A113278 - OEIS

%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