A104982 Column 3 of triangle A104980, omitting leading zeros.

1, 4, 21, 133, 977, 8135, 75609, 775667, 8707057, 106185715, 1398451353, 19786121467, 299384925569, 4825081148819, 82531968286569, 1493412479919371, 28504390805515921, 572363196501249667, 12061937537478658809

Offset: 0

Author

Paul D. Hanna, Apr 10 2005

Keywords

• nonn

Comments

Equals one-half of column 0 (after initial term) in triangle A104988, which equals the matrix square of triangle A104980.

Links

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

Crossrefs

Cf. A104980, A104981, A104988.

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}]]]];
Table[T[n+3, 3], {n, 0, 30}] (* G. C. Greubel, Jun 07 2021 *)

PARI

{a(n) = if(n<0, 0, (matrix(n+4, n+4, 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+4,4])}

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+3,3) for n in (0..30)] # G. C. Greubel, Jun 07 2021

Formula

a(n) = A104988(n+1, 0)/2 for n>=0.