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).

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

Offset: 0

Author

Gerald McGarvey, Aug 12 2004

Keywords

• nonn
• tabl

Comments

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

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

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

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

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

This is the reverse of A091491 - see A091491 for more information. The sequence of antidiagonal sums gives A124642. - Gerald McGarvey, Dec 09 2006

Examples

			Triangle begins as:
  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, 26, 67, 131, 196, 197, 1;
		

Links

• Reinhard Zumkeller, Rows n=0..150 of triangle, flattened

Crossrefs

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

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

Programs

Haskell

a096465 n k = a096465_tabl !! n !! k
a096465_row n = a096465_tabl !! n
a096465_tabl = map reverse a091491_tabl
-- Reinhard Zumkeller, Jul 12 2012

Magma

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]]) >;
[A096465(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 30 2021

Maple

A096465:= (n,k)-> `if`(k=n, 1, (n-k)*add(binomial(n+k-2*j, n-j)/(n+k-2*j), j=0..k));
seq(seq(A096465(n,k), k=0..n), n=0..12) # G. C. Greubel, Apr 30 2021

Mathematica

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 *)

Sage

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))
flatten([[A096465(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 30 2021

Formula

From G. C. Greubel, Apr 30 2021: (Start)

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.

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

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

Extensions

Offset changed by Reinhard Zumkeller, Jul 12 2012