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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.

%I A214622 #31 May 25 2024 12:12:58
%S A214622 1,-1,1,3,-2,1,-10,9,-3,1,45,-40,18,-4,1,-256,225,-100,30,-5,1,1743,
%T A214622 -1536,675,-200,45,-6,1,-13840,12201,-5376,1575,-350,63,-7,1,125625,
%U A214622 -110720,48804,-14336,3150,-560,84,-8,1,-1282816,1130625,-498240,146412,-32256,5670,-840,108,-9,1
%N 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.
%F A214622 T(n,k) = matrix inverse of A109449(n,k)*(-1)^floor((k-n+5)/2).
%F A214622 T(n,0) = A003704(n+1).
%F A214622 E.g.f.: exp(x*z)/(sech(x)+tanh(x)). - _Peter Luschny_, Aug 01 2012
%e A214622 Triangle begins:
%e A214622      1;
%e A214622     -1,     1;
%e A214622      3,    -2,    1;
%e A214622    -10,     9,   -3,    1;
%e A214622     45,   -40,   18,   -4,  1;
%e A214622   -256,   225, -100,   30, -5,  1;
%e A214622   1743, -1536,  675, -200, 45, -6, 1;
%e A214622   ...
%p A214622 A214622_row := proc(n) local s,t,k;
%p A214622   s := series(exp(z*x)/(sech(x)+tanh(x)),x,n+2);
%p A214622   t := factorial(n)*coeff(s,x,n); seq(coeff(t,z,k), k=(0..n)) end:
%p A214622 for n from 0 to 7 do A214622_row(n) od; # _Peter Luschny_, Aug 01 2012
%t A214622 A214622row[n_] := Module[{s, t},
%t A214622    s = Series[Exp[z*x]/(Sech[x] + Tanh[x]), {x, 0, n+2}];
%t A214622    t = n!*Coefficient[s, x, n];
%t A214622    Table[Coefficient[t, z, k], {k, 0, n}]];
%t A214622 Table[A214622row[n], {n, 0, 9}] // Flatten (* _Jean-François Alcover_, May 25 2024, after _Peter Luschny_ *)
%o A214622 (Sage)
%o A214622 R = PolynomialRing(ZZ, 'x')
%o A214622 @CachedFunction
%o A214622 def skp(n, x) : # Swiss-Knife polynomials A153641.
%o A214622     if n == 0 : return 1
%o A214622     return add(skp(k, 0)*binomial(n, k)*(x^(n-k)-(n+1)%2) for k in range(n)[::2])
%o A214622 def A109449_signed(n, k) : return 0 if k > n else R(skp(n, x)-skp(n, x-1)+x^n)[k]
%o A214622 T = matrix(ZZ, 9, A109449_signed).inverse(); T
%K A214622 sign,tabl
%O A214622 0,4
%A A214622 _Peter Luschny_, Jul 23 2012