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

1, 2, 6, 24, 122, 750, 5376, 43856, 400518, 4046334, 44808104, 539850984, 7032370302, 98516491214, 1477264979352, 23612920280976, 400847064718166, 7202901369491694, 136596819590256984, 2726503675380494408

Offset: 0

Author

Paul D. Hanna, Apr 10 2005

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..440

Crossrefs

Cf. A104980.

Programs

Mathematica

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 *)
Table[Sum[T[n, k], {k,0,n}], {n,0,30}] (* G. C. Greubel, Jun 07 2021 *)

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]))};

Sage

@CachedFunction
def T(n,k):
    if (k<0 or k>n): return 0
    elif (k==n): return 1
    elif (k==n-1): return n
    else: return k*T(n, k+1) + sum( T(j, 0)*T(n, j+k+1) for j in (0..n-k-1) )
[sum(T(n,k) for k in (0..n)) for n in (0..30)] # G. C. Greubel, Jun 07 2021

Formula

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