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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). 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

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 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 October 22 03:43 EDT 2017. Contains 293756 sequences.