A383715 Triangle T(n,k), n >= 0, 0 <= k <= n, read by rows, where T(n,k) = (-1)^floor((k+1)/2) * A099927(n,k).

1, 1, -1, 1, -2, -1, 1, -5, -5, 1, 1, -12, -30, 12, 1, 1, -29, -174, 174, 29, -1, 1, -70, -1015, 2436, 1015, -70, -1, 1, -169, -5915, 34307, 34307, -5915, -169, 1, 1, -408, -34476, 482664, 1166438, -482664, -34476, 408, 1, 1, -985, -200940, 6791772, 39618670, -39618670, -6791772, 200940, 985, -1

Offset: 0

Author

Seiichi Manyama, May 07 2025

Keywords

• sign
• tabl

Examples

			Triangle starts:
  1;
  1,   -1;
  1,   -2,    -1;
  1,   -5,    -5,     1;
  1,  -12,   -30,    12,     1;
  1,  -29,  -174,   174,    29,    -1;
  1,  -70, -1015,  2436,  1015,   -70,   -1;
  1, -169, -5915, 34307, 34307, -5915, -169, 1;
  ...
		

Crossrefs

Cf. A055870, A099927.

Programs

PARI

pell(n) = ([2, 1; 1, 0]^n)[2, 1];
p(n, k) = prod(j=0, k-1, pell(n-j));
a099927(n, k) = p(n, k)/p(k, k);
T(n, k) = (-1)^((k+1)\2)*a099927(n, k);

Formula

Let f(n, x) be defined as f(n, x) = Sum_{k=0..n} T(n,k) * x^k.

f(n, x) = exp( -Sum_{k>=1} Pell(n*k)/Pell(k) * x^k/k ).

Sum_{k>=0} A099927(n+k,n) * x^k = 1/f(n+1, x).