A247453 T(n,k) = binomial(n,k)*A000111(n-k)*(-1)^(n-k), 0 <= k <= n.

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

Offset: 0

Author

Reinhard Zumkeller, Sep 17 2014

Keywords

• sign
• tabl

Comments

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

Examples

			.   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  .
		

Links

• 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

Crossrefs

Cf. A000111, A007318, A062162, A109449.

Programs

Haskell

a247453 n k = a247453_tabl !! n !! k
a247453_row n = a247453_tabl !! n
a247453_tabl = zipWith (zipWith (*)) a109449_tabl a097807_tabl

Mathematica

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 *)

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

Formula

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.

Extensions

Edited by M. F. Hasler, Oct 06 2017