A106340 - OEIS

%I #12 Jun 27 2019 06:11:12

%S 1,-1,1,1,-3,1,-1,9,-7,1,1,-45,55,-15,1,-1,585,-835,285,-31,1,1,

%T -21105,30835,-11025,1351,-63,1,-1,1858185,-2719675,977445,-121891,

%U 6069,-127,1,1,-367958745,538607755,-193649085,24187051,-1213065,26335,-255,1,-1,157169540745,-230061795355,82717588485

%N Triangle T, read by rows, equal to the matrix inverse of the triangle defined by [T^-1](n,k) = (n-k)!*A008278(n+1,k+1), for n>=k>=0, where A008278 is a triangle of Stirling numbers of 2nd kind.

%C Row sums are {1,0,-1,2,-3,4,-5,6,...}. Column 1 is A106341.

%F T(n, k) = A106338(n, k)/k!, for n>=k>=0.

%e Triangle T begins:

%e   1;

%e   -1,1;

%e   1,-3,1;

%e   -1,9,-7,1;

%e   1,-45,55,-15,1;

%e   -1,585,-835,285,-31,1;

%e   1,-21105,30835,-11025,1351,-63,1;

%e   -1,1858185,-2719675,977445,-121891,6069,-127,1;

%e   1,-367958745,538607755,-193649085,24187051,-1213065,26335,-255,1;

%e   ...

%e Matrix inverse begins:

%e   1;

%e   1,1;

%e   2,3,1;

%e   6,12,7,1;

%e   24,60,50,15,1;

%e   120,360,390,180,31,1;

%e   ...

%e where [T^-1](n,k) = (n-k)!*A008278(n+1,k+1).

%t rows = 10;

%t M = Table[If[r >= c, (r-c)! Sum[(-1)^(r-c-m+1) m^r/m!/(r-c-m+1)!, {m, 0, r-c+1}], 0], {r, rows}, {c, rows}] // Inverse;

%t T[n_, k_] := M[[n+1, k+1]];

%t Table[T[n, k], {n, 0, rows-1}, {k, 0, n}] (* _Jean-François Alcover_, Jun 27 2019, from PARI *)

%o (PARI) {T(n,k)=(matrix(n+1,n+1,r,c,if(r>=c,(r-c)!* sum(m=0,r-c+1,(-1)^(r-c+1-m)*m^r/m!/(r-c+1-m)!)))^-1)[n+1,k+1]}

%o (Sage)

%o def A106340_matrix(d):

%o     def A130850(n, k):   # EulerianNumber = A173018

%o         return add(EulerianNumber(n,j)*binomial(n-j,k) for j in (0..n))

%o     return matrix(ZZ, d, A130850).inverse()

%o A106340_matrix(8)  # _Peter Luschny_, May 21 2013

%Y Cf. A106338, A008278, A106341.

%K sign,tabl

%O 0,5

%A _Paul D. Hanna_, May 01 2005