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A335312 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.

%I A335312 #14 Jun 24 2020 16:00:55
%S A335312 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 A335312 64,1,8,29,96,265,560,729,128,1,9,34,123,384,989,1912,2187,256,1,10,
%U A335312 39,152,521,1536,3691,6528,6561,512,1,11,44,183,676,2207,6144,13775,22288,19683,1024
%N A335312 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 A335312 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 A335312 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 A335312 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 A335312 [0] 1, 2, 4,    8,  16,   32,    64,   128,    256,     512, ...  [A000079]
%e A335312 [1] 1, 3, 9,   27,  81,  243,   729,  2187,   6561,   19683, ...  [A000244]
%e A335312 [2] 1, 4, 14,  48, 164,  560,  1912,  6528,  22288,   76096, ...  [A007070]
%e A335312 [3] 1, 5, 19,  71, 265,  989,  3691, 13775,  51409,  191861, ...  [A001834]
%e A335312 [4] 1, 6, 24,  96, 384, 1536,  6144, 24576,  98304,  393216, ...  [A164908]
%e A335312 [5] 1, 7, 29, 123, 521, 2207,  9349, 39603, 167761,  710647, ...  [A048876]
%e A335312 [6] 1, 8, 34, 152, 676, 3008, 13384, 59552, 264976, 1179008, ...  [A335749]
%p A335312 Arow := proc(n, len) local H; H := (x, y) -> exp(2*x)*(y*sinh(x*y) + cosh(x*y)):
%p A335312 series(H(x, sqrt(n)), x, len+1): seq(k!*coeff(%, x, k), k=0..len-1) end:
%p A335312 A := (n, k) -> Arow(n, k+2)[k+1]: seq(lprint(Arow(n, 9)), n=0..6);
%p A335312 # Alternative:
%p A335312 A := proc(n, k) option remember; if k = 0 then return 1 fi;
%p A335312 if k = 1 then return n+2 fi; 4*A(n, k-1) + (n-4)*A(n, k-2) end;
%Y A335312 Cf. A000079 (n=0), A000244 (n=1), A007070 (n=2), A001834 (n=3), A164908 (n=4), A048876 (n=5), A335749 (n=6).
%Y A335312 Cf. A335537, A118800.
%K A335312 nonn,tabl
%O A335312 0,3
%A A335312 _Peter Luschny_, Jun 24 2020