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A110555 Triangle of partial sums of alternating binomial coefficients: T(n,k) = Sum_{k=0..n} binomial(n,k)*(-1)^k. 21
1, 1, 0, 1, -1, 0, 1, -2, 1, 0, 1, -3, 3, -1, 0, 1, -4, 6, -4, 1, 0, 1, -5, 10, -10, 5, -1, 0, 1, -6, 15, -20, 15, -6, 1, 0, 1, -7, 21, -35, 35, -21, 7, -1, 0, 1, -8, 28, -56, 70, -56, 28, -8, 1, 0, 1, -9, 36, -84, 126, -126, 84, -36, 9, -1, 0, 1, -10, 45, -120, 210, -252, 210, -120 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,8

COMMENTS

T(n,0)=1, T(n,n)=0^n, T(n,k) = -T(n-1,k-1) + T(n-1,k), 0 < k < n;

T(n,n-k-1) = -T(n,k), 0 < k < n;

A071919(n,k) = abs(T(n,k)), T(n,k) = A071919(n,k)*(-1)^k;

row sums give A000007; central terms give A110556;

T(n,1)  = -n + 1 for n>0;

T(n,2)  =  A000217(n-2) for n > 1;

T(n,3)  = -A000292(n-4) for n > 2;

T(n,4)  =  A000332(n-1) for n > 3;

T(n,5)  = -A000389(n-1) for n > 5;

T(n,6)  =  A000579(n-1) for n > 6;

T(n,7)  = -A000580(n-1) for n > 7;

T(n,8)  =  A000581(n-1) for n > 8;

T(n,9)  = -A000582(n-1) for n > 9;

T(n,10) =  A001287(n-1) for n > 10;

T(n,11) = -A001288(n-1) for n > 11;

T(n,12) =  A010965(n-1) for n > 12;

T(n,13) = -A010966(n-1) for n > 13;

T(n,14) =  A010967(n-1) for n > 14;

T(n,15) = -A010968(n-1) for n > 15;

T(n,16) =  A010969(n-1) for n > 16.

Triangle T(n,k), 0 <= k <= n, read by rows, given by [1, 0, 0, 0, 0, 0, 0, 0, ...] DELTA [0, -1, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Sep 05 2005

LINKS

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Ângela Mestre, José Agapito, A Family of Riordan Group Automorphisms, J. Int. Seq., Vol. 22 (2019), Article 19.8.5.

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

FORMULA

T(n, k) = binomial(n-1, k)*(-1)^k, 0 <= k < n, T(n, n)=0^n.

G.f.: (1+x*y)/(1+x*y-x). - R. J. Mathar, Aug 11 2015

EXAMPLE

  1;

  1,  0;

  1, -1,  0;

  1, -2,  1,  0;

  1, -3,  3, -1,  0;

  1, -4,  6, -4,  1,  0;

  1, -5, 10,-10,  5, -1,  0;

  1, -6, 15,-20, 15, -6,  1,  0;

  1, -7, 21,-35, 35,-21,  7, -1,  0;

MATHEMATICA

T[0, 0] := 1;  T[n_, n_] := 0; T[n_, k_] := (-1)^k*Binomial[n - 1, k]; Table[T[n, k], {n, 0, 20}, {k, 0, n}] // Flatten (* G. C. Greubel, Aug 31 2017 *)

PROG

(PARI) concat(1, for(n=1, 10, for(k=0, n, print1(if(k != n, (-1)^k*binomial(n-1, k), 0), ", ")))) \\ G. C. Greubel, Aug 31 2017

CROSSREFS

Cf. A008949, A007318.

Sequence in context: A213888 A119337 A213889 * A097805 A071919 A321791

Adjacent sequences:  A110552 A110553 A110554 * A110556 A110557 A110558

KEYWORD

sign,easy,tabl

AUTHOR

Reinhard Zumkeller, Jul 27 2005

STATUS

approved

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Last modified October 25 06:05 EDT 2020. Contains 338011 sequences. (Running on oeis4.)