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%I #32 Jun 07 2021 18:50:07
%S 0,1,2,7,33,191,1297,10063,87669,847015,8989301,103996703,1303132269,
%T 17589153719,254509227541,3931158238735,64573130459613,
%U 1124144767682215,20677664894412965,400760695386194687,8163539437728923181
%N Column 1 of triangle A104980; also equals column 0 of triangle A104986, which equals the matrix logarithm of A104980.
%H G. C. Greubel, <a href="/A104981/b104981.txt">Table of n, a(n) for n = 0..440</a>
%F From _Gary W. Adamson_, Jul 14 2011: (Start)
%F Let M = triangle A128175 as an infinite square production matrix (deleting the first "1"):
%F 1, 1, 0, 0, 0, ...
%F 2, 2, 1, 0, 0, ...
%F 4, 4, 3, 1, 0, ...
%F 8, 8, 7, 4, 1, ...
%F ...
%F a(n) = sum of top row terms of M^(n-1). Example: top row of M^4 = (71, 71, 38, 10, 1), sum = 191 = a(5). (End)
%F a(0) = 1, a(n) = n * a(n-1) + Sum_{j=1..n} A003319(j) * a(n - j), with offset 0 for the term 1. - _F. Chapoton_, Feb 26 2018
%t T[n_, k_]:= T[n, k]= If[n<k || k<0, 0, If[n==k, 1, If[n==k+1, n, k*T[n, k+1] + Sum[T[j, 0]*T[n, j+k+1], {j, 0, n-k-1}]]]];
%t a[n_]:= T[n, 1];
%t Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Aug 09 2018 *)
%o (PARI) {a(n) = if(n<0, 0, (matrix(n+2, n+2, 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+1,2])}
%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,1) for n in (0..30)] # _G. C. Greubel_, Jun 07 2021
%Y Cf. A104980, A104982, A104986, A128175.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Apr 10 2005
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