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A357721 a(n) = Sum_{k=0..floor(n/2)} (-n)^k * Stirling1(n,2*k). 2

%I #9 Oct 12 2022 08:58:27

%S 1,0,-2,9,-28,0,1200,-16464,167904,-1393200,7429240,43124400,

%T -2404571904,55590286752,-1027511503200,16489054310400,

%U -222885864448000,1994839594780032,14489184835474272,-1470395490046560000,54581408106475622400,-1608207353670788640000

%N a(n) = Sum_{k=0..floor(n/2)} (-n)^k * Stirling1(n,2*k).

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

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

%F a(n) = (-1)^n * ( (sqrt(n) * i)_n + (-sqrt(n) * i)_n )/2, where (x)_n is the Pochhammer symbol and i is the imaginary unit.

%o (PARI) a(n) = sum(k=0, n\2, (-n)^k*stirling(n, 2*k, 1));

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

%o (PARI) a(n) = (-1)^n*round((prod(k=0, n-1, sqrt(n)*I+k)+prod(k=0, n-1, -sqrt(n)*I+k)))/2;

%Y Main diagonal of A357720.

%Y Cf. A357683, A357729.

%K sign

%O 0,3

%A _Seiichi Manyama_, Oct 10 2022

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Last modified May 23 11:57 EDT 2024. Contains 372763 sequences. (Running on oeis4.)