A214622 Triangle read by rows, matrix inverse of [x^(n-k)](skp(n,x)-skp(n,x-1)+x^n) where skp denotes the Swiss-Knife polynomials A153641.

1, -1, 1, 3, -2, 1, -10, 9, -3, 1, 45, -40, 18, -4, 1, -256, 225, -100, 30, -5, 1, 1743, -1536, 675, -200, 45, -6, 1, -13840, 12201, -5376, 1575, -350, 63, -7, 1, 125625, -110720, 48804, -14336, 3150, -560, 84, -8, 1, -1282816, 1130625, -498240, 146412

Offset

0,4

Links

Table of n, a(n) for n=0..48.

Formula

T(n,k) matrix inverse of A109449(n,k)*(-1)^floor((k-n+5)/2).

T(n, 0) = A003704(n+1).

E.g.f. exp(x*z)/(sech(x)+tanh(x)). - Peter Luschny, Aug 01 2012

Example

   1,
  -1,     1,
   3,    -2,    1,
 -10,     9,   -3,    1,
  45,   -40,   18,   -4,  1,
-256,   225, -100,   30, -5,  1,
1743, -1536,  675, -200, 45, -6, 1.

Maple

A214622_row := proc(n) local s, t, k;
  s := series(exp(z*x)/(sech(x)+tanh(x)), x, n+2);
  t := factorial(n)*coeff(s, x, n); seq(coeff(t, z, k), k=(0..n)) end:
for n from 0 to 7 do A214622_row(n) od; # Peter Luschny, Aug 01 2012

Prog

(Sage)
R = PolynomialRing(ZZ, 'x')
@CachedFunction
def skp(n, x) : # Swiss-Knife polynomials A153641.
    if n == 0 : return 1
    return add(skp(k, 0)*binomial(n, k)*(x^(n-k)-(n+1)%2) for k in range(n)[::2])
def A109449_signed(n, k) : return 0 if k > n else R(skp(n, x)-skp(n, x-1)+x^n)[k]
T = matrix(ZZ, 9, A109449_signed).inverse(); T

Crossrefs

Sequence in context: A101894 A187105 A116071 * A327801 A320578 A267836

Adjacent sequences: A214619 A214620 A214621 * A214623 A214624 A214625

Keyword

sign,tabl

Author

Peter Luschny, Jul 23 2012

Status

approved