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A247453
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T(n,k) = binomial(n,k)*A000111(n-k)*(-1)^(n-k), 0 <= k <= n.
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5
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1, -1, 1, 1, -2, 1, -2, 3, -3, 1, 5, -8, 6, -4, 1, -16, 25, -20, 10, -5, 1, 61, -96, 75, -40, 15, -6, 1, -272, 427, -336, 175, -70, 21, -7, 1, 1385, -2176, 1708, -896, 350, -112, 28, -8, 1, -7936, 12465, -9792, 5124, -2016, 630, -168, 36, -9, 1, 50521
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OFFSET
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0,5
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COMMENTS
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Matrix inverse of A109449, the unsigned version of this sequence. More precisely, consider both of these triangles as the nonzero lower left of an infinite square array / matrix, filled with zeros above/right of the diagonal. Then these are mutually inverse of each other; in matrix notation: A247453 . A109449 = A109449 . A247453 = Identity matrix. In more conventional notation, for any m,n >= 0, Sum_{k=0..n} A247453(n,k)*A109449(k,m) = Sum_{k=0..n} A109449(n,k)*A247453(k,m) = delta(m,n), the Kronecker delta (= 1 if m = n, 0 else). - M. F. Hasler, Oct 06 2017
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LINKS
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Reinhard Zumkeller, Rows n = 0..125 of table, flattened
Peter Luschny, An old operation on sequences: the Seidel transform
J. Millar, N. J. A. Sloane and N. E. Young, A new operation on sequences: the Boustrophedon transform, J. Combin. Theory, 17A 44-54 1996 (Abstract, pdf, ps).
OEIS Wiki, Boustrophedon transform.
Wikipedia, Boustrophedon transform
Index entries for sequences related to boustrophedon transform
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FORMULA
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T(n,k) = (-1)^(n-k) * A007318(n,k) * A000111(n-k), k = 0..n;
T(n,k) = (-1)^(n-k) * A109449(n,k); A109449(n,k) = abs(T(n,k));
abs(sum of row n) = A062162(n);
Sum_{k=0..n} T(n,k)*A000111(k) = 0^n.
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EXAMPLE
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. 0: 1
. 1: -1 1
. 2: 1 -2 1
. 3: -2 3 -3 1
. 4: 5 -8 6 -4 1
. 5: -16 25 -20 10 -5 1
. 6: 61 -96 75 -40 15 -6 1
. 7: -272 427 -336 175 -70 21 -7 1
. 8: 1385 -2176 1708 -896 350 -112 28 -8 1
. 9: -7936 12465 -9792 5124 -2016 630 -168 36 -9 1
. 10: 50521 -79360 62325 -32640 12810 -4032 1050 -240 45 -10 1 .
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MATHEMATICA
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a111[n_] := n! SeriesCoefficient[(1+Sin[x])/Cos[x], {x, 0, n}];
T[n_, k_] := (-1)^(n-k) Binomial[n, k] a111[n-k];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Aug 03 2018 *)
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PROG
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(Haskell)
a247453 n k = a247453_tabl !! n !! k
a247453_row n = a247453_tabl !! n
a247453_tabl = zipWith (zipWith (*)) a109449_tabl a097807_tabl
(PARI) A247453(n, k)=(-1)^(n-k)*binomial(n, k)*if(n>k, 2*abs(polylog(k-n, I)), 1) \\ M. F. Hasler, Oct 06 2017
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CROSSREFS
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Cf. A000111, A007318, A062162, A109449.
Sequence in context: A291980 A238281 A080850 * A109449 A129570 A238385
Adjacent sequences: A247450 A247451 A247452 * A247454 A247455 A247456
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KEYWORD
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sign,tabl
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AUTHOR
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Reinhard Zumkeller, Sep 17 2014
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EXTENSIONS
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Edited by M. F. Hasler, Oct 06 2017
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STATUS
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approved
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