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A110555 Triangle of partial sums of alternating binomial coefficients: T(n,k) = Sum(binomial(n,k)*(-1)^k: 0<=k<=n). 20
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

Table of n, a(n) for n=1..74.

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;

CROSSREFS

Cf. A008949, A007318.

Sequence in context: A213888 A119337 A213889 * A097805 A071919 A167763

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 August 22 12:39 EDT 2017. Contains 290946 sequences.