A029618 Numbers in (3,2)-Pascal triangle (by row).

1, 3, 2, 3, 5, 2, 3, 8, 7, 2, 3, 11, 15, 9, 2, 3, 14, 26, 24, 11, 2, 3, 17, 40, 50, 35, 13, 2, 3, 20, 57, 90, 85, 48, 15, 2, 3, 23, 77, 147, 175, 133, 63, 17, 2, 3, 26, 100, 224, 322, 308, 196, 80, 19, 2, 3, 29, 126, 324, 546, 630, 504, 276, 99, 21, 2, 3, 32, 155, 450, 870

Offset: 0

Author

Mohammad K. Azarian

Keywords

• nonn
• easy
• tabl

Comments

Reverse of A029600. - Philippe Deléham, Nov 21 2006

Triangle T(n,k), read by rows, given by (3,-2,0,0,0,0,0,0,0,...) DELTA (2,-1,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 10 2011

Row n: expansion of (3+2x)*(1+x)^(n-1), n>0. - Philippe Deléham, Oct 10 2011

For a closed-form formula for generalized Pascal's triangle see A228576. - Boris Putievskiy, Sep 04 2013

Examples

			Triangle begins as:
  1;
  3,  2;
  3,  5,  2;
  3,  8,  7,  2;
  3, 11, 15,  9,  2;
  ...
		

Links

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

Crossrefs

Cf. A007318, A029600, A084938, A228196, A228576, A016789 (2nd column), A005449 (3rd column), A006002 (4th column).

Programs

GAP

T:= function(n,k)
    if n=0 and k=0 then return 1;
    elif k=0 then return 3;
    elif k=n then return 2;
    else return T(n-1,k-1) + T(n-1,k);
    fi;
  end;
Flat(List([0..12], n-> List([0..n], k-> T(n,k) ))); # G. C. Greubel, Nov 12 2019

Maple

A029618 := proc(n,k)
    if k < 0 or k > n then
        0;
    elif  n = 0 then
        1;
    elif k=0 then
        3;
    elif k = n then
        2;
    else
        procname(n-1,k-1)+procname(n-1,k) ;
    end if;
end proc: # R. J. Mathar, Jul 08 2015

Mathematica

T[n_, k_]:= T[n, k]= If[n==0 && k==0, 1, If[k==0, 3, If[k==n, 2, T[n-1, k-1] + T[n-1, k] ]]]; Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Nov 13 2019 *)

PARI

T(n,k) = if(n==0 && k==0, 1, if(k==0, 3, if(k==n, 2, T(n-1, k-1) + T(n-1, k) ))); \\ G. C. Greubel, Nov 12 2019

Sage

@CachedFunction
def T(n, k):
    if (n==0 and k==0): return 1
    elif (k==0): return 3
    elif (k==n): return 2
    else: return T(n-1,k-1) + T(n-1, k)
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Nov 12 2019

Formula

T(n,k) = T(n-1,k-1) + T(n-1,k) with T(0,0)=1, T(n,0)=3, T(n,n)=2; n, k > 0. - Boris Putievskiy, Sep 04 2013

G.f.: (-1-x*y-2*x)/(-1+x*y+x). - R. J. Mathar, Aug 11 2015