A104985 - OEIS

%I #8 Sep 22 2025 16:00:46

%S 1,0,-2,-6,-20,-92,-554,-4002,-33096,-306440,-3135766,-35134670,

%T -427878628,-5628940084,-79572364498,-1203168642362,-19379896959776,

%U -331331041788640,-5993029816637262,-114348894263852326,-2295445815821635932,-48362099044178487564

%N Row sums of triangle A104984.

%C A104984 equals the matrix inverse of A104980.

%H G. C. Greubel, <a href="/A104985/b104985.txt">Table of n, a(n) for n = 0..445</a>

%F a(n) = Sum_{k=0..n} A104984(n, k).

%t A003319[n_]:= A003319[n]= If[n==0, 0, n! -Sum[j!*A003319[n-j], {j,n-1}]];

%t A104984[n_, k_]:= If[k==n, 1, If[k==n-1, -n, -A003319[n-k]]];

%t a[n_]:= Sum[A104984[n, k], {k,0,n}];

%t Table[a[n], {n, 0, 30}] (* _G. C. Greubel_, Jun 07 2021 *)

%o (PARI) {a(n)=sum(k=0,n,if(k==n,1,if(k==n-1,-n, -polcoeff((1-1/sum(i=0,n-k,i!*x^i))/x+O(x^(n-k)),n-k-1) )))}

%o (SageMath)

%o @CachedFunction

%o def T(n,k):

%o     if (k==n): return 1

%o     elif (k==n-1): return -n

%o     else: return -factorial(n-k) - sum( factorial(j)*T(n-k-j, 0) for j in (1..n-k-1) )

%o [sum(T(n,k) for k in (0..n)) for n in (0..30)] # _G. C. Greubel_, Jun 07 2021

%Y Cf. A104984, A104980.

%K sign

%O 0,3

%A _Paul D. Hanna_, Apr 10 2005