A238731 Riordan array ((1-2*x)/(1-3*x+x^2), x/(1-3*x+x^2)).

1, 1, 1, 2, 4, 1, 5, 13, 7, 1, 13, 40, 33, 10, 1, 34, 120, 132, 62, 13, 1, 89, 354, 483, 308, 100, 16, 1, 233, 1031, 1671, 1345, 595, 147, 19, 1, 610, 2972, 5561, 5398, 3030, 1020, 203, 22, 1, 1597, 8495, 17984, 20410, 13893, 5943, 1610, 268, 25, 1, 4181

Offset

0,4

Comments

Unsigned version of A124037 and A126126.

Subtriangle of the triangle given by (0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 2, -2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Row sums are A001075(n).

Diagonal sums are A133494(n).

Sum_{k=0..n} T(n,k)*x^k = A001519(n), A001075(n), A002320(n), A038723(n), A033889(n) for x = 0, 1, 2, 3, 4 respectively. - Philippe Deléham, Mar 05 2014

Links

Table of n, a(n) for n=0..55.

Formula

T(n,k) = 3*T(n-1,k) + T(n-1,k-1) - T(n-2,k), T(0,0) = T(1,0) = T(1,1) = 1, T(n,k) = 0 if k<0 or if k>n.

G.f.: (1-2*x)/(1-(y+3)*x+x^2). - Philippe Deléham, Mar 05 2014

Example

Triangle begins:
1;
1, 1;
2, 4, 1;
5, 13, 7, 1;
13, 40, 33, 10, 1;
34, 120, 132, 62, 13, 1;
89, 354, 483, 308, 100, 16, 1;
233, 1031, 1671, 1345, 595, 147, 19, 1;...
Triangle (0, 1, 1, 1, 0, 0, 0, ...) DELTA (1, 0, 2, -2, 0, 0, ...) begins:
1;
0, 1;
0, 1, 1;
0, 2, 4, 1;
0, 5, 13, 7, 1;
0, 13, 40, 33, 10, 1;
0, 34, 120, 132, 62, 13, 1;
0, 89, 354, 483, 308, 100, 16, 1;
0, 233, 1031, 1671, 1345, 595, 147, 19, 1;...

Mathematica

(* The function RiordanArray is defined in A256893. *)
RiordanArray[(1-2#)/(1-3#+#^2)&, x/(1-3#+#^2)&, 10] // Flatten (* Jean-François Alcover, Jul 16 2019 *)

Crossrefs

Cf. A001519, A000012, A016777, A062708.

Cf. A001906, A124037, A126126.

Sequence in context: A085059 A336164 A181336 * A124037 A126126 A090285

Adjacent sequences: A238728 A238729 A238730 * A238732 A238733 A238734

Keyword

nonn,tabl

Author

Philippe Deléham, Mar 03 2014

Status

approved