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A099555 Triangle, read by rows, where T(n,k) = (n-[k/2])^k and [k/2] is the integer floor of k/2, for 2n>=k>=0. 3
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, 1, 8, 49, 343, 1296, 7776, 15625, 78125, 65536 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

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.

LINKS

Table of n, a(n) for n=0..65.

FORMULA

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

EXAMPLE

The asymptotic behavior can be demonstrated at the 4th 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.

PROG

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

CROSSREFS

Cf. A003992, A099554, A099556.

Sequence in context: A122867 A124775 A140075 * A262082 A299045 A124530

Adjacent sequences:  A099552 A099553 A099554 * A099556 A099557 A099558

KEYWORD

nonn,tabl

AUTHOR

Paul D. Hanna, Oct 22 2004

STATUS

approved

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Last modified November 13 04:20 EST 2019. Contains 329085 sequences. (Running on oeis4.)