OFFSET
0,5
COMMENTS
The third column is A034856 (binomial(n+1, 2) + n-1).
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
LINKS
Reinhard Zumkeller, Rows n=0..150 of triangle, flattened
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).
EXAMPLE
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;
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 *)
PROG
(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
(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
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Gerald McGarvey, Aug 12 2004
EXTENSIONS
Offset changed by Reinhard Zumkeller, Jul 12 2012
STATUS
approved