A141541 A triangular sequence of coefficients in a renormalized fractional factorial recursion ( a neo -combinatorial process): a(n) = A000045[n]; Renormalized factorial; f(n) = a(n)*n*f(n - 1)/a(n - 1); Neo-combination: t(n, m) = f(n)/(f(n - m)*f(m)).

1, 1, 1, 1, 4, 1, 1, 8, 8, 1, 1, 10, 18, 10, 1, 1, 12, 29, 29, 12, 1, 1, 14, 44, 56, 44, 14, 1, 1, 17, 61, 99, 99, 61, 17, 1, 1, 19, 82, 158, 198, 158, 82, 19, 1, 1, 22, 105, 236, 357, 357, 236, 105, 22, 1, 1, 24, 131, 338, 594, 713, 594, 338, 131, 24, 1

Offset

1,5

Comments

Row sums:

{1, 2, 6, 18, 40, 84, 174, 356, 718, 1442, 2889}.

Links

Table of n, a(n) for n=1..66.

Formula

a(n) = A000129[n]; Renormalized factorial; f(n) = a(n)*n*f(n - 1)/a(n - 1); Neo-combination: t(n, m) = f(n)/(f(n - m)*f(m))

Example

{1},
{1, 1},
{1, 4, 1},
{1, 8, 8, 1},
{1, 10, 18, 10, 1},
{1, 12, 29, 29, 12, 1},
{1, 14, 44, 56, 44, 14, 1},
{1, 17, 61, 99, 99, 61, 17, 1},
{1, 19, 82, 158, 198, 158, 82, 19, 1},
{1, 22, 105, 236, 357, 357, 236, 105, 22, 1},
{1, 24, 131, 338, 594, 713, 594, 338, 131, 24, 1}

Mathematica

Clear[a, n, f, g] a[0] = 0; a[1] = 1; a[n_] := a[n] = 2*a[n - 1] + a[n - 2] Table[a[n], {n, 0, 30}] (* renormalized fractional factorial recursion*) f[0] = 1; f[1] = 1; f[n_] := f[n] = a[n]*n*f[n - 1]/a[n - 1]; Table[f[n], {n, 0, 10}]; g[n_, m_] := g[n, m] = f[n]/(f[n - m]*f[m]); Table[Table[Round[g[n, m]], {m, 0, n}], {n, 0, 10}]; Flatten[%]

Crossrefs

Cf. A000129.

Sequence in context: A346875 A289511 A158687 * A334552 A347676 A177947

Adjacent sequences: A141538 A141539 A141540 * A141542 A141543 A141544

Keyword

nonn,uned,tabl

Author

Roger L. Bagula and Gary W. Adamson, Aug 15 2008

Status

approved