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A104983 Row sums of triangular matrix T = A104980 which satisfies: SHIFT_LEFT(column 0 of T^p) = p*(column p+1 of T).

%I A104983 #9 Jul 22 2024 05:40:28
%S A104983 1,2,6,24,122,750,5376,43856,400518,4046334,44808104,539850984,
%T A104983 7032370302,98516491214,1477264979352,23612920280976,400847064718166,
%U A104983 7202901369491694,136596819590256984,2726503675380494408
%N A104983 Row sums of triangular matrix T = A104980 which satisfies: SHIFT_LEFT(column 0 of T^p) = p*(column p+1 of T).
%H A104983 G. C. Greubel, <a href="/A104983/b104983.txt">Table of n, a(n) for n = 0..440</a>
%F A104983 a(n) = Sum_{k=0..n} A104980(n, k).
%t A104983 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=A104980 *)
%t A104983 Table[Sum[T[n, k], {k,0,n}], {n,0,30}] (* _G. C. Greubel_, Jun 07 2021 *)
%o A104983 (PARI) {a(n) = if(n<0, 0, sum(k=0, n, (matrix(n+1, n+1, 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,k+1]))};
%o A104983 (Sage)
%o A104983 @CachedFunction
%o A104983 def T(n,k):
%o A104983     if (k<0 or k>n): return 0
%o A104983     elif (k==n): return 1
%o A104983     elif (k==n-1): return n
%o A104983     else: return k*T(n, k+1) + sum( T(j, 0)*T(n, j+k+1) for j in (0..n-k-1) )
%o A104983 [sum(T(n,k) for k in (0..n)) for n in (0..30)] # _G. C. Greubel_, Jun 07 2021
%Y A104983 Cf. A104980.
%K A104983 nonn
%O A104983 0,2
%A A104983 _Paul D. Hanna_, Apr 10 2005