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%I #8 Jun 08 2021 14:57:53
%S 1,4,21,133,977,8135,75609,775667,8707057,106185715,1398451353,
%T 19786121467,299384925569,4825081148819,82531968286569,
%U 1493412479919371,28504390805515921,572363196501249667,12061937537478658809
%N Column 3 of triangle A104980, omitting leading zeros.
%C Equals one-half of column 0 (after initial term) in triangle A104988, which equals the matrix square of triangle A104980.
%H G. C. Greubel, <a href="/A104982/b104982.txt">Table of n, a(n) for n = 0..440</a>
%F a(n) = A104988(n+1, 0)/2 for n>=0.
%t T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[k==n, 1, If[k==n-1, n, k*T[n, k+1] + Sum[T[j, 0]*T[n, j+k+1], {j,0,n-k-1}]]]];
%t Table[T[n+3, 3], {n, 0, 30}] (* _G. C. Greubel_, Jun 07 2021 *)
%o (PARI) {a(n) = if(n<0, 0, (matrix(n+4, n+4, m, j, if(m==j, 1, if(m==j+1, -m+1, -polcoeff((1-1/sum(i=0, m, i!*x^i))/x +O(x^m), m-j-1))))^-1)[n+4,4])}
%o (Sage)
%o @CachedFunction
%o def T(n,k):
%o if (k<0 or k>n): return 0
%o elif (k==n): return 1
%o elif (k==n-1): return n
%o else: return k*T(n, k+1) + sum( T(j, 0)*T(n, j+k+1) for j in (0..n-k-1) )
%o [T(n+3,3) for n in (0..30)] # _G. C. Greubel_, Jun 07 2021
%Y Cf. A104980, A104981, A104988.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Apr 10 2005
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