A176331 Number triangle T(n,k) = Sum_{j=0..n} C(j,n-k)*C(j,k)*(-1)^(n-j).

1, 1, 1, 1, 3, 1, 1, 7, 7, 1, 1, 13, 28, 13, 1, 1, 21, 79, 79, 21, 1, 1, 31, 181, 315, 181, 31, 1, 1, 43, 361, 971, 971, 361, 43, 1, 1, 57, 652, 2511, 3876, 2511, 652, 57, 1, 1, 73, 1093, 5713, 12606, 12606, 5713, 1093, 73, 1, 1, 91, 1729, 11789, 35246, 50358, 35246, 11789, 1729, 91, 1

Offset

0,5

Comments

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

Row sums are A176332.

Diagonal sums are A176334.

Central coefficients T(2n,n) are A176335.

Links

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Example

Triangle begins
  1;
  1,  1;
  1,  3,    1;
  1,  7,    7,     1;
  1, 13,   28,    13,     1;
  1, 21,   79,    79,    21,     1;
  1, 31,  181,   315,   181,    31,     1;
  1, 43,  361,   971,   971,   361,    43,     1;
  1, 57,  652,  2511,  3876,  2511,   652,    57,    1;
  1, 73, 1093,  5713, 12606, 12606,  5713,  1093,   73,  1;
  1, 91, 1729, 11789, 35246, 50358, 35246, 11789, 1729, 91, 1;

Maple

T:= proc(n, k) option remember; add( (-1)^(n-j)*binomial(j, n-k)*binomial(j, k), j=0..n); end:
seq(seq(T(n, k), k=0..n), n=0..10); # G. C. Greubel, Dec 07 2019

Mathematica

T[n_, k_]:= Sum[(-1)^(n-j)*Binomial[j, k]*Binomial[j, n-k], {j, 0, n}]; Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Dec 07 2019 *)

Prog

(PARI) T(n, k) = sum(j=0, n, (-1)^(n-j)*binomial(j, n-k)*binomial(j, k)); \\ G. C. Greubel, Dec 07 2019
(Magma) T:= func< n, k | &+[(-1)^(n-j)*Binomial(j, n-k)*Binomial(j, k): j in [0..n]] >;
[T(n, k): k in [0..n], n in [0..10]]; // G. C. Greubel, Dec 07 2019
(Sage)
@CachedFunction
def T(n, k): return sum( (-1)^(n-j)*binomial(j, n-k)*binomial(j, k) for j in (0..n))
[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Dec 07 2019
(GAP)
T:= function(n, k)
    return Sum([0..n], j-> (-1)^(n-j)*Binomial(j, k)*Binomial(j, n-k) );
  end;
Flat(List([0..10], n-> List([0..n], k-> T(n, k) ))); # G. C. Greubel, Dec 07 2019

Crossrefs

Cf. A176332, A176334, A176335.

Sequence in context: A347971 A082039 A367505 * A157836 A205497 A063394

Adjacent sequences: A176328 A176329 A176330 * A176332 A176333 A176334

Keyword

easy,nonn,tabl

Author

Paul Barry, Apr 15 2010

Status

approved