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%I #31 Apr 10 2022 13:28:50
%S 0,1,-1,-1,0,9,-45,279,-2072,17265,-160065,1638031,-18353544,
%T 223578809,-2943054205,41639195623,-630238419600,10162622387809,
%U -173942578536993,3149754003442847,-60163773962649200,1208991988527548137
%N E.g.f. log(1+x)*cos(x).
%H Robert Israel, <a href="/A009410/b009410.txt">Table of n, a(n) for n = 0..450</a>
%F a(n) ~ (n-1)! * (-1)^(n+1) * cos(1). - _Vaclav Kotesovec_, Jan 23 2015
%F a(n) = -i*n*(i^n*3F1(1,1,1-n;2;-i)-(-i)^n*3F1(1,1,1-n;2;i))/2, n>0. - _Benedict W. J. Irwin_, May 30 2016
%F (4*(n+4))*(n+3)*(n+2)*(n+1)*a(n)+(16*(n+4))*(n+3)*(n+2)*a(n+1)+(n+4)*(n+3)*(8*n^2+32*n+51)*a(n+2)+(2*(n+4))*(16*n^2+92*n+135)*a(n+3)+(4*n^4+48*n^3+254*n^2+722*n+869)*a(n+4)+(4*(4*n^3+42*n^2+151*n+189))*a(n+5)+(n+4)*(23*n+89)*a(n+6)+(2*(7*n+30))*a(n+7)+3*a(n+8) = 0. - _Robert Israel_, May 30 2016
%F Recurrence: (4*n^2 - 32*n + 67)*a(n) = -2*(4*n^3 - 40*n^2 + 131*n - 138)*a(n-1) - (n-4)*(4*n^3 - 36*n^2 + 115*n - 139)*a(n-2) - 4*(n-3)*(4*n^2 - 30*n + 57)*a(n-3) - (2*n - 9)*(4*n^3 - 38*n^2 + 124*n - 141)*a(n-4) - 2*(n-4)*(4*n^2 - 28*n + 51)*a(n-5) - (n-5)*(n-4)*(4*n^2 - 24*n + 39)*a(n-6). - _Vaclav Kotesovec_, May 30 2016
%F a(n) = Sum_{k=0..floor((n-1)/2)} (-1)^(n-k-1) * binomial(n,2*k) * (n-2*k-1)!. - _Ilya Gutkovskiy_, Apr 10 2022
%p S:= series(log(1+x)*cos(x),x,31):
%p seq(coeff(S,x,j)*j!, j=0..30); # _Robert Israel_, May 30 2016
%t CoefficientList[Series[Cos[x]*Log[1 + x], {x, 0, 20}], x] * Range[0, 20]! (* _Vaclav Kotesovec_, Jan 23 2015 *)
%t Table[- I n (I^n HypergeometricPFQ[{1, 1, 1 - n}, {2}, -I] - (-I)^n HypergeometricPFQ[{1, 1, 1 - n}, {2}, I])/2, {n, 1, 20}] (* _Benedict W. J. Irwin_, May 30 2016 *)
%Y Cf. A009416.
%K sign,easy
%O 0,6
%A _R. H. Hardin_
%E Extended with signs by _Olivier Gérard_, Mar 15 1997
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