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A104981 Column 1 of triangle A104980; also equals column 0 of triangle A104986, which equals the matrix logarithm of A104980. 7
0, 1, 2, 7, 33, 191, 1297, 10063, 87669, 847015, 8989301, 103996703, 1303132269, 17589153719, 254509227541, 3931158238735, 64573130459613, 1124144767682215, 20677664894412965, 400760695386194687, 8163539437728923181 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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

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Last modified June 22 21:29 EDT 2021. Contains 345393 sequences. (Running on oeis4.)