A214622 - OEIS

%I #31 May 25 2024 12:12:58

%S 1,-1,1,3,-2,1,-10,9,-3,1,45,-40,18,-4,1,-256,225,-100,30,-5,1,1743,

%T -1536,675,-200,45,-6,1,-13840,12201,-5376,1575,-350,63,-7,1,125625,

%U -110720,48804,-14336,3150,-560,84,-8,1,-1282816,1130625,-498240,146412,-32256,5670,-840,108,-9,1

%N 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.

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

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

%F E.g.f.: exp(x*z)/(sech(x)+tanh(x)). - _Peter Luschny_, Aug 01 2012

%e Triangle begins:

%e      1;

%e     -1,     1;

%e      3,    -2,    1;

%e    -10,     9,   -3,    1;

%e     45,   -40,   18,   -4,  1;

%e   -256,   225, -100,   30, -5,  1;

%e   1743, -1536,  675, -200, 45, -6, 1;

%e   ...

%p A214622_row := proc(n) local s,t,k;

%p   s := series(exp(z*x)/(sech(x)+tanh(x)),x,n+2);

%p   t := factorial(n)*coeff(s,x,n); seq(coeff(t,z,k), k=(0..n)) end:

%p for n from 0 to 7 do A214622_row(n) od; # _Peter Luschny_, Aug 01 2012

%t A214622row[n_] := Module[{s, t},

%t    s = Series[Exp[z*x]/(Sech[x] + Tanh[x]), {x, 0, n+2}];

%t    t = n!*Coefficient[s, x, n];

%t    Table[Coefficient[t, z, k], {k, 0, n}]];

%t Table[A214622row[n], {n, 0, 9}] // Flatten (* _Jean-François Alcover_, May 25 2024, after _Peter Luschny_ *)

%o (Sage)

%o R = PolynomialRing(ZZ, 'x')

%o @CachedFunction

%o def skp(n, x) : # Swiss-Knife polynomials A153641.

%o     if n == 0 : return 1

%o     return add(skp(k, 0)*binomial(n, k)*(x^(n-k)-(n+1)%2) for k in range(n)[::2])

%o def A109449_signed(n, k) : return 0 if k > n else R(skp(n, x)-skp(n, x-1)+x^n)[k]

%o T = matrix(ZZ, 9, A109449_signed).inverse(); T

%K sign,tabl

%O 0,4

%A _Peter Luschny_, Jul 23 2012