A131441 Row sums of triangle A130757 (coefficients of scaled Laguerre-Sonin polynomials n!(2^(n-m))*L(n,1/2,x)).

1, 2, 6, 20, 28, -936, -23672, -469456, -9112560, -182135008, -3804634784, -83297957568, -1906560847424, -45349267830400, -1110454747949952, -27582769902812416, -677408818380914432, -15581576995770441216, -284593895830642711040

Offset: 0

Author

Wolfdieter Lang, Aug 07 2007

Keywords

• sign
• easy
• changed

Links

• G. C. Greubel, Table of n, a(n) for n = 0..400

Crossrefs

Cf. A130757.

Programs

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

Mathematica

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 *)

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

Formula

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.

D-finite with recurrence: 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