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A190156 Expansion of (1-x-3*x^2-sqrt(1-2*x-5*x^2+2*x^3+x^4))/(2*x^3*(1+2*x)).

%I A190156 #39 Jun 06 2023 18:10:14
%S A190156 1,1,4,8,24,61,175,486,1405,4059,11924,35223,105007,314867,950018,
%T A190156 2880620,8775638,26843704,82420464,253916555,784672011,2431695541,
%U A190156 7555381574,23531026853,73448858179,229730744171,719914525210,2260031465504,7106721944206
%N A190156 Expansion of (1-x-3*x^2-sqrt(1-2*x-5*x^2+2*x^3+x^4))/(2*x^3*(1+2*x)).
%C A190156 Diagonal sums of the Riordan matrix (g(x),x*g(x)), where g(x) = (1-x-x^2-sqrt(1-2*x-5*x^2+2*x^3+x^4))/(2*x^2) (A132276).
%H A190156 G. C. Greubel, <a href="/A190156/b190156.txt">Table of n, a(n) for n = 0..1000</a>
%H A190156 Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL26/Barry/barry601.html">On Motzkin-Schröder Paths, Riordan Arrays, and Somos-4 Sequences</a>, J. Int. Seq. (2023) Vol. 26, Art. 23.4.7.
%F A190156 G.f.: (1-x-3*x^2-sqrt(1-2*x-5*x^2+2*x^3+x^4))/(2*x^3*(1+2*x)).
%F A190156 D-finite with recurrence: (n+3)*a(n) +3*a(n-1) +3*(-3*n-2)*a(n-2) +(-8*n-3)*a(n-3) +(5*n-9) *a(n-4) +2*(n-3)*a(n-5)=0. - _R. J. Mathar_, Oct 08 2016
%F A190156 a(n) = Sum_{m=0..n} Sum_{j=0..n-m+2} 2^(m-j+1)*C(n-m,j-1)*C(n-m+2,j)*C(n-m-j+1,m-j+1)/(n-m+2). - _Vladimir Kruchinin_, Oct 19 2020
%F A190156 a(n) = Sum_{k=0..floor(n/2)} 2^k*binomial(n-k, k)*hypergeom([-k, k-n-1], [2], 1/2). - _Peter Luschny_, Oct 19 2020
%p A190156 T := (n, k) -> simplify(2^k*binomial(n-k, k)*hypergeom([-k, k-n-1], [2], 1/2)):
%p A190156 seq(add(T(n, k), k=0..floor(n/2)), n=0..28); # _Peter Luschny_, Oct 19 2020
%t A190156 CoefficientList[Series[(1-x-3x^2-Sqrt[1-2x-5x^2+2x^3+x^4])/(2x^3(1+2x)),{x,0,28}],x]
%o A190156 (PARI)  x='x+O('x^66); Vec((1-x-3*x^2-sqrt(1-2*x-5*x^2+2*x^3+x^4))/(2*x^3*(1+2*x))) /* _Joerg Arndt_, May 15 2011 */
%o A190156 (Magma) m:=50; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!((1-x-3*x^2-Sqrt(1-2*x-5*x^2+2*x^3+x^4))/(2*x^3*(1+2*x)))); // _G. C. Greubel_, Oct 22 2018
%o A190156 (Maxima)
%o A190156 a(n):=sum(sum(2^(m-j+1)*binomial(n-m,j-1)*binomial(n-m+2,j)*binomial(n-m-j+1,m-j+1),j,0,n-m+2)/(n-m+2),m,0,n); /* _Vladimir Kruchinin_, Oct 19 2020 */
%Y A190156 Cf. A132276.
%K A190156 nonn
%O A190156 0,3
%A A190156 _Emanuele Munarini_, May 05 2011