%I #14 Jun 24 2020 16:00:55
%S 1,1,2,1,3,4,1,4,9,8,1,5,14,27,16,1,6,19,48,81,32,1,7,24,71,164,243,
%T 64,1,8,29,96,265,560,729,128,1,9,34,123,384,989,1912,2187,256,1,10,
%U 39,152,521,1536,3691,6528,6561,512,1,11,44,183,676,2207,6144,13775,22288,19683,1024
%N A(n, k) = k! [x^k] exp(2*x)*(y*sinh(x*y) + cosh(x*y)) and y = sqrt(n). Square array read by ascending antidiagonals, for n >= 0 and k >= 0.
%F The Taylor series of exp(2*x)*(y*sinh(x*y) + cosh(x*y)) starts: 1 + x*(y^2 + 2) + x^2*((5*y^2)/2 + 2) + (1/6)*x^3*(y^4 + 18*y^2 + 8) + x^4*((3*y^4)/8 + (7*y^2)/3 + 2/3) + O(x^5). The coefficient polynomials expand in even powers (cf. A118800).
%F A(n, k) = k! [x^k] (c*exp(x*(1 + c)) + d*exp(x*(1 + d)))/2 where c = 1 + sqrt(n) and d = 1 - sqrt(n).
%F A(n, k) = 4*A(n, k-1) + (n-4)*A(n, k-2) if k >= 2. A(n, 0) = 1, A(n, 1) = n + 2.
%e [0] 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, ... [A000079]
%e [1] 1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, ... [A000244]
%e [2] 1, 4, 14, 48, 164, 560, 1912, 6528, 22288, 76096, ... [A007070]
%e [3] 1, 5, 19, 71, 265, 989, 3691, 13775, 51409, 191861, ... [A001834]
%e [4] 1, 6, 24, 96, 384, 1536, 6144, 24576, 98304, 393216, ... [A164908]
%e [5] 1, 7, 29, 123, 521, 2207, 9349, 39603, 167761, 710647, ... [A048876]
%e [6] 1, 8, 34, 152, 676, 3008, 13384, 59552, 264976, 1179008, ... [A335749]
%p Arow := proc(n, len) local H; H := (x, y) -> exp(2*x)*(y*sinh(x*y) + cosh(x*y)):
%p series(H(x, sqrt(n)), x, len+1): seq(k!*coeff(%, x, k), k=0..len-1) end:
%p A := (n, k) -> Arow(n, k+2)[k+1]: seq(lprint(Arow(n, 9)), n=0..6);
%p # Alternative:
%p A := proc(n, k) option remember; if k = 0 then return 1 fi;
%p if k = 1 then return n+2 fi; 4*A(n, k-1) + (n-4)*A(n, k-2) end;
%Y Cf. A000079 (n=0), A000244 (n=1), A007070 (n=2), A001834 (n=3), A164908 (n=4), A048876 (n=5), A335749 (n=6).
%Y Cf. A335537, A118800.
%K nonn,tabl
%O 0,3
%A _Peter Luschny_, Jun 24 2020