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%I #17 Jan 29 2022 09:41:13
%S 1,1,1,-1,2,1,3,-3,3,1,-15,12,-6,4,1,105,-75,30,-10,5,1,-945,630,-225,
%T 60,-15,6,1,10395,-6615,2205,-525,105,-21,7,1,-135135,83160,-26460,
%U 5880,-1050,168,-28,8,1,2027025,-1216215,374220,-79380,13230,-1890,252,-36,9,1
%N Triangle T, read by rows, such that the matrix square, T^2, forms a simple 2-diagonal triangle where [T^2](n,n) = 1 and [T^2](n+1,n) = 2*(n+1) for n>=0.
%H Peter Bala, <a href="/A035342/a035342_Bala.txt">Generalized Dobinski formulas</a>
%F Exponential Riordan array [sqrt(1 + 2*x),x] with e.g.f. sqrt(1+2*x)*exp(t*x) = 1 + (1+t)*x + (-1+2*t+t^2)*x^2/2! + ... . The n-th row polynomial R(n,x) is given by the type B Dobinski formula R(n,x) = exp(-x/2)*sum {k = 0..inf} (2*k+1)*(2*k-1)*...*(2*k+1-2*(n-1))*(x/2)^k/k!. Cf. A122848. - _Peter Bala_, Jun 23 2014
%e Triangle begins:
%e 1;
%e 1,1;
%e -1,2,1;
%e 3,-3,3,1;
%e -15,12,-6,4,1;
%e 105,-75,30,-10,5,1;
%e -945,630,-225,60,-15,6,1;
%e 10395,-6615,2205,-525,105,-21,7,1;
%e ...
%e where T(n,k) = (-1)^(n-1-k)*A001147(n-1)*C(n,k).
%e The matrix square equals:
%e 1;
%e 2,1;
%e 0,4,1;
%e 0,0,6,1;
%e 0,0,0,8,1;
%e 0,0,0,0,10,1;
%e 0,0,0,0,0,12,1;
%e ...
%e The matrix log, L, begins:
%e 0;
%e 1,0;
%e -2,2,0;
%e 8,-6,3,0;
%e -48,32,-12,4,0;
%e 384,-240,80,-20,5,0;
%e -3840,2304,-720,160,-30,6,0;
%e ...
%e where L(n,k) = (-1)^(n-1-k)*A000165(n-1)*C(n,k).
%t (* The function RiordanArray is defined in A256893. *)
%t RiordanArray[Sqrt[1 + 2 #]&, #&, 10, True] // Flatten (* _Jean-François Alcover_, Jul 19 2019 *)
%o (PARI) {T(n,k)=local(M=matrix(n+1,n+1,r,c,if(r==c,1,if(r==c+1,2*c)))); (sum(i=0,n+1,(sum(j=1,n+1,-(M^0-M)^j/j)/2)^i/i!))[n+1,k+1]}
%Y Cf. A039683, A122848.
%Y Cf. A001147 (odd double factorials), A000165 (even double factorials).
%K sign,tabl
%O 0,5
%A _Paul D. Hanna_, Oct 22 2005
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