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

%I A029600 #57 Jul 02 2025 16:01:56
%S A029600 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,
%T A029600 15,48,85,90,57,20,3,2,17,63,133,175,147,77,23,3,2,19,80,196,308,322,
%U A029600 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
%N A029600 Numbers in the (2,3)-Pascal triangle (by row).
%C A029600 Reverse of A029618. - _Philippe Deléham_, Nov 21 2006
%C A029600 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
%C A029600 Row n: expansion of (2+3x)*(1+x)^(n-1), n>0. - _Philippe Deléham_, Oct 10 2011.
%C A029600 For n > 0: T(n,k) = A029635(n,k) + A007318(n,k), 0 <= k <= n. - _Reinhard Zumkeller_, Apr 16 2012
%C A029600 For a closed-form formula for generalized Pascal's triangle see A228576. - _Boris Putievskiy_, Sep 04 2013
%C A029600 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
%H A029600 Reinhard Zumkeller, <a href="/A029600/b029600.txt">Rows n=0..150 of triangle, flattened</a>
%F A029600 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
%F A029600 G.f.: (-1-2*x*y-x)/(-1+x*y+x). - _R. J. Mathar_, Aug 11 2015
%e A029600 First few rows are:
%e A029600   1;
%e A029600   2, 3;
%e A029600   2, 5,  3;
%e A029600   2, 7,  8,  3;
%e A029600   2, 9, 15, 11, 3;
%e A029600 ...
%p A029600 T:= proc(n, k) option remember;
%p A029600       if k=0 and n=0 then 1
%p A029600     elif k=0 then 2
%p A029600     elif k=n then 3
%p A029600     else T(n-1, k-1) + T(n-1, k)
%p A029600       fi
%p A029600     end:
%p A029600 seq(seq(T(n, k), k=0..n), n=0..12); # _G. C. Greubel_, Nov 12 2019
%t A029600 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 *)
%o A029600 (Haskell)
%o A029600 a029600 n k = a029600_tabl !! n !! k
%o A029600 a029600_row n = a029600_tabl !! n
%o A029600 a029600_tabl = [1] : iterate
%o A029600    (\row -> zipWith (+) ([0] ++ row) (row ++ [0])) [2,3]
%o A029600 -- _Reinhard Zumkeller_, Apr 08 2012
%o A029600 (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
%o A029600 (Sage)
%o A029600 @CachedFunction
%o A029600 def T(n, k):
%o A029600     if (n==0 and k==0): return 1
%o A029600     elif (k==0): return 2
%o A029600     elif (k==n): return 3
%o A029600     else: return T(n-1,k-1) + T(n-1, k)
%o A029600 [[T(n, k) for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, Nov 12 2019
%o A029600 (GAP)
%o A029600 T:= function(n,k)
%o A029600     if n=0 and k=0 then return 1;
%o A029600     elif k=0 then return 2;
%o A029600     elif k=n then return 3;
%o A029600     else return T(n-1,k-1) + T(n-1,k);
%o A029600     fi;
%o A029600   end;
%o A029600 Flat(List([0..12], n-> List([0..n], k-> T(n,k) ))); # _G. C. Greubel_, Nov 12 2019
%Y A029600 Cf. A007318 (Pascal's triangle), A029618, A084938, A228196, A228576.
%K A029600 nonn,tabl,easy
%O A029600 0,2
%A A029600 _Mohammad K. Azarian_
%E A029600 More terms from _James Sellers_