cp's OEIS Frontend

A026725 Triangular array, T, read by rows: T(n,0) = T(n,n) = 1. For n >= 2 and 1<=k<=n-1, T(n,k) = T(n-1,k-1) + T(n-2,k-1) + T(n-1,k) if n is odd and k=n/2, otherwise T(n,k) = T(n-1,k-1) + T(n-1,k).

%I A026725 #39 Oct 27 2019 05:18:25
%S A026725 1,1,1,1,2,1,1,4,3,1,1,5,7,4,1,1,6,16,11,5,1,1,7,22,27,16,6,1,1,8,29,
%T A026725 65,43,22,7,1,1,9,37,94,108,65,29,8,1,1,10,46,131,267,173,94,37,9,1,1,
%U A026725 11,56,177,398,440,267,131,46,10,1,1,12,67,233
%N A026725 Triangular array, T, read by rows: T(n,0) = T(n,n) = 1. For n >= 2 and 1<=k<=n-1, T(n,k) = T(n-1,k-1) + T(n-2,k-1) + T(n-1,k) if n is odd and k=n/2, otherwise T(n,k) = T(n-1,k-1) + T(n-1,k).
%C A026725 T(n+2,n) = A134869(n+1). - _Philippe Deléham_, Feb 01 2014
%H A026725 G. C. Greubel, <a href="/A026725/b026725.txt">Rows n = 0..99 of triangle, flattened</a>
%H A026725 Rob Arthan, <a href="/A026674/a026674.txt">Comments on A026674, A026725, A026670</a>
%F A026725 T(n, k) = number of paths from (0, 0) to (n-k, k) in directed graph having vertices (i, j) and edges (i, j)-to-(i+1, j) and (i, j)-to-(i, j+1) for i, j >= 0 and edges (i, i+1)-to-(i+1, i+2) for i >= 0.
%F A026725 Comment from _Rick L. Shepherd_, Aug 05 2002: Probably this should be changed to "and edges (i+1, i)-to-(i+2, i+1) for i >= 0."
%e A026725 Triangle begins:
%e A026725 1
%e A026725 1  1
%e A026725 1  2  1
%e A026725 1  4  3   1
%e A026725 1  5  7   4   1
%e A026725 1  6 16  11   5    1
%e A026725 1  7 22  27  16    6   1
%e A026725 1  8 29  65  43   22   7   1
%e A026725 1  9 37  94 108   65  29   8   1
%e A026725 1 10 46 131 267  173  94  37   9  1
%e A026725 1 11 56 177 398  440 267 131  46 10  1
%e A026725 1 12 67 233 575 1105 707 398 177 56 11 1
%e A026725 ... - _Philippe Deléham_, Feb 01 2014
%p A026725 A026725 := proc(n,k)
%p A026725     option remember;
%p A026725     if n < 0 or k < 0 then
%p A026725         0;
%p A026725     elif k=0 or k=n then
%p A026725         1;
%p A026725     elif 2*k = n-1 then
%p A026725       procname(n-1,k-1)+procname(n-2,k-1)+procname(n-1,k) ;
%p A026725    else
%p A026725        procname(n-1,k-1)+procname(n-1,k) ;
%p A026725     end if;
%p A026725 end proc: # _R. J. Mathar_, Oct 21 2019
%t A026725 T[n_, k_]:= T[n, k]= If[k==0||k==n, 1, If[OddQ[n] && k==(n-1)/2, T[n-1, k -1] + T[n-2, k-1] + T[n-1, k], T[n-1, k-1] + T[n-1, k]]];
%t A026725 Table[T[n, k], {n,0,14}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jul 16 2019 *)
%o A026725 (PARI) T(n,k) = if(k==n || k==0, 1, if(2*k==n-1, T(n-1, k-1) + T(n-2, k-1) + T(n-1, k), T(n-1, k-1) + T(n-1, k) ));
%o A026725 for(n=0,11, for(k=0,n, print1(T(n,k), ", "))) \\ _G. C. Greubel_, Jul 16 2019
%o A026725 (Sage)
%o A026725 @CachedFunction
%o A026725 def T(n, k):
%o A026725     if (k==0 or k==n): return 1
%o A026725     elif (mod(n,2)==0 and k==(n-1)/2): return T(n-1, k-1) + T(n-2, k-1) + T(n-1, k)
%o A026725     else: return T(n-1, k-1) + T(n-1, k)
%o A026725 [[T(n, k) for k in (0..n)] for n in (0..14)] # _G. C. Greubel_, Jul 16 2019
%o A026725 (GAP)
%o A026725 T:= function(n,k)
%o A026725     if k=0 or k=n then return 1;
%o A026725     elif 2*k=n-1 then return T(n-1, k-1) + T(n-2, k-1) + T(n-1, k);
%o A026725     else return T(n-1, k-1) + T(n-1, k);
%o A026725     fi;
%o A026725   end;
%o A026725 Flat(List([0..14], n-> List([0..n], k-> T(n,k) ))); # _G. C. Greubel_, Jul 16 2019
%Y A026725 Cf. A026674.
%K A026725 nonn,tabl
%O A026725 0,5
%A A026725 _Clark Kimberling_
%E A026725 Title and offset corrected by _G. C. Greubel_, Jul 16 2019, again by _R. J. Mathar_, Oct 21 2019, again by _Sean A. Irvine_, Oct 25 2019