A202327 Triangle read by rows, T(n, k) is the coefficient of x^n in expansion of ((-1 - x + sqrt(1 + 2*x + 5*x^2)) /2)^k.

1, -1, 1, 0, -2, 1, 2, 1, -3, 1, -3, 4, 3, -4, 1, -1, -10, 5, 6, -5, 1, 11, 4, -21, 4, 10, -6, 1, -15, 28, 21, -35, 0, 15, -7, 1, -13, -64, 42, 56, -50, -8, 21, -8, 1, 77, 9, -162, 36, 114, -63, -21, 28, -9, 1, -86, 230, 135, -312, -15, 198, -70, -40, 36, -10, 1

Offset: 1

Author

Vladimir Kruchinin, Dec 17 2011

Keywords

• sign
• tabl

Examples

			   1;
  -1,   1;
   0,  -2,   1;
   2,   1,  -3,   1;
  -3,   4,   3,  -4,   1;
  -1, -10,   5,   6,  -5,   1;
  11,   4, -21,   4,  10,  -6,   1;
		

Crossrefs

Cf. A007440 (1st column), A108624 (row sums).

Programs

Maple

# Assuming offset = 0.
T := (n,k) -> (-1)^(n-k)*binomial(n,k)*hypergeom([(k-n)/2, (k-n+1)/2], [k+2], -4): for n from 0 to 9 do seq(simplify(T(n, k)), k=0..n) od; # Peter Luschny, May 19 2021

Maxima

T(n,k):=(k*sum(binomial(j,-n-k+2*j)*(-1)^(j-k)*binomial(n,j),j,0,n))/n;

Formula

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

T(n, k) = binomial(n, k)*hypergeom([(k - n)/2, (k - n + 1)/2], [k + 2], -4)*(-1)^(n - k), assuming offset = 0. - Peter Luschny, May 19 2021

Extensions

More terms from Sean A. Irvine, Mar 03 2021