login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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. 4
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

LINKS

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

P. 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)

and A001147 forms the odd double factorials.

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)

and A000165 forms the even double factorials.

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.

Sequence in context: A152534 A136018 A138022 * A132382 A048865 A058754

Adjacent sequences:  A113275 A113276 A113277 * A113279 A113280 A113281

KEYWORD

sign,tabl

AUTHOR

Paul D. Hanna, Oct 22 2005

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified February 16 08:26 EST 2019. Contains 320159 sequences. (Running on oeis4.)