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%I #21 Sep 08 2022 08:46:19
%S 3,-3,20,-210,3024,-55440,1235520,-32432400,980179200,-33522128640,
%T 1279935820800,-53970627110400,2490952020480000,-124903451312640000,
%U 6761440164390912000,-393008709555221760000,24412776311194951680000,-1613955767240110694400000,113146793787569865523200000,-8384177419658927035269120000
%N Expansion of the exponential generating function (1/2)*(1 + 4*x)*(1 - (1 + 4*x)^(-3/2))/x.
%C This gives one half of the z-sequence for the generalized unsigned Lah number Sheffer matrix Lah[4,3] = A292219.
%C For Sheffer a- and z-sequences see a W. Lang link under A006232 with the references for the Riordan case, and also the present link.
%C a(n) = (-1)^n * A006963(n+2) for all n >= 0. - _Michael Somos_, Jul 02 2018
%H G. C. Greubel, <a href="/A292221/b292221.txt">Table of n, a(n) for n = 0..365</a>
%H Wolfdieter Lang, <a href="/A290597/a290597.log.txt">Note on a- and z-sequences of Sheffer number triangles for certain generalized Lah numbers.</a>
%F a(n) = [x^n/n!] (1/2)*(1 + 4*x)*(1 - (1 + 4*x)^(-3/2))/x.
%F a(0) = 3, a(n) = (-2)^n*Product_{j=1..n} (1 + 2*j)/(n+1) = ((-2)^n/(n+1))*A001147(n+1), n >= 1.
%F 0 = a(n)*(-2880*a(n+2) +2760*a(n+3) +952*a(n+4) +45*a(n+5)) +a(n+1)*(+216*a(n+2) -328*a(n+3) -81*a(n+4) -2*a(n+5)) +a(n+2)*(+12*a(n+3) +2*a(n+4)) for all n>0. - _Michael Somos_, Jul 02 2018
%e The sequence z(4,3;n) = 2*a(n) begins: {6, -6, 40, -420, 6048, -110880, 2471040, -64864800, 1960358400, -67044257280, 2559871641600, ...}.
%p seq(coeff(series(factorial(n)*(1/2)*(1+4*x)*(1-(1+4*x)^(-3/2))/x, x,n+1),x,n),n=0..20); # _Muniru A Asiru_, Jul 29 2018
%t a[ n_] := If[ n < 1, 3 Boole[n == 0], (-2)^n (2 n + 1)!! / (n + 1)]; (* _Michael Somos_, Jul 02 2018 *)
%t With[{nn=20},CoefficientList[Series[1/2 (1+4x) (1-(1+4x)^(-3/2))/x,{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Dec 24 2018 *)
%o (PARI) {a(n) = if( n<1, 3*(n==0), (-1)^n * (2*n + 1)! / (n + 1)!)}; /* _Michael Somos_, Jul 02 2018 */
%o (Magma) [3,-3] cat [(-1)^n*Factorial(2*n+1)/Factorial(n+1): n in [2..30]]; // _G. C. Greubel_, Jul 28 2018
%Y Cf. A001147, A006232, A006963, A292219.
%K sign,easy
%O 0,1
%A _Wolfdieter Lang_, Sep 23 2017
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