cp's OEIS Frontend

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.

%I A366858 #15 Nov 12 2023 13:30:13
%S A366858 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,
%T A366858 177,41,7,1,408,2131,2344,1097,296,55,8,1,985,7953,11719,6829,2181,
%U A366858 457,71,9,1,2378,29681,58594,42565,16186,3889,666,89,10,1,5741,110771,292969,265401,120421,33415,6413,929,109,11
%N 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.
%F A366858 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.
%F A366858 A(n, 2) = A000129(n+1).
%F A366858 A(2, n) = A028387(n-1).
%e A366858 The array begins:
%e A366858   1,   2,    3,     4,     5,      6, ...
%e A366858   1,   5,   11,    19,    29,     41, ...
%e A366858   1,  12,   41,    94,   177,    296, ...
%e A366858   1,  29,  153,   469,  1097,   2181, ...
%e A366858   1,  70,  571,  2344,  6829,  16186, ...
%e A366858   1, 169, 2131, 11719, 42565, 120421, ...
%e A366858   ...
%t A366858 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 *)
%t A366858 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
%Y A366858 Cf. A000012 (k=1), A000129 (k=2), A001835 (k=3), A083065 (k=4), A163073 (k=5).
%Y A366858 Cf. A000027 (n=1), A028387 (n=2).
%Y A366858 Cf. A366859 (antidiagonal sums).
%K A366858 nonn,tabl
%O A366858 1,3
%A A366858 _Stefano Spezia_, Oct 25 2023