A138022 Triangular array read by rows: e.g.f. sqrt(1-z^2)*exp(x*z)/(1+z).

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, 893025, -992250, 496125, -189000, 47250, -11340, 1890, -360, 45, -10, 1

Offset: 1

Author

Roger L. Bagula, May 01 2008

Keywords

• tabl
• sign

Examples

			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;
  ...
		

Crossrefs

Cf. A130915 (row sums).

Programs

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 *)

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

Extensions

Edited by Peter Luschny and Joerg Arndt, Aug 21 2014