A096465 - OEIS

%I #19 May 01 2021 22:02:27

%S 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 26,67,131,196,197,1,1,8,34,101,232,428,625,626,1,1,9,43,144,376,804,

%U 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 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 The third column is A034856 (binomial(n+1, 2) + n-1).

%C The row sums are A014137 (partial sums of Catalan numbers (A000108)).

%C The "1st subdiagonal" ((i+1,i) entries) are also A014137.

%C The "2nd subdiagonal" ((i+2,i) entries) is A014138 ( Partial sums of Catalan numbers (starting 1,2,5,...)).

%C The "3rd subdiagonal" ((i+3,i) entries) is A001453 (Catalan numbers - 1.)

%C This is the reverse of A091491 - see A091491 for more information. The sequence of antidiagonal sums gives A124642. - _Gerald McGarvey_, Dec 09 2006

%H Reinhard Zumkeller, <a href="/A096465/b096465.txt">Rows n=0..150 of triangle, flattened</a>

%F From _G. C. Greubel_, Apr 30 2021: (Start)

%F 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 T(n, k) = A091491(n, n-k).

%F Sum_{k=0..n} T(n,k) = Sum_{j=0..n} A000108(j) = A014137(n). (End)

%e Triangle begins as:

%e   1;

%e   1, 1;

%e   1, 2,  1;

%e   1, 3,  4,  1;

%e   1, 4,  8,  9,   1;

%e   1, 5, 13, 22,  23,   1;

%e   1, 6, 19, 41,  64,  65,   1;

%e   1, 7, 26, 67, 131, 196, 197, 1;

%p 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 seq(seq(A096465(n,k), k=0..n), n=0..12) # _G. C. Greubel_, Apr 30 2021

%t 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 (Haskell)

%o a096465 n k = a096465_tabl !! n !! k

%o a096465_row n = a096465_tabl !! n

%o a096465_tabl = map reverse a091491_tabl

%o -- _Reinhard Zumkeller_, Jul 12 2012

%o (Magma)

%o 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(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Apr 30 2021

%o (Sage)

%o 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 flatten([[A096465(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Apr 30 2021

%Y Cf. A000108, A001453, A014137, A014138, A034856.

%Y Cf. A006134, A024718, A030237, A078478, A091491, A100066, A105848, A124642.

%K nonn,tabl

%O 0,5

%A _Gerald McGarvey_, Aug 12 2004

%E Offset changed by _Reinhard Zumkeller_, Jul 12 2012