A104981 Column 1 of triangle A104980; also equals column 0 of triangle A104986, which equals the matrix logarithm of A104980.

0, 1, 2, 7, 33, 191, 1297, 10063, 87669, 847015, 8989301, 103996703, 1303132269, 17589153719, 254509227541, 3931158238735, 64573130459613, 1124144767682215, 20677664894412965, 400760695386194687, 8163539437728923181

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, A104982, A104986, A128175.

Programs

Mathematica

T[n_, k_]:= T[n, k]= If[nJean-François Alcover, Aug 09 2018 *)

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])}

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) )
[T(n,1) for n in (0..30)] # G. C. Greubel, Jun 07 2021

Formula

From Gary W. Adamson, Jul 14 2011: (Start)

Let M = triangle A128175 as an infinite square production matrix (deleting the first "1"):

1, 1, 0, 0, 0, ...

2, 2, 1, 0, 0, ...

4, 4, 3, 1, 0, ...

8, 8, 7, 4, 1, ...

...

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)

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