cp's OEIS Frontend

A051631 Triangle formed using Pascal's rule except begin and end n-th row with n-1.

%I A051631 #31 Oct 27 2023 22:00:44
%S A051631 -1,0,0,1,0,1,2,1,1,2,3,3,2,3,3,4,6,5,5,6,4,5,10,11,10,11,10,5,6,15,
%T A051631 21,21,21,21,15,6,7,21,36,42,42,42,36,21,7,8,28,57,78,84,84,78,57,28,
%U A051631 8,9,36,85,135,162,168,162,135,85,36,9
%N A051631 Triangle formed using Pascal's rule except begin and end n-th row with n-1.
%C A051631 Row sums give A000918(n).
%C A051631 Central terms for n>0: T(2*n,n)=A024483(n+1), T(n,[n/2])=A116385(n-1); for n>1: T(n,1) = T(n,n-1) = A000217(n-2). - _Reinhard Zumkeller_, Nov 13 2011
%H A051631 Reinhard Zumkeller, <a href="/A051631/b051631.txt">Rows n=0..100 of triangle, flattened</a>
%H A051631 <a href="/index/Pas#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>
%F A051631 T(n,k) = T(n-1,k) + T(n-1,k-1), 0 < k < n, T(n,0) = T(n,n) = n - 1.
%F A051631 T(n,k) = C(n+2,k+1) - 3*C(n,k). - _Charlie Neder_, Jan 10 2019
%e A051631 Triangle begins
%e A051631   -1;
%e A051631    0, 0;
%e A051631    1, 0, 1;
%e A051631    2, 1, 1, 2;
%e A051631    3, 3, 2, 3, 3;
%e A051631    4, 6, 5, 5, 6, 4; ...
%t A051631 Clear[t]; t[n_, k_] := t[n, k] = t[n-1, k] + t[n-1, k-1]; t[n_, 0] := n-1; t[n_, n_] := n-1; Table[t[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Apr 11 2013 *)
%o A051631 (Haskell)
%o A051631 a051631 n k = a051631_tabl !! n !! k
%o A051631 a051631_row n = a051631_tabl !! n
%o A051631 a051631_list = concat a051631_tabl
%o A051631 a051631_tabl = iterate (\row -> zipWith (+) ([1] ++ row) (row ++[1])) [-1]
%o A051631 -- _Reinhard Zumkeller_, Nov 13 2011
%o A051631 (Magma) /* As triangle */ [[Binomial(n+2,k+1) - 3*Binomial(n,k): k in [0..n]]: n in [0.. 10]]; // _Vincenzo Librandi_, Jan 11 2019
%Y A051631 Cf. A007318.
%K A051631 easy,nice,sign,tabl
%O A051631 0,7
%A A051631 _Asher Auel_
%E A051631 Definition modified and keyword tabl added by _Reinhard Zumkeller_, Nov 13 2011