A110291 Riordan array (1/(1-x), x*(1+2*x)).

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

Offset: 0

Author

Paul Barry, Jul 18 2005

Keywords

• easy
• nonn
• tabl

Comments

Inverse is A110292.

Examples

			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;
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Crossrefs

Cf. A000244, A001045, A001047, A005408, A014335.

Cf. A066810, A077890, A077912, A110292, A180146.

Cf. A000975 (row sums), A052947 (diagonal sums).

Programs

Magma

R:=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

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 *)

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

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)

Extensions

a(30) and following corrected by Georg Fischer, Oct 11 2022