=0, n) ;

    coeftayl( %, y=0, k) ;

end proc: # R. J. Mathar, Sep 07 2016

MATHEMATICA

T[n_, k_]:= If[k==n, 1, If[k==n-1, -(1-(-1)^n)/2, If[n==2*k+2, -4, 0]]];

Table[T[n, k], {n, 0, 18}, {k, 0, n}]//Flatten (* G. C. Greubel, Nov 23 2021 *)

PROG

(Sage)

def A115713(n, k):

    if (k==n): return 1

    elif (k==n-1): return -(n%2)

    elif (n==2*k+2): return -4

    else: return 0

flatten([[A115713(n, k) for k in (0..n)] for n in (0..18)]) # G. C. Greubel, Nov 23 2021

CROSSREFS

Cf. A115634 (row sums), A115714 (diagonal sums), A115715 (inverse).

Sequence in context: A321188 A322076 A115633 * A199571 A036859 A036861

Adjacent sequences:  A115710 A115711 A115712 * A115714 A115715 A115716

KEYWORD

easy,sign,tabl,changed

AUTHOR

Paul Barry, Jan 29 2006

STATUS

approved

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Last modified November 28 06:42 EST 2021. Contains 349401 sequences. (Running on oeis4.)