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

%I A029618 #40 Nov 13 2019 01:49:27
%S A029618 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,
%T A029618 20,57,90,85,48,15,2,3,23,77,147,175,133,63,17,2,3,26,100,224,322,308,
%U A029618 196,80,19,2,3,29,126,324,546,630,504,276,99,21,2,3,32,155,450,870
%N A029618 Numbers in (3,2)-Pascal triangle (by row).
%C A029618 Reverse of A029600. - _Philippe Deléham_, Nov 21 2006
%C A029618 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
%C A029618 Row n: expansion of (3+2x)*(1+x)^(n-1), n>0. - _Philippe Deléham_, Oct 10 2011
%C A029618 For a closed-form formula for generalized Pascal's triangle see A228576. - _Boris Putievskiy_, Sep 04 2013
%H A029618 G. C. Greubel, <a href="/A029618/b029618.txt">Rows n = 0..100 of triangle, flattened</a>
%F A029618 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
%F A029618 G.f.: (-1-x*y-2*x)/(-1+x*y+x). - _R. J. Mathar_, Aug 11 2015
%e A029618 Triangle begins as:
%e A029618   1;
%e A029618   3,  2;
%e A029618   3,  5,  2;
%e A029618   3,  8,  7,  2;
%e A029618   3, 11, 15,  9,  2;
%e A029618   ...
%p A029618 A029618 := proc(n,k)
%p A029618     if k < 0 or k > n then
%p A029618         0;
%p A029618     elif  n = 0 then
%p A029618         1;
%p A029618     elif k=0 then
%p A029618         3;
%p A029618     elif k = n then
%p A029618         2;
%p A029618     else
%p A029618         procname(n-1,k-1)+procname(n-1,k) ;
%p A029618     end if;
%p A029618 end proc: # _R. J. Mathar_, Jul 08 2015
%t A029618 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 *)
%o A029618 (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
%o A029618 (Sage)
%o A029618 @CachedFunction
%o A029618 def T(n, k):
%o A029618     if (n==0 and k==0): return 1
%o A029618     elif (k==0): return 3
%o A029618     elif (k==n): return 2
%o A029618     else: return T(n-1,k-1) + T(n-1, k)
%o A029618 [[T(n, k) for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, Nov 12 2019
%o A029618 (GAP)
%o A029618 T:= function(n,k)
%o A029618     if n=0 and k=0 then return 1;
%o A029618     elif k=0 then return 3;
%o A029618     elif k=n then return 2;
%o A029618     else return T(n-1,k-1) + T(n-1,k);
%o A029618     fi;
%o A029618   end;
%o A029618 Flat(List([0..12], n-> List([0..n], k-> T(n,k) ))); # _G. C. Greubel_, Nov 12 2019
%Y A029618 Cf. A007318, A029600, A084938, A228196, A228576, A016789 (2nd column), A005449 (3rd column), A006002 (4th column).
%K A029618 nonn,easy,tabl
%O A029618 0,2
%A A029618 _Mohammad K. Azarian_