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

0,5

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

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

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.

Example

.   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  .

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 *)

Prog

(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

Crossrefs

Cf. A000111, A007318, A062162, A109449.

Sequence in context: A291980 A238281 A080850 * A109449 A129570 A238385

Adjacent sequences: A247450 A247451 A247452 * A247454 A247455 A247456

Keyword

sign,tabl

Author

Reinhard Zumkeller, Sep 17 2014

Extensions

Edited by M. F. Hasler, Oct 06 2017

Status

approved