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A110555
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Triangle of partial sums of alternating binomial coefficients: T(n,k) = Sum_{j=0..k} binomial(n,j)*(-1)^j; n >= 0, 0 <= k <= n.
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21
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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
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OFFSET
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1,8
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COMMENTS
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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
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LINKS
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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
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FORMULA
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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
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EXAMPLE
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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;
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Cf. A008949, A007318.
Sequence in context: A213888 A119337 A213889 * A097805 A071919 A321791
Adjacent sequences: A110552 A110553 A110554 * A110556 A110557 A110558
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KEYWORD
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sign,easy,tabl
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AUTHOR
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Reinhard Zumkeller, Jul 27 2005
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EXTENSIONS
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Typo in name corrected by Andrey Zabolotskiy, Feb 22 2022
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STATUS
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approved
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