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

0,3

Links

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

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

Mathematica

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}]]]];
a[n_]:= T[n, 1];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Aug 09 2018 *)

Prog

(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

Crossrefs

Cf. A104980, A104982, A104986, A128175.

Sequence in context: A186760 A162661 A299043 * A058797 A121965 A337058

Adjacent sequences: A104978 A104979 A104980 * A104982 A104983 A104984

Keyword

nonn

Author

Paul D. Hanna, Apr 10 2005

Status

approved