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%I #11 Nov 30 2022 08:25:55
%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.
%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.
%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
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