A182001 Riordan array ((2*x+1)/(1-x-x^2), x/(1-x-x^2)).

1, 3, 1, 4, 4, 1, 7, 9, 5, 1, 11, 20, 15, 6, 1, 18, 40, 40, 22, 7, 1, 29, 78, 95, 68, 30, 8, 1, 47, 147, 213, 185, 105, 39, 9, 1, 76, 272, 455, 466, 320, 152, 49, 10, 1, 123, 495, 940, 1106, 891, 511, 210, 60, 11, 1, 199, 890, 1890, 2512, 2317, 1554, 770, 280, 72, 12, 1

Offset

0,2

Comments

Subtriangle of the triangle given by (0, 3, -5/3, -1/3, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, -2/3, 2/3, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Antidiagonal sums are A001045(n+2).

Links

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Formula

G.f.: (1+2*x)/(1-x-y*x-x^2).

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

Sum_{k=0..nn} T(n,k)*x^k = A000034(n), A000032(n+1), A048654(n), A108300(n), A048875(n) for x = -1, 0, 1, 2, 3 respectively.

Example

Triangle begins :
    1;
    3,   1;
    4,   4,    1;
    7,   9,    5,    1;
   11,  20,   15,    6,    1;
   18,  40,   40,   22,    7,    1;
   29,  78,   95,   68,   30,    8,   1;
   47, 147,  213,  185,  105,   39,   9,   1;
   76, 272,  455,  466,  320,  152,  49,  10, 1;
  123, 495,  940, 1106,  891,  511, 210,  60, 11,  1;
  199, 890, 1890, 2512, 2317, 1554, 770, 280, 72, 12, 1;
(0, 3, -5/3, -1/3, 0, 0, ...) DELTA (1, 0, -2/3, 2/3, 0, 0, ...) begins:
  1;
  0,  1;
  0,  3,  1;
  0,  4,  4,  1;
  0,  7,  9,  5,  1;
  0, 11, 20, 15,  6, 1;
  0, 18, 40, 40, 22, 7, 1;

Maple

with(combinat);
T:= proc(n, k) option remember;
      if k<0 or k>n then 0
    elif k=n then 1
    elif k=0 then fibonacci(n+2) + fibonacci(n)
    else T(n-1, k) + T(n-1, k-1) + T(n-2, k)
      fi; end:
seq(seq(T(n, k), k=0..n), n=0..10); # G. C. Greubel, Feb 18 2020

Mathematica

With[{m = 10}, CoefficientList[CoefficientList[Series[(1+2*x)/(1-x-y*x-x^2), {x, 0, m}, {y, 0, m}], x], y]] // Flatten (* Georg Fischer, Feb 18 2020 *)
T[n_, k_]:= T[n, k]= If[k<0||k>n, 0, If[k==n, 1, If[k==0, LucasL[n+1], T[n-1, k] + T[n-1, k-1] + T[n-2, k] ]]]; Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Feb 18 2020 *)

Prog

(Magma)
function T(n, k)
  if k lt 0 or k gt n then return 0;
  elif k eq n then return 1;
  elif k eq 0 then return Lucas(n+1);
  else return T(n-1, k) + T(n-1, k-1) + T(n-2, k);
  end if; return T; end function;
[T(n, k): k in [0..n], n in [0..10]]; // G. C. Greubel, Feb 18 2020

Crossrefs

Cf. Columns : A000032, A023607, A152881

Sequence in context: A286625 A129246 A125608 * A099813 A141300 A260502

Adjacent sequences: A181998 A181999 A182000 * A182002 A182003 A182004

Keyword

easy,nonn,tabl

Author

Philippe Deléham, Apr 05 2012

Extensions

a(29) corrected by and a(55)-a(65) from Georg Fischer, Feb 18 2020

Status

approved