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A125694 Riordan array ((1+3x-sqrt(1+2x+9x^2))/(2x),(1+3x-sqrt(1+2x+9x^2))/2).

%I A125694 #13 Jun 13 2019 10:01:28
%S A125694 1,-2,1,2,-4,1,2,8,-6,1,-10,-4,18,-8,1,6,-24,-26,32,-10,1,42,60,-18,
%T A125694 -72,50,-12,1,-102,24,162,48,-150,72,-14,1,-82,-388,-214,248,230,-268,
%U A125694 98,-16,1,782,536,-546,-800,158,600,-434,128,-18,1
%N A125694 Riordan array ((1+3x-sqrt(1+2x+9x^2))/(2x),(1+3x-sqrt(1+2x+9x^2))/2).
%C A125694 First column is A125695. Row sums are A091593. Inverse of A125693.
%F A125694 T(n-1,k-1) = (k/n)*sum_{i=0..n-k} binomial(n,n-k-i) *(-3)^(n-k-i) *binomial(i+n-1,n-1). - Vladimir Kruchinin, Feb 12 2011
%F A125694 T(n, k) = (-3)^(n-k)*C(n,k)*hypergeometric([k-n, n+1], [k+2], 1/3). - _Peter Luschny_, Sep 17 2014
%e A125694 Triangle begins
%e A125694 1,
%e A125694 -2, 1,
%e A125694 2, -4, 1,
%e A125694 2, 8, -6, 1,
%e A125694 -10, -4, 18, -8, 1,
%e A125694 6, -24, -26, 32, -10, 1,
%e A125694 42, 60, -18, -72, 50, -12, 1
%t A125694 m = 9;
%t A125694 T[0, 0] = 1; T[1, 0] = 2; T[1, 1] = 1; T[n_, k_] /; 0 <= k <= n := T[n, k] = 3 T[n - 1, k] + T[n - 1, k - 1] - T[n - 2, k - 1]; T[_, _] = 0;
%t A125694 M = Table[T[n, k], {n, 0, m}, {k, 0, m}] // Inverse;
%t A125694 A[n_, k_] := M[[n + 1, k + 1]];
%t A125694 Table[A[n, k], {n, 0, m}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jun 13 2019 *)
%o A125694 (Sage)
%o A125694 A125694 = lambda n,k : (-3)^(n-k)*binomial(n,k)*hypergeometric([k-n, n+1], [k+2], 1/3)
%o A125694 for n in (0..6): [round(A125694(n,k).n(100)) for k in (0..n)] # _Peter Luschny_, Sep 17 2014
%K A125694 easy,sign,tabl
%O A125694 0,2
%A A125694 _Paul Barry_, Nov 30 2006