A225413 Triangle read by rows: T(n,k) = (A101164(n,k) - A014473(n,k))/2.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 3, 3, 0, 0, 0, 0, 6, 12, 6, 0, 0, 0, 0, 10, 30, 30, 10, 0, 0, 0, 0, 15, 60, 91, 60, 15, 0, 0, 0, 0, 21, 105, 215, 215, 105, 21, 0, 0, 0, 0, 28, 168, 435, 590, 435, 168, 28, 0, 0, 0, 0, 36, 252, 791, 1365, 1365, 791, 252, 36, 0, 0

Offset: 0

Author

Jeremy Gardiner, Jul 28 2013

Keywords

• nonn
• easy
• tabl

Comments

Has opposite parity to A140356, A155454.

Examples

			Triangle begins as:
  0;
  0,  0;
  0,  0,  0;
  0,  0,  0,   0;
  0,  0,  1,   0,    0;
  0,  0,  3,   3,    0,    0;
  0,  0,  6,  12,    6,    0,    0;
  0,  0, 10,  30,   30,   10,    0,    0;
  0,  0, 15,  60,   91,   60,   15,    0,    0;
  0,  0, 21, 105,  215,  215,  105,   21,    0,    0;
  0,  0, 28, 168,  435,  590,  435,  168,   28,    0,   0;
  0,  0, 36, 252,  791, 1365, 1365,  791,  252,   36,   0,  0;
  0,  0, 45, 360, 1330, 2800, 3571, 2800, 1330,  360,  45,  0,  0;
  0,  0, 55, 495, 2106, 5250, 8197, 8197, 5250, 2106, 495, 55,  0,  0;
		

Links

• Reinhard Zumkeller, Rows n = 0..100 of table, flattened

Crossrefs

Cf. A000129, A007318, A008288, A014473, A101164, A121262, A140356, A155454.

3rd column = A000217 (triangular numbers).

4th column = A027480 (n(n+1)(n+2)/2).

Programs

Haskell

a225413 n k = a225413_tabl !! n !! k
a225413_row n = a225413_tabl !! n
a225413_tabl = map (map (`div` 2)) $
               zipWith (zipWith (-)) a101164_tabl a014473_tabl
-- Reinhard Zumkeller, Jul 30 2013

Magma

A008288:= func< n,k | (&+[Binomial(n-j, j)*Binomial(n-2*j, k-j): j in [0..k]]) >;
A225413:= func< n,k | (A008288(n,k) - 2*Binomial(n,k) + 1)/2 >;
[A225413(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 08 2024

Mathematica

T[n_, k_]:= ((-1)^(n-k)*Hypergeometric2F1[-n+k,k+1,1,2] - 2*Binomial[n, k] +1)/2;
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Apr 08 2024 *)

SageMath

def A008288(n,k): return sum(binomial(n-j,j)*binomial(n-2*j,k-j) for j in range(k+1))
def A225413(n,k): return (A008288(n,k) -2*binomial(n,k) +1)//2
flatten([[A225413(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Apr 08 2024

Formula

T(n, k) = (A101164(n,k) - A014473(n,k))/2.

T(n, k) = (A008288(n,k) - 2*A007318(n,k) + 1)/2.

From G. C. Greubel, Apr 08 2024: (Start)

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

Sum_{k=0..n} T(n, k) = (A000129(n+1) + n + 1 - 2^(n+1))/2.

Sum_{k=0..n} (-1)^k*T(n, k) = A121262(n) - [n=0]. (End)