A176602 Triangle defined by T(n, m) = -b(n) + b(m) + b(n-m), where b(n) = binomial(2*n, n)/(n + 1) = A000108(n), read by rows.

1, 1, 1, 1, 0, 1, 1, -2, -2, 1, 1, -8, -10, -8, 1, 1, -27, -35, -35, -27, 1, 1, -89, -116, -122, -116, -89, 1, 1, -296, -385, -410, -410, -385, -296, 1, 1, -1000, -1296, -1383, -1402, -1383, -1296, -1000, 1, 1, -3431, -4431, -4725, -4806, -4806, -4725, -4431, -3431, 1

Offset

0,8

Comments

Row sums are: {1, 2, 2, -2, -24, -122, -530, -2180, -8758, -34784, -137328, ...}.

Links

Indranil Ghosh, Rows 0..120, flattened

Formula

T(n, m) = - Catalan(n) + Catalan(m) + Catalan(n-m), where Catalan(n) = binomial(2*n, n)/(n + 1) = A000108(n).

Example

Triangle begins as:
  1;
  1,     1;
  1,     0,     1;
  1,    -2,    -2,     1;
  1,    -8,   -10,    -8,     1;
  1,   -27,   -35,   -35,   -27,     1;
  1,   -89,  -116,  -122,  -116,   -89,     1;
  1,  -296,  -385,  -410,  -410,  -385,  -296,     1;
  1, -1000, -1296, -1383, -1402, -1383, -1296, -1000,     1;
  1, -3431, -4431, -4725, -4806, -4806, -4725, -4431, -3431, 1;

Mathematica

T[n_, m_] = -CatalanNumber[n] + CatalanNumber[m] + CatalanNumber[n-m]; Table[T[n, m], {n, 0, 12}, {m, 0, n}]//Flatten (* modified by G. C. Greubel, May 07 2019 *)

Prog

(PARI)
{catalan(n) = binomial(2*n, n)/(n+1)};
{T(n, k) = -catalan(n) +catalan(k) +catalan(n-k)};
for(n=0, 12, for(k=0, n, print1(T(n, k), ", "))) \\ G. C. Greubel, May 07 2019
(Magma)
C:= func< n | Binomial(2*n, n)/(n+1) >;
[[-C(n) +C(k) +C(n-k): k in [0..n]]: n in [0..12]]; // G. C. Greubel, May 07 2019
(Sage)
[[-catalan_number(n) +catalan_number(k) +catalan_number(n-k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, May 07 2019
(GAP)
Catalan2 := n -> Binomial(2*n, n)/(n + 1);
Flat(List([0..12], n-> List([0..n], k-> -Catalan2(n) +Catalan2(k) +Catalan2(n-k) ))); # G. C. Greubel, May 07 2019

Crossrefs

Cf. A000108.

Sequence in context: A365077 A260360 A011296 * A322194 A174120 A240939

Adjacent sequences: A176599 A176600 A176601 * A176603 A176604 A176605

Keyword

sign,tabl

Author

Roger L. Bagula, Apr 21 2010

Extensions

Edited by G. C. Greubel, May 07 2019

Status

approved