A164925 Array, binomial(j-i,j), read by rising antidiagonals.

1, 1, 1, 1, 0, 1, 1, -1, 0, 1, 1, -2, 0, 0, 1, 1, -3, 1, 0, 0, 1, 1, -4, 3, 0, 0, 0, 1, 1, -5, 6, -1, 0, 0, 0, 1, 1, -6, 10, -4, 0, 0, 0, 0, 1, 1, -7, 15, -10, 1, 0, 0, 0, 0, 1, 1, -8, 21, -20, 5, 0, 0, 0, 0, 0, 1, 1, -9, 28, -35, 15, -1, 0, 0, 0, 0, 0, 1, 1, -10, 36, -56, 35, -6, 0, 0, 0, 0, 0, 0, 1

Offset

0,12

Comments

Inverse of A052509, or A004070???

Links

G. C. Greubel, Antidiagonals n = 0..50, flattened

Formula

Sum_{k=0..n} T(n, k) = A164965(n). - Mark Dols, Sep 02 2009

From G. C. Greubel, Feb 10 2023: (Start)

A(n, k) = binomial(k-n, k), with A(0, k) = A(n, 0) = 1 (array).

T(n, k) = binomial(2*k-n, k), with T(n, 0) = T(n, n) = 1 (antidiagonal triangle).

Sum_{k=0..n} (-1)^k*T(n, k) = A008346(n).

Sum_{k=0..n} (-2)^k*T(n, k) = (-1)^n*A052992(n). (End)

Example

Array, A(n, k), begins as:
  1,  1,  1,   1,  1,   1,  1,  1,  1, ...
  1,  0,  0,   0,  0,   0,  0,  0,  0, ...
  1, -1,  0,   0,  0,   0,  0,  0,  0, ...
  1, -2,  1,   0,  0,   0,  0,  0,  0, ...
  1, -3,  3,  -1,  0,   0,  0,  0,  0, ...
  1, -4,  6,  -4,  1,   0,  0,  0,  0, ...
  1, -5, 10, -10,  5,  -1,  0,  0,  0, ...
  1, -6, 15, -20, 15,  -6,  1,  0,  0, ...
  1, -7, 21, -35, 35, -21,  7, -1,  0, ...
Antidiagonal triangle, T(n, k), begins as:
  1;
  1,  1;
  1,  0,  1;
  1, -1,  0,  1;
  1, -2,  0,  0,  1;
  1, -3,  1,  0,  0,  1;
  1, -4,  3,  0,  0,  0,  1;
  1, -5,  6, -1,  0,  0,  0,  1;
  1, -6, 10, -4,  0,  0,  0,  0,  1;

Mathematica

T[n_, k_]:= If[k==0 || k==n, 1, Binomial[2*k-n, k]];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Feb 10 2023 *)

Prog

(PARI) {A(i, j) = if( i<0, 0, if(i==0 || j==0, 1, binomial(j-i, j)))}; /* Michael Somos, Jan 25 2012 */
(Magma)
A164925:= func< n, k | k eq 0 or k eq n select 1 else Binomial(2*k-n, k) >;
[A164925(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 10 2023
(SageMath)
def A164925(n, k): return 1 if (k==0 or k==n) else binomial(2*k-n, k)
flatten([[A164925(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Feb 10 2023

Crossrefs

Cf. A004070, A008346, A010892, A052509, A052992.

Cf. A109466, A130595, A164899, A164915, A164965.

Sequence in context: A277537 A323179 A320508 * A334546 A035671 A190164

Adjacent sequences: A164922 A164923 A164924 * A164926 A164927 A164928

Keyword

sign,easy,tabl

Author

Mark Dols, Aug 31 2009

Extensions

Edited by Michael Somos, Jan 26 2012

Offset changed by G. C. Greubel, Feb 10 2023

Status

approved