0,5

Row functions in y are given by: R_n(y)=Sum_{k=0..2n} (n-[k/2])^k*y^k/k!. Evaluated at y=1, the asymptotic behavior of the rows is given by: R_n(1) ~ c*r^n where c = (r+sqrt(r))/(1+2*sqrt(r)) = 0.8957126... and r = 2.0207473586... satisfies r = exp(1/sqrt(r)) - see A099554 for the decimal expansion of this constant.

E.g.f.: ((1-x*cosh(sqrt(x)*y))+sqrt(x)*sinh(sqrt(x)*y))/(1+x^2-2*x*cosh(sqrt(x)*y)).

The asymptotic behavior can be demonstrated at the 4-th row function:

R_4(y) = 1+4y+9y^2/2!+27y^3/3!+16y^4/4!+32y^5/5!+y^6/6!+y^7/7!;

R_4(1) = 14.93492... = (0.895684...)*r^4, where r = 2.0207473586...

Rows begin:

[1]

[1,1],

[1,2,1,1],

[1,3,4,8,1,1],

[1,4,9,27,16,32,1,1],

[1,5,16,64,81,243,64,128,1,1],

[1,6,25,125,256,1024,729,2187,256,512,1,1],

[1,7,36,216,625,3125,4096,16384,6561,19683,1024,2048,1,1],...

which can be derived from the square array A003992:

[1,0,0,0,0,0,0,0,...],

[1,1,1,1,1,1,1,1,...],

[1,2,4,8,16,32,64,...],

[1,3,9,27,81,243,729,...],

[1,4,16,64,256,1024,4096,...],

[1,5,25,125,625,3125,15625,...],...

by shifting each column k down by [k/2] rows,

and omitting the zeros coming from row 0 of A003992.

(PARI) T(n, k)=(n-k\2)^k

Cf. A003992, A099554, A099556.

Sequence in context: A122867 A124775 A140075 this_sequence A124530 A070914 A124560

Adjacent sequences: A099552 A099553 A099554 this_sequence A099556 A099557 A099558

nonn,tabl

Paul D. Hanna (pauldhanna(AT)juno.com), Oct 22 2004