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A357729 a(n) = Sum_{k=0..floor(n/2)} (-n)^k * Stirling2(n,2*k). 3

%I #18 Jan 25 2024 12:47:43

%S 1,0,-2,-9,-12,175,1938,9506,-24248,-1065663,-12021610,-56195425,

%T 677072220,19979234080,251733387514,1135594212255,-29317384858352,

%U -901607623649489,-13233854770928514,-68574233644270566,2258648937829442660,81748108921355457777

%N a(n) = Sum_{k=0..floor(n/2)} (-n)^k * Stirling2(n,2*k).

%H Andrew Howroyd, <a href="/A357729/b357729.txt">Table of n, a(n) for n = 0..200</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BellPolynomial.html">Bell Polynomial</a>.

%F a(n) = n! * [x^n] cos( sqrt(n) * (exp(x) - 1) ).

%F a(n) = ( Bell_n(sqrt(n) * i) + Bell_n(-sqrt(n) * i) )/2, where Bell_n(x) is n-th Bell polynomial and i is the imaginary unit.

%o (PARI) a(n) = sum(k=0, n\2, (-n)^k*stirling(n, 2*k, 2));

%o (PARI) a(n) = round(n!*polcoef(cos(sqrt(n)*(exp(x+x*O(x^n))-1)), n));

%o (PARI) Bell_poly(n, x) = exp(-x)*suminf(k=0, k^n*x^k/k!);

%o a(n) = round((Bell_poly(n, sqrt(n)*I)+Bell_poly(n, -sqrt(n)*I)))/2;

%Y Main diagonal of A357728.

%Y Cf. A357682, A357721.

%K sign

%O 0,3

%A _Seiichi Manyama_, Oct 11 2022

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Last modified April 16 03:28 EDT 2024. Contains 371696 sequences. (Running on oeis4.)