A029600 Numbers in the (2,3)-Pascal triangle (by row).

1, 2, 3, 2, 5, 3, 2, 7, 8, 3, 2, 9, 15, 11, 3, 2, 11, 24, 26, 14, 3, 2, 13, 35, 50, 40, 17, 3, 2, 15, 48, 85, 90, 57, 20, 3, 2, 17, 63, 133, 175, 147, 77, 23, 3, 2, 19, 80, 196, 308, 322, 224, 100, 26, 3, 2, 21, 99, 276, 504, 630, 546, 324, 126, 29, 3, 2, 23, 120, 375, 780, 1134, 1176, 870, 450, 155, 32, 3

Offset

0,2

Comments

Reverse of A029618. - Philippe Deléham, Nov 21 2006

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

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

For n > 0: T(n,k) = A029635(n,k) + A007318(n,k), 0 <= k <= n. - Reinhard Zumkeller, Apr 16 2012

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

For n>0, row sums = 5*2^(n-1). Generally, for all (a,b)-Pascal triangles, row sums are (a+b)*2^(n-1), n>0. - Bob Selcoe, Mar 28 2015

Links

Reinhard Zumkeller, Rows n=0..150 of triangle, flattened

Formula

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

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

Example

First few rows are:
  1;
  2, 3;
  2, 5,  3;
  2, 7,  8,  3;
  2, 9, 15, 11, 3;
...

Maple

T:= proc(n, k) option remember;
      if k=0 and n=0 then 1
    elif k=0 then 2
    elif k=n then 3
    else T(n-1, k-1) + T(n-1, k)
      fi
    end:
seq(seq(T(n, k), k=0..n), n=0..12); # G. C. Greubel, Nov 12 2019

Mathematica

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

Prog

(Haskell)
a029600 n k = a029600_tabl !! n !! k
a029600_row n = a029600_tabl !! n
a029600_tabl = [1] : iterate
   (\row -> zipWith (+) ([0] ++ row) (row ++ [0])) [2, 3]
-- Reinhard Zumkeller, Apr 08 2012
(PARI) T(n, k) = if(n==0 && k==0, 1, if(k==0, 2, if(k==n, 3, 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 2
    elif (k==n): return 3
    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
(GAP)
T:= function(n, k)
    if n=0 and k=0 then return 1;
    elif k=0 then return 2;
    elif k=n then return 3;
    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

Crossrefs

Cf. A007318 (Pascal's triangle), A029618, A084938, A228196, A228576.

Sequence in context: A049805 A104887 A064886 * A169616 A344448 A111076

Adjacent sequences: A029597 A029598 A029599 * A029601 A029602 A029603

Keyword

nonn,tabl,easy

Author

Mohammad K. Azarian

Extensions

More terms from James A. Sellers

Status

approved