OFFSET
0,5
COMMENTS
Inverse is A110292.
LINKS
G. C. Greubel, Rows n = 0..50 of the triangle, flattened
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).
From G. C. Greubel, Jan 05 2023: (Start)
T(n, 0) = T(n, n) = 1.
T(n, n-1) = A005408(n-1).
T(2*n, n) = T(2*n+1), n) = A000244(n).
T(2*n, n+1) = A066810(n+1).
T(2*n, n-1) = A000244(n-1).
T(2*n+1, n+1) = A001047(n+1).
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];
A110291[n_, k_]:= F[k][[n+1]];
Table[A110291[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Jan 05 2023 *)
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] >;
[A110291(n, k): k in [0..n], n in [0..10]]; // G. C. Greubel, Jan 05 2023
(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]
flatten([[A110291(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jan 05 2023
CROSSREFS
KEYWORD
AUTHOR
Paul Barry, Jul 18 2005
EXTENSIONS
a(30) and following corrected by Georg Fischer, Oct 11 2022
STATUS
approved