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.

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, 64, 1, 8, 29, 96, 265, 560, 729, 128, 1, 9, 34, 123, 384, 989, 1912, 2187, 256, 1, 10, 39, 152, 521, 1536, 3691, 6528, 6561, 512, 1, 11, 44, 183, 676, 2207, 6144, 13775, 22288, 19683, 1024

Offset

0,3

Links

Table of n, a(n) for n=0..65.

Formula

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).

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).

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.

Example

[0] 1, 2, 4,    8,  16,   32,    64,   128,    256,     512, ...  [A000079]
[1] 1, 3, 9,   27,  81,  243,   729,  2187,   6561,   19683, ...  [A000244]
[2] 1, 4, 14,  48, 164,  560,  1912,  6528,  22288,   76096, ...  [A007070]
[3] 1, 5, 19,  71, 265,  989,  3691, 13775,  51409,  191861, ...  [A001834]
[4] 1, 6, 24,  96, 384, 1536,  6144, 24576,  98304,  393216, ...  [A164908]
[5] 1, 7, 29, 123, 521, 2207,  9349, 39603, 167761,  710647, ...  [A048876]
[6] 1, 8, 34, 152, 676, 3008, 13384, 59552, 264976, 1179008, ...  [A335749]

Maple

Arow := proc(n, len) local H; H := (x, y) -> exp(2*x)*(y*sinh(x*y) + cosh(x*y)):
series(H(x, sqrt(n)), x, len+1): seq(k!*coeff(%, x, k), k=0..len-1) end:
A := (n, k) -> Arow(n, k+2)[k+1]: seq(lprint(Arow(n, 9)), n=0..6);
# Alternative:
A := proc(n, k) option remember; if k = 0 then return 1 fi;
if k = 1 then return n+2 fi; 4*A(n, k-1) + (n-4)*A(n, k-2) end;

Crossrefs

Cf. A000079 (n=0), A000244 (n=1), A007070 (n=2), A001834 (n=3), A164908 (n=4), A048876 (n=5), A335749 (n=6).

Cf. A335537, A118800.

Sequence in context: A210219 A125103 A171275 * A284873 A107616 A055208

Adjacent sequences: A335309 A335310 A335311 * A335313 A335314 A335315

Keyword

nonn,tabl

Author

Peter Luschny, Jun 24 2020

Status

approved