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A108561 Triangle read by rows: T(n,0)=1, T(n,n)=(-1)^n, T(n+1,k)=T(n,k-1)+T(n,k) for 0<k<n. 23
1, 1, -1, 1, 0, 1, 1, 1, 1, -1, 1, 2, 2, 0, 1, 1, 3, 4, 2, 1, -1, 1, 4, 7, 6, 3, 0, 1, 1, 5, 11, 13, 9, 3, 1, -1, 1, 6, 16, 24, 22, 12, 4, 0, 1, 1, 7, 22, 40, 46, 34, 16, 4, 1, -1, 1, 8, 29, 62, 86, 80, 50, 20, 5, 0, 1, 1, 9, 37, 91, 148, 166, 130, 70, 25, 5, 1, -1, 1, 10, 46, 128, 239, 314, 296, 200, 95, 30, 6, 0, 1, 1, 11, 56, 174, 367 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,12

COMMENTS

Sum(T(n,k): 0<=k<=n) = A078008(n);

Sum(abs(T(n,k)): 0<=k<=n) = A052953(n-1) for n>0;

T(n,1) = n - 2 for n>1;

T(n,2) = A000124(n-3) for n>2;

T(n,3) = A003600(n-4) for n>4;

T(n,n-6) = A001753(n-6) for n>6;

T(n,n-5) = A001752(n-5) for n>5;

T(n,n-4) = A002623(n-4) for n>4;

T(n,n-3) = A002620(n-1) for n>3;

T(n,n-2) = A008619(n-2) for n>2;

T(n,n-1) = n mod 2 for n>0;

T(2*n,n) = A072547(n+1).

Sum_{k=0..n}T(n,k)*x^k = A232015(n), A078008(n), A000012(n), A040000(n), A001045(n+2), A140725(n+1) for x = 2, 1, 0, -1, -2, -3 respectively. - Philippe Deléham, Nov 17 2013, Nov 19 2013

(1,a^n)Pascal triangle with a = -1. - Philippe Deléham, Dec 27 2013

T(n,k) = A112465(n,n-k). - Reinhard Zumkeller, Jan 03 2014

LINKS

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened

Index entries for triangles and arrays related to Pascal's triangle

FORMULA

G.f.: (1-y*x)/(1-x-(y+y^2)*x). - Philippe Deléham, Nov 17 2013

T(n,k)=T(n-1,k)+T(n-2,k-1)+T(n-2,k-2), T(0,0)=T(1,0)=1, T(1,1)=-1, T(n,k)=0 if k<0 or if k>n. - Philippe Deléham, Nov 17 2013

EXAMPLE

Triangle begins:

1

1, -1

1, 0, 1

1, 1, 1, -1

1, 2, 2, 0, 1

1, 3, 4, 2, 1, -1

1, 4, 7, 6, 3, 0, 1 - Philippe Deléham, Nov 17 2013

MATHEMATICA

Clear[t]; t[n_, 0] = 1; t[n_, n_] := t[n, n] = (-1)^Mod[n, 2]; t[n_, k_] := t[n, k] = t[n-1, k] + t[n-1, k-1]; Table[t[n, k], {n, 0, 13}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 06 2013 *)

PROG

(Haskell)

a108561 n k = a108561_tabl !! n !! k

a108561_row n = a108561_tabl !! n

a108561_tabl = map reverse a112465_tabl

-- Reinhard Zumkeller, Jan 03 2014

(Sage)

def A108561_row(n):

    @cached_function

    def prec(n, k):

        if k==n: return 1

        if k==0: return 0

        return -prec(n-1, k-1)-sum(prec(n, k+i-1) for i in (2..n-k+1))

    return [(-1)^k*prec(n, k) for k in (1..n-1)]+[(-1)^(n+1)]

for n in (1..12): print A108561_row(n) # Peter Luschny, Mar 16 2016

CROSSREFS

Cf. A007318 (a=1), A008949(a=2), A164844(a=10).

Similar to the triangles A035317, A059259, A080242, A112555.

Cf. A228196

Cf. A072547 (central terms).

Sequence in context: A172371 A279006 A112555 * A174626 A264909 A104579

Adjacent sequences:  A108558 A108559 A108560 * A108562 A108563 A108564

KEYWORD

sign,tabl

AUTHOR

Reinhard Zumkeller, Jun 10 2005

EXTENSIONS

Definition corrected by Philippe Deléham, Dec 26 2013

STATUS

approved

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Last modified May 30 06:51 EDT 2017. Contains 287302 sequences.