OFFSET
1,5
FORMULA
The unsigned version has the e.g.f. exp(x*z)/sqrt((1-z)/(1+z)). - Peter Luschny, Aug 21 2014
T(n+3,k+1) = T(n+2,k) - T(n+2,k+1) + (n+1)*(n+2)*(T(n+1,k+1)-T(n,k)) with T(n,n) = 1, T(n,n-1) = -n, T(n+2,0) = T(n+1,0) + (n^2+n)*T(n,0). - Robert Israel, Aug 21 2014
T(n, k) = 1 if n = k, otherwise (-1)^(n+k)*(n-k)!*Sum_{i = 1..n-k} (Sum_{j = i..n-k} 2^(j-i)*Stirling1(j, i)*binomial(n-k-1, j-1)/j!)*binomial(n, k). - Detlef Meya, Jan 16 2024
EXAMPLE
Triangle starts:
1;
-1, 1;
1, -2, 1;
-3, 3, -3, 1;
9, -12, 6, -4, 1;
-45, 45, -30, 10, -5, 1;
225, -270, 135, -60, 15, -6, 1;
-1575, 1575, -945, 315, -105, 21, -7, 1;
11025, -12600, 6300, -2520, 630, -168, 28, -8, 1;
-99225, 99225, -56700, 18900, -5670, 1134, -252, 36, -9, 1;
...
MAPLE
g := sqrt(1-z^2)*exp(x*z)/(1+z); gser := n -> series(g, z, n+2):
seq(print(seq(coeff(n!*coeff(gser(n), z, n), x, i), i=0..n)), n=0..10); # Peter Luschny, Aug 21 2014
MATHEMATICA
max=10; g=Exp[x*z]*Sqrt[(1-z)/(1+z)]; gser=Series[g, {z, 0, max}]; p[n_]:=n!*Coefficient[gser, z, n]; T[n_, k_]:=Coefficient[p[n], x, k]; Flatten[Table[T[n, k], {n, 0, max}, {k, 0, n}]]
T[n_, k_] := If[n==k, 1, (-1)^(n + k)*(n - k)!*Sum[Sum[2^(j - i)*StirlingS1[j, i]*Binomial[n - k - 1, j - 1]/j!, {j, i, n - k}], {i, 1, n - k}]*Binomial[n, k]]; Flatten[Table[T[n, k], {n, 0, 10}, {k, 0, n}]] (* Detlef Meya, Jan 16 2024 *)
CROSSREFS
KEYWORD
tabl,sign
AUTHOR
Roger L. Bagula, May 01 2008
EXTENSIONS
Edited by Peter Luschny and Joerg Arndt, Aug 21 2014
STATUS
approved