A099555 - OEIS

%I #14 Nov 21 2024 16:22:27

%S 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,

%T 1,1,1,6,25,125,256,1024,729,2187,256,512,1,1,1,7,36,216,625,3125,

%U 4096,16384,6561,19683,1024,2048,1,1,1,8,49,343,1296,7776,15625,78125,65536

%N Triangle, read by rows, where T(n,k) = (n-floor(k/2))^k for k = 0..2*n - 1, with T(0,0) = 1.

%C 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.

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

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

%e 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!;

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

%e Rows begin:

%e   [1]

%e   [1, 1],

%e   [1, 2,  1,   1],

%e   [1, 3,  4,   8,   1,    1],

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

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

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

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

%e   ...

%e which can be derived from the square array A003992:

%e   [1, 0,  0,   0,   0,    0,     0, ...],

%e   [1, 1,  1,   1,   1,    1,     1, ...],

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

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

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

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

%e   ...

%e by shifting each column k down by floor(k/2) rows, and omitting the zeros coming from row 0 of A003992.

%p 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

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

%Y Cf. A003992, A099554, A099556.

%K nonn,tabf

%O 0,5

%A _Paul D. Hanna_, Oct 22 2004

%E Definition corrected by _Georg Fischer_, Nov 21 2024