A176602 - OEIS

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

(list; table; graph; refs; listen; history; text; internal format)

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, 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);