A366858 Array read by ascending antidiagonals: A(n, k) = n! * [x^n] exp((k-1)*x)*(k*cosh(sqrt(k)*x) + sqrt(k)*sinh(sqrt(k)*x))/k, with 1 <= k <= n.

1, 1, 2, 1, 5, 3, 1, 12, 11, 4, 1, 29, 41, 19, 5, 1, 70, 153, 94, 29, 6, 1, 169, 571, 469, 177, 41, 7, 1, 408, 2131, 2344, 1097, 296, 55, 8, 1, 985, 7953, 11719, 6829, 2181, 457, 71, 9, 1, 2378, 29681, 58594, 42565, 16186, 3889, 666, 89, 10, 1, 5741, 110771, 292969, 265401, 120421, 33415, 6413, 929, 109, 11

Offset: 1

Author

Stefano Spezia, Oct 25 2023

Keywords

• nonn
• tabl

Examples

			The array begins:
  1,   2,    3,     4,     5,      6, ...
  1,   5,   11,    19,    29,     41, ...
  1,  12,   41,    94,   177,    296, ...
  1,  29,  153,   469,  1097,   2181, ...
  1,  70,  571,  2344,  6829,  16186, ...
  1, 169, 2131, 11719, 42565, 120421, ...
  ...
		

Crossrefs

Cf. A000012 (k=1), A000129 (k=2), A001835 (k=3), A083065 (k=4), A163073 (k=5).

Cf. A000027 (n=1), A028387 (n=2).

Cf. A366859 (antidiagonal sums).

Programs

Mathematica

A[n_,k_]:=n! SeriesCoefficient[E^((k-1) x)(k Cosh[Sqrt[k]x]+Sqrt[k]Sinh[Sqrt[k]*x])/k,{x,0,n}]; Table[A[n-k+1,k],{n,11},{k,n}]//Flatten (* or *)
A[n_,k_]:=(Sqrt[k]((k+Sqrt[k]-1)^n+(k-Sqrt[k]-1)^n)+(k+Sqrt[k]-1)^n-(k-Sqrt[k]-1)^n)/(2Sqrt[k]); Simplify[Table[A[n-k+1,k],{n,11},{k,n}]]//Flatten

Formula

A(n, k) = (sqrt(k)*(b(k)^n + c(k)^n) + b(k)^n - c(k)^n)/(2*sqrt(k)), with b(k) = k + sqrt(k) - 1 and c(k) = k - sqrt(k) - 1.

A(n, 2) = A000129(n+1).

A(2, n) = A028387(n-1).