OFFSET
0,5
COMMENTS
Row functions in y are given by: R_n(y) = Sum_{k=0..2n} (n-floor(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.
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 + 4*y + 9*y^2/2! + 27*y^3/3! + 16*y^4/4! + 32*y^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, ...],
[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 floor(k/2) rows, and omitting the zeros coming from row 0 of A003992.
MAPLE
seq(print(`if`(n=0, 1, seq((n - floor(k/2))^k, k=0..2*n-1))), n=0..10); # Georg Fischer, Nov 21 2024
PROG
(PARI) T(n, k)=(n-k\2)^k
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Paul D. Hanna, Oct 22 2004
EXTENSIONS
Definition corrected by Georg Fischer, Nov 21 2024
STATUS
approved