A104983 - OEIS

%I #9 Jul 22 2024 05:40:28

%S 1,2,6,24,122,750,5376,43856,400518,4046334,44808104,539850984,

%T 7032370302,98516491214,1477264979352,23612920280976,400847064718166,

%U 7202901369491694,136596819590256984,2726503675380494408

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

%H G. C. Greubel, <a href="/A104983/b104983.txt">Table of n, a(n) for n = 0..440</a>

%F a(n) = Sum_{k=0..n} A104980(n, k).

%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=A104980 *)

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

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

%Y Cf. A104980.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Apr 10 2005