|
|
A208153
|
|
Convolution triangle based on A006053.
|
|
1
|
|
|
1, 1, 1, 3, 2, 1, 4, 7, 3, 1, 9, 14, 12, 4, 1, 14, 35, 31, 18, 5, 1, 28, 70, 87, 56, 25, 6, 1, 47, 154, 207, 175, 90, 33, 7, 1, 89, 306, 504, 476, 310, 134, 42, 8, 1, 155, 633, 1137, 1274, 941, 504, 189, 52, 9, 1
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,4
|
|
COMMENTS
|
Riordan array (1/(1-x-2*x^2+x^3), x/(1-x-2*x^2+x^3).
Subtriangle of triangle given by (0, 1, 2, -5/2, 1/10, 2/5, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.
Mirror image of triangle in A188107.
|
|
LINKS
|
|
|
FORMULA
|
T(n,k) = T(n-1,k-1) + T(n-1,k) + 2*T(n-2,k) - T(n-3,k).
G.f.: 1/(1-x-2*x^2+x^3-y*x).
Sum_{k, k>=0} T(n-2*k,k) = A001045(n+1).
Sum_{k, 0<=k<=n} T(n,k)*x^k = (-1)^n*A008346(n), A006053(n+2), A001654(n+1) for x = -1, 0, 1 respectively.
|
|
EXAMPLE
|
Triangle begins :
1
1, 1
3, 2, 1
4, 7, 3, 1
9, 14, 12, 4, 1
14, 35, 31, 18, 5, 1
Triangle (0, 1 ,2, -5/2, 1/10, 2/5, 0, 0,...) DELTA (1, 0, 0, 0,...) begins :
1
0, 1
0, 1, 1
0, 3, 2, 1
0, 4, 7, 3, 1
0, 9, 14, 12, 4, 1
0, 14, 35, 31, 18, 5, 1
|
|
MATHEMATICA
|
nmax=9; Flatten[CoefficientList[Series[CoefficientList[Series[1/(1 - x - 2*x^2 + x^3 - y*x), {x, 0, nmax}], x], {y, 0, nmax}], y]] (* Indranil Ghosh, Mar 10 2017 *)
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|