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

%I A138022 #31 Jan 17 2024 14:29:45
%S A138022 1,-1,1,1,-2,1,-3,3,-3,1,9,-12,6,-4,1,-45,45,-30,10,-5,1,225,-270,135,
%T A138022 -60,15,-6,1,-1575,1575,-945,315,-105,21,-7,1,11025,-12600,6300,-2520,
%U A138022 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
%N A138022 Triangular array read by rows: e.g.f. sqrt(1-z^2)*exp(x*z)/(1+z).
%F A138022 The unsigned version has the e.g.f. exp(x*z)/sqrt((1-z)/(1+z)). - _Peter Luschny_, Aug 21 2014
%F A138022 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
%F A138022 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
%e A138022 Triangle starts:
%e A138022        1;
%e A138022       -1,      1;
%e A138022        1,     -2,      1;
%e A138022       -3,      3,     -3,     1;
%e A138022        9,    -12,      6,    -4,     1;
%e A138022      -45,     45,    -30,    10,    -5,    1;
%e A138022      225,   -270,    135,   -60,    15,   -6,    1;
%e A138022    -1575,   1575,   -945,   315,  -105,   21,   -7,  1;
%e A138022    11025, -12600,   6300, -2520,   630, -168,   28, -8,  1;
%e A138022   -99225,  99225, -56700, 18900, -5670, 1134, -252, 36, -9, 1;
%e A138022   ...
%p A138022 g := sqrt(1-z^2)*exp(x*z)/(1+z); gser := n -> series(g, z, n+2):
%p A138022 seq(print(seq(coeff(n!*coeff(gser(n),z,n),x,i),i=0..n)),n=0..10); # _Peter Luschny_, Aug 21 2014
%t A138022 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 A138022 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 *)
%Y A138022 Cf. A130915 (row sums).
%K A138022 tabl,sign
%O A138022 1,5
%A A138022 _Roger L. Bagula_, May 01 2008
%E A138022 Edited by _Peter Luschny_ and _Joerg Arndt_, Aug 21 2014