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Search: id:A110555
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| A110555 |
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Triangle of partial sums of alternating binomial coefficients: T(n,k) = Sum(binomial(n,k)*(-1)^k: 0<=k<=n). |
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20
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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
(list; table; graph; listen)
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OFFSET
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1,8
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COMMENT
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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 DELEHAM (kolotoko(AT)wanadoo.fr), Sep 05 2005
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LINKS
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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.
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CROSSREFS
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Cf. A008949, A007318.
Sequence in context: A082601 A077593 A119337 this_sequence A071919 A097805 A167763
Adjacent sequences: A110552 A110553 A110554 this_sequence A110556 A110557 A110558
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KEYWORD
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sign,tabl
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AUTHOR
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Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jul 27 2005
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