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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
Inverse is A110292.
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).
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
Cf. A000975 (row sums), A052947 (diagonal sums).
Sequence in context: A271451 A131248 A116445 * A152027 A077308 A075001
KEYWORD
easy,nonn,tabl
AUTHOR
Paul Barry, Jul 18 2005
EXTENSIONS
a(30) and following corrected by Georg Fischer, Oct 11 2022
STATUS
approved

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Last modified March 28 17:25 EDT 2024. Contains 371254 sequences. (Running on oeis4.)