login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

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