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%I #13 Sep 08 2022 08:45:31
%S 1,2,6,20,28,-936,-23672,-469456,-9112560,-182135008,-3804634784,
%T -83297957568,-1906560847424,-45349267830400,-1110454747949952,
%U -27582769902812416,-677408818380914432,-15581576995770441216,-284593895830642711040
%N Row sums of triangle A130757 (coefficients of scaled Laguerre-Sonin polynomials n!(2^(n-m))*L(n,1/2,x)).
%H G. C. Greubel, <a href="/A131441/b131441.txt">Table of n, a(n) for n = 0..400</a>
%F a(n) = Sum_{m=0..n} A130757(n,m), n>=0, with A130757(n,m) = n!*2^(n-m) *(-1)^m*binomial(n+1/2,n-m)/m!, n>=m>=0, else 0.
%F Conjecture: a(n) +2*(1-2*n)*a(n-1) +2*(2*n-1)*(n-1)*a(n-2)=0. - _R. J. Mathar_, Oct 02 2013
%t T[n_,k_]:= (-1)^k*n!*2^(n-k)*Binomial[n +1/2, n-k]/k!; Table[Sum[T[n, k], {k, 0, n}], {n, 0, 40}] (* _G. C. Greubel_, May 14 2018 *)
%o (PARI) for(n=0,30, print1(sum(k=0,n, (-1)^k*n!*2^(n-k)*binomial(n+1/2, n-k)/k!), ", ")) \\ _G. C. Greubel_, May 14 2018
%o (Magma) [Round(Factorial(n)*(&+[(-1)^k*2^(n-k)*Gamma(n+3/2)/(Gamma(k+1) *Gamma(n -k+1)*Gamma(k+3/2)): k in [0..n]])): n in [0..20]]; // _G. C. Greubel_, May 14 2018
%Y Cf. A130757.
%K sign,easy
%O 0,2
%A _Wolfdieter Lang_, Aug 07 2007
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