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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, 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
1,3
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).
EXAMPLE
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, ...
...
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
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).
Sequence in context: A106513 A054446 A164981 * A047858 A125171 A280784
KEYWORD
nonn,tabl
AUTHOR
Stefano Spezia, Oct 25 2023
STATUS
approved