|
|
A110291
|
|
Riordan array (1/(1-x), x*(1+2*x)).
|
|
2
|
|
|
1, 1, 1, 1, 3, 1, 1, 3, 5, 1, 1, 3, 9, 7, 1, 1, 3, 9, 19, 9, 1, 1, 3, 9, 27, 33, 11, 1, 1, 3, 9, 27, 65, 51, 13, 1, 1, 3, 9, 27, 81, 131, 73, 15, 1, 1, 3, 9, 27, 81, 211, 233, 99, 17, 1, 1, 3, 9, 27, 81, 243, 473, 379, 129, 19, 1, 1, 3, 9, 27, 81, 243, 665, 939, 577, 163, 21, 1
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,5
|
|
COMMENTS
|
|
|
LINKS
|
|
|
FORMULA
|
T(n, k) = [x^n]( x^k*(1+2*x)^k/(1-x) ).
Sum_{k=0..n} T(n, k) = A000975(n+1)).
Sum_{k=0..floor(n/2)} T(n-k, k) = A052947(n+1).
T(n, 0) = T(n, n) = 1.
T(2*n, n) = T(2*n+1), n) = A000244(n).
Sum_{k=0..n} (-1)^k * T(n, k) = A077912(n).
Sum_{k=0..n} 2^k * T(n, k) = A014335(n+2).
Sum_{k=0..n} 3^k * T(n, k) = A180146(n).
Sum_{k=0..floor(n/2)} (-1)^k * T(n-k, k) = A077890(n). (End)
|
|
EXAMPLE
|
Rows begin
1;
1, 1;
1, 3, 1;
1, 3, 5, 1;
1, 3, 9, 7, 1;
1, 3, 9, 19, 9, 1;
1, 3, 9, 27, 33, 11, 1;
1, 3, 9, 27, 65, 51, 13, 1;
1, 3, 9, 27, 81, 131, 73, 15, 1;
|
|
MATHEMATICA
|
F[k_]:= CoefficientList[Series[x^k*(1+2*x)^k/(1-x), {x, 0, 40}], x];
|
|
PROG
|
(Magma)
R<x>:=PowerSeriesRing(Rationals(), 30);
F:= func< k | Coefficients(R!( x^k*(1+2*x)^k/(1-x) )) >;
A110291:= func< n, k | F(k)[n-k+1] >;
(SageMath)
def p(k, x): return x^k*(1+2*x)^k/(1-x)
def A110291(n, k): return ( p(k, x) ).series(x, 30).list()[n]
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|