A155556 Multi-bifurcating recursion of a factorial type based on the Eulerian numbers A008292 as a triangle sequence: t(n,k)=Sum[(-1)^j Binomial[n + 1, j](k + 1 - j)^n, {j, 0, k + 1}]; f(n, m) = If[m <= Floor[n/2], f(m, 1)*f(n - m, 1)*t(n + 1, m)];

1, 1, 4, 44, 1144, 1056, 65208, 53152, 7824960, 5450016, 4677376, 1932765120, 1119751776, 786197984, 970248090240, 457228062720, 253156757568, 204411475840, 982861315413120, 369853933363200, 156721804462080

Offset

0,3

Comments

Row sums are:

{1, 1, 4, 44, 2200, 118360, 17952352, 3838714880, 1885044386368,

1607186778033408, 2934910973174349312,...}.

The Eulerian numbers factored as factorial like to middle Floor[n/2]:

t(n,m)=f(n,m)/(f[m,1]*f[n-m,1]).

The idea is to factor the Eulerian numbers as

if the coefficients were made up of equivalents to factorials.

The result is a multi-bifurcating recursion thast fits the Eulerian numbers.

Links

Table of n, a(n) for n=0..20.

Example

Half Eulerian numbers: Table[Table[f[n, m]/(f[m, 1]*f[n - m, 1]), {m, 0, Floor[n/2]}], {n, 0, 10}];
{1},
{1},
{1, 4},
{1, 11},
{1, 26, 66},
{1, 57, 302},
{1, 120, 1191, 2416},
{1, 247, 4293, 15619},
{1, 502, 14608, 88234, 156190},
{1, 1013, 47840, 455192, 1310354},
{1, 2036, 152637, 2203488, 9738114, 15724248}...
Factorial type triangle is:
{1},
{1},
{4},
{44},
{1144, 1056},
{65208, 53152},
{7824960, 5450016, 4677376},
{1932765120, 1119751776, 786197984},
{970248090240, 457228062720, 253156757568, 204411475840},
{982861315413120, 369853933363200, 156721804462080, 97749724795008},
{2001105638181112320, 592383030999851520, 187388288944496640, 87173203289103360, 66860811759785472}

Mathematica

Clear[t, n, m, f, x];
t[n_, k_] = Sum[(-1)^j Binomial[n + 1, j](k + 1 - j)^n, {j, 0, k + 1}];
f[0, 1] = 1; f[1, 1] = 1; f[2, 1] = 4;
f[n_, m_] := f[n, m] = If[m <= Floor[n/2], f[m, 1]*f[n - m, 1]*t[n + 1, m]];
a = Join[{{1}}, {{1}}, Table[Table[f[n, m], {m, 1, Floor[n/2]}], {n, 2, 10}]];
Flatten[%]

Crossrefs

A008292

Sequence in context: A255928 A137783 A136552 * A188456 A306372 A217473

Adjacent sequences: A155553 A155554 A155555 * A155557 A155558 A155559

Keyword

nonn,tabl,uned

Author

Roger L. Bagula, Jan 24 2009

Status

approved