A113278 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.

1, 1, 1, -1, 2, 1, 3, -3, 3, 1, -15, 12, -6, 4, 1, 105, -75, 30, -10, 5, 1, -945, 630, -225, 60, -15, 6, 1, 10395, -6615, 2205, -525, 105, -21, 7, 1, -135135, 83160, -26460, 5880, -1050, 168, -28, 8, 1, 2027025, -1216215, 374220, -79380, 13230, -1890, 252, -36, 9, 1

Offset

0,5

Links

Table of n, a(n) for n=0..54.

Peter Bala, Generalized Dobinski formulas

Formula

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

Example

Triangle begins:
  1;
  1,1;
  -1,2,1;
  3,-3,3,1;
  -15,12,-6,4,1;
  105,-75,30,-10,5,1;
  -945,630,-225,60,-15,6,1;
  10395,-6615,2205,-525,105,-21,7,1;
  ...
where T(n,k) = (-1)^(n-1-k)*A001147(n-1)*C(n,k).
The matrix square equals:
  1;
  2,1;
  0,4,1;
  0,0,6,1;
  0,0,0,8,1;
  0,0,0,0,10,1;
  0,0,0,0,0,12,1;
  ...
The matrix log, L, begins:
  0;
  1,0;
  -2,2,0;
  8,-6,3,0;
  -48,32,-12,4,0;
  384,-240,80,-20,5,0;
  -3840,2304,-720,160,-30,6,0;
  ...
where L(n,k) = (-1)^(n-1-k)*A000165(n-1)*C(n,k).

Mathematica

(* The function RiordanArray is defined in A256893. *)
RiordanArray[Sqrt[1 + 2 #]&, #&, 10, True] // Flatten (* Jean-François Alcover, Jul 19 2019 *)

Prog

(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]}

Crossrefs

Cf. A039683, A122848.

Cf. A001147 (odd double factorials), A000165 (even double factorials).

Sequence in context: A152534 A136018 A138022 * A132382 A366627 A325495

Adjacent sequences: A113275 A113276 A113277 * A113279 A113280 A113281

Keyword

sign,tabl

Author

Paul D. Hanna, Oct 22 2005

Status

approved