|
| |
|
|
A156186
|
|
A generalized recursion triangle sequence : m=3; e(n,k,n)=(k + m - 1)*e(n - 1, k, m) + (m*n - k + 1 - m)*e(n - 1, k - 1, m); t(n,k)=e(n,k,m)+e(n,n-k,m).
|
|
0
|
|
|
|
2, 1, 1, 1, 6, 1, 1, 30, 30, 1, 1, 159, 360, 159, 1, 1, 1119, 3639, 3639, 1119, 1, 1, 10932, 41262, 57414, 41262, 10932, 1, 1, 136764, 582642, 898632, 898632, 582642, 136764, 1, 1, 2031933, 9957168, 16634718, 17182152, 16634718, 9957168, 2031933, 1, 1
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,1
|
|
|
COMMENTS
|
Row sums are:
{2, 8, 62, 680, 9518, 161804, 3236078, 74429792, 1935174590, 56120063108,...}.
The sequence comes from a generalization of the recurrence for A008517.
|
|
|
REFERENCES
|
Weisstein, Eric W. "Second-Order Eulerian Triangle." http://mathworld.wolfram.com/Second-OrderEulerianTriangle.html
|
|
|
LINKS
|
Table of n, a(n) for n=0..45.
|
|
|
FORMULA
|
m=3; e(n,k,n)=(k + m - 1)*e(n - 1, k, m) + (m*n - k + 1 - m)*e(n - 1, k - 1, m);
t(n,k)=e(n,k,m)+e(n,n-k,m).
|
|
|
EXAMPLE
|
{2},
{1, 1},
{1, 6, 1},
{1, 30, 30, 1},
{1, 159, 360, 159, 1},
{1, 1119, 3639, 3639, 1119, 1},
{1, 10932, 41262, 57414, 41262, 10932, 1},
{1, 136764, 582642, 898632, 898632, 582642, 136764, 1},
{1, 2031933, 9957168, 16634718, 17182152, 16634718, 9957168, 2031933, 1},
{1, 34474173, 194894781, 369132246, 369086094, 369086094, 369132246, 194894781, 34474173, 1},
{1, 654773346, 4228768422, 9285005715, 9780535908, 8221896324, 9780535908, 9285005715, 4228768422, 654773346, 1}
|
|
|
MATHEMATICA
|
m = 3; e[n_, 0, m_] := 1;
e[n_, k_, m_] := 0 /; k >= n;
e[n_, k_, 1] := 1 /; k >= n;
e[n_, k_, m_] := (k + m - 1)e[n - 1, k, m] + (m*n - k + 1 - m)e[n - 1, k - 1, m];
Table[Table[e[n, k, m], {k, 0, n - 1}], {n, 1, 10}];
Table[Table[e[n, k, m] + e[n, n - k, m], {k, 0, n}], {n, 0, 10}];
Flatten[%]
|
|
|
CROSSREFS
|
A054091, A054090, A008517, A156141
Sequence in context: A320637 A280491 A157118 * A156233 A251725 A292977
Adjacent sequences: A156183 A156184 A156185 * A156187 A156188 A156189
|
|
|
KEYWORD
|
nonn,tabl,uned
|
|
|
AUTHOR
|
Roger L. Bagula, Feb 05 2009
|
|
|
STATUS
|
approved
|
| |
|
|
