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

Author

Mohammad K. Azarian

Keywords

• nonn
• tabl
• easy

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

Examples

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

Links

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

Crossrefs

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

Programs

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

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

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 *)

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

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

Extensions

More terms from James Sellers