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%I #15 Aug 03 2018 08:14:44
%S 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,
%T 15,-6,1,-272,427,-336,175,-70,21,-7,1,1385,-2176,1708,-896,350,-112,
%U 28,-8,1,-7936,12465,-9792,5124,-2016,630,-168,36,-9,1,50521
%N T(n,k) = binomial(n,k)*A000111(n-k)*(-1)^(n-k), 0 <= k <= n.
%C 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
%H Reinhard Zumkeller, <a href="/A247453/b247453.txt">Rows n = 0..125 of table, flattened</a>
%H Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/SeidelTransform">An old operation on sequences: the Seidel transform</a>
%H 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 (<a href="http://neilsloane.com/doc/bous.txt">Abstract</a>, <a href="http://neilsloane.com/doc/bous.pdf">pdf</a>, <a href="http://neilsloane.com/doc/bous.ps">ps</a>).
%H OEIS Wiki, <a href="/wiki/Boustrophedon_transform">Boustrophedon transform</a>.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Boustrophedon_transform">Boustrophedon transform</a>
%H <a href="/index/Bo#boustrophedon">Index entries for sequences related to boustrophedon transform</a>
%F T(n,k) = (-1)^(n-k) * A007318(n,k) * A000111(n-k), k = 0..n;
%F T(n,k) = (-1)^(n-k) * A109449(n,k); A109449(n,k) = abs(T(n,k));
%F abs(sum of row n) = A062162(n);
%F Sum_{k=0..n} T(n,k)*A000111(k) = 0^n.
%e . 0: 1
%e . 1: -1 1
%e . 2: 1 -2 1
%e . 3: -2 3 -3 1
%e . 4: 5 -8 6 -4 1
%e . 5: -16 25 -20 10 -5 1
%e . 6: 61 -96 75 -40 15 -6 1
%e . 7: -272 427 -336 175 -70 21 -7 1
%e . 8: 1385 -2176 1708 -896 350 -112 28 -8 1
%e . 9: -7936 12465 -9792 5124 -2016 630 -168 36 -9 1
%e . 10: 50521 -79360 62325 -32640 12810 -4032 1050 -240 45 -10 1 .
%t a111[n_] := n! SeriesCoefficient[(1+Sin[x])/Cos[x], {x, 0, n}];
%t T[n_, k_] := (-1)^(n-k) Binomial[n, k] a111[n-k];
%t Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Aug 03 2018 *)
%o (Haskell)
%o a247453 n k = a247453_tabl !! n !! k
%o a247453_row n = a247453_tabl !! n
%o a247453_tabl = zipWith (zipWith (*)) a109449_tabl a097807_tabl
%o (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
%Y Cf. A000111, A007318, A062162, A109449.
%K sign,tabl
%O 0,5
%A _Reinhard Zumkeller_, Sep 17 2014
%E Edited by _M. F. Hasler_, Oct 06 2017
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