%I M2856 #71 Jul 30 2024 15:26:40

%S 1,0,-1,3,-10,40,-190,1050,-6620,46800,-365300,3103100,-28269800,

%T 271627200,-2691559000,26495469000,-238131478000,1394099824000,

%U 15194495654000,-936096296850000,29697351895900000,-819329864480400000,21683886333440500000,-570263312237604700000

%N Expansion of e.g.f. cos(log(1+x)).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Vaclav Kotesovec, <a href="/A003703/b003703.txt">Table of n, a(n) for n = 0..400</a> (first 100 terms from T. D. Noe)

%H Vaclav Kotesovec, <a href="/A003703/a003703.jpg">Graph a(n+1)/a(n)</a>

%H Vladimir Victorovich Kruchinin, <a href="http://arxiv.org/abs/1009.2565">Composition of ordinary generating functions</a>, arXiv:1009.2565 [math.CO], 2010.

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

%F a(n) = Sum_{k=0..n-1} (-1)^(k+1)*T(n-k, k)*sin(Pi*(n-k-1)/2) + 0^n; T(n, k)=abs(A008276(n, k)). - _Paul Barry_, Apr 18 2005

%F abs(a(n)) = abs(f(n)) with f(n)=Product_{k=1..n} i+k (where i^2=-1). - _Yalcin Aktar_, Jul 13 2009

%F a(n) = Sum_{k=0..floor(n/2)} Stirling1(n,2*k)*(-1)^k. - _Vladimir Kruchinin_, Jan 29 2011

%F a(n+2)= -a(n+1)*(2*n+1) - a(n)*(1+n^2), a(0)=1, a(1)=0. - _Sergei N. Gladkovskii_, Aug 17 2012

%F a(n) = (-1)^n * ( (i)_n + (-i)_n )/2, where (x)_n is the Pochhammer symbol and i is the imaginary unit. - _Seiichi Manyama_, Oct 10 2022

%F a(n) = Re(gamma(i+1)/gamma(i+1-n)). The imaginary part is A009454. - _Colin Beveridge_, Jul 30 2024

%e 1 - x^2 + 3*x^3 - 10*x^4 + 40*x^5 - 190*x^6 + 1050*x^7 - 6620*x^8 + ...

%p a:= n-> add(Stirling1(n, 2*k) * (-1)^(k), k=0..floor(n/2)):

%p seq(a(n), n=0..20);

%t CoefficientList[Series[Cos[Log[1 + x]], {x, 0, 20}], x] * Range[0, 20]! (* _Vaclav Kotesovec_, Feb 16 2015 *)

%t Table[(-1)^n Im[Pochhammer[1-I, n-1]], {n, 0, 20}] (* _Vladimir Reshetnikov_, Sep 13 2016 *)

%o (PARI) {a(n) = if( n<0, 0, n! * polcoeff( cos( log( 1 + x + x * O(x^n))), n))} /* _Michael Somos_, Jul 26 2012 */

%o (PARI) a(n) = (-1)^n*(prod(k=0, n-1, I+k)+prod(k=0, n-1, -I+k))/2; \\ _Seiichi Manyama_, Oct 10 2022

%o (Python)

%o from sympy.functions.combinatorial.numbers import stirling

%o def A003703(n): return sum(stirling(n,k<<1,kind=1,signed=True)*(-1 if k&1 else 1) for k in range((n>>1)+1)) # _Chai Wah Wu_, Feb 22 2024

%Y Cf. A009024, A009454.

%K easy,sign

%O 0,4

%A _R. H. Hardin_, _Simon Plouffe_