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A357728 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. cos( sqrt(k) * (exp(x) - 1) ).

%I A357728 #20 Feb 16 2025 08:34:04
%S A357728 1,1,0,1,0,0,1,0,-1,0,1,0,-2,-3,0,1,0,-3,-6,-6,0,1,0,-4,-9,-10,-5,0,1,
%T A357728 0,-5,-12,-12,10,33,0,1,0,-6,-15,-12,45,190,266,0,1,0,-7,-18,-10,100,
%U A357728 465,1106,1309,0,1,0,-8,-21,-6,175,852,2394,4438,4905,0,1,0,-9,-24,0,270,1345,4004,7827,9978,11516,0
%N A357728 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. cos( sqrt(k) * (exp(x) - 1) ).
%H A357728 Andrew Howroyd, <a href="/A357728/b357728.txt">Table of n, a(n) for n = 0..1325</a> (first 51 antidiagonals)
%H A357728 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BellPolynomial.html">Bell Polynomial</a>.
%F A357728 T(n,k) = Sum_{j=0..floor(n/2)} (-k)^j * Stirling2(n,2*j).
%F A357728 T(n,k) = ( Bell_n(sqrt(k) * i) + Bell_n(-sqrt(k) * i) )/2, where Bell_n(x) is n-th Bell polynomial and i is the imaginary unit.
%e A357728 Square array begins:
%e A357728   1,  1,   1,   1,   1,   1, ...
%e A357728   0,  0,   0,   0,   0,   0, ...
%e A357728   0, -1,  -2,  -3,  -4,  -5, ...
%e A357728   0, -3,  -6,  -9, -12, -15, ...
%e A357728   0, -6, -10, -12, -12, -10, ...
%e A357728   0, -5,  10,  45, 100, 175, ...
%o A357728 (PARI) T(n, k) = sum(j=0, n\2, (-k)^j*stirling(n, 2*j, 2));
%o A357728 (PARI) Bell_poly(n, x) = exp(-x)*suminf(k=0, k^n*x^k/k!);
%o A357728 T(n, k) = round((Bell_poly(n, sqrt(k)*I)+Bell_poly(n, -sqrt(k)*I)))/2;
%Y A357728 Columns k=0-4 give: A000007, A121867, A357725, A357726, A357727.
%Y A357728 Main diagonal gives A357729.
%Y A357728 Cf. A357681, A357720.
%K A357728 sign,tabl
%O A357728 0,13
%A A357728 _Seiichi Manyama_, Oct 11 2022