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A096465 Triangle (read by rows) formed by setting all entries in the first column and in the main diagonal ((i,i) entries) to 1 and the rest of the entries by the recursion T(n, k) = T(n-1, k) + T(n, k-1).

%I A096465 #19 May 01 2021 22:02:27
%S A096465 1,1,1,1,2,1,1,3,4,1,1,4,8,9,1,1,5,13,22,23,1,1,6,19,41,64,65,1,1,7,
%T A096465 26,67,131,196,197,1,1,8,34,101,232,428,625,626,1,1,9,43,144,376,804,
%U A096465 1429,2055,2056,1,1,10,53,197,573,1377,2806,4861,6917,6918,1,1,11,64,261,834,2211,5017,9878,16795,23713,23714,1
%N A096465 Triangle (read by rows) formed by setting all entries in the first column and in the main diagonal ((i,i) entries) to 1 and the rest of the entries by the recursion T(n, k) = T(n-1, k) + T(n, k-1).
%C A096465 The third column is A034856 (binomial(n+1, 2) + n-1).
%C A096465 The row sums are A014137 (partial sums of Catalan numbers (A000108)).
%C A096465 The "1st subdiagonal" ((i+1,i) entries) are also A014137.
%C A096465 The "2nd subdiagonal" ((i+2,i) entries) is A014138 ( Partial sums of Catalan numbers (starting 1,2,5,...)).
%C A096465 The "3rd subdiagonal" ((i+3,i) entries) is A001453 (Catalan numbers - 1.)
%C A096465 This is the reverse of A091491 - see A091491 for more information. The sequence of antidiagonal sums gives A124642. - _Gerald McGarvey_, Dec 09 2006
%H A096465 Reinhard Zumkeller, <a href="/A096465/b096465.txt">Rows n=0..150 of triangle, flattened</a>
%F A096465 From _G. C. Greubel_, Apr 30 2021: (Start)
%F A096465 T(n, k) = (n-k) * Sum_{j=0..k} binomial(n+k-2*j, n-j)/(n+k-2*j) with T(n,n) = 1.
%F A096465 T(n, k) = A091491(n, n-k).
%F A096465 Sum_{k=0..n} T(n,k) = Sum_{j=0..n} A000108(j) = A014137(n). (End)
%e A096465 Triangle begins as:
%e A096465   1;
%e A096465   1, 1;
%e A096465   1, 2,  1;
%e A096465   1, 3,  4,  1;
%e A096465   1, 4,  8,  9,   1;
%e A096465   1, 5, 13, 22,  23,   1;
%e A096465   1, 6, 19, 41,  64,  65,   1;
%e A096465   1, 7, 26, 67, 131, 196, 197, 1;
%p A096465 A096465:= (n,k)-> `if`(k=n, 1, (n-k)*add(binomial(n+k-2*j, n-j)/(n+k-2*j), j=0..k));
%p A096465 seq(seq(A096465(n,k), k=0..n), n=0..12) # _G. C. Greubel_, Apr 30 2021
%t A096465 T[_, 0]= 1; T[n_, n_]= 1; T[n_, m_]:= T[n, m]= T[n-1, m] + T[n, m-1]; T[n_, m_] /; n < 0 || m > n = 0; Table[T[n, m], {n, 0, 12}, {m, 0, n}]//Flatten (* _Jean-François Alcover_, Dec 17 2012 *)
%o A096465 (Haskell)
%o A096465 a096465 n k = a096465_tabl !! n !! k
%o A096465 a096465_row n = a096465_tabl !! n
%o A096465 a096465_tabl = map reverse a091491_tabl
%o A096465 -- _Reinhard Zumkeller_, Jul 12 2012
%o A096465 (Magma)
%o A096465 A096465:= func< n,k | k eq n select 1 else (n-k)*(&+[Binomial(n+k-2*j, n-j)/(n+k-2*j): j in [0..k]]) >;
%o A096465 [A096465(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Apr 30 2021
%o A096465 (Sage)
%o A096465 def A096465(n,k): return 1 if (k==n) else (n-k)*sum( binomial(n+k-2*j, n-j)/(n+k-2*j) for j in (0..k))
%o A096465 flatten([[A096465(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Apr 30 2021
%Y A096465 Cf. A000108, A001453, A014137, A014138, A034856.
%Y A096465 Cf. A006134, A024718, A030237, A078478, A091491, A100066, A105848, A124642.
%K A096465 nonn,tabl
%O A096465 0,5
%A A096465 _Gerald McGarvey_, Aug 12 2004
%E A096465 Offset changed by _Reinhard Zumkeller_, Jul 12 2012