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Row sums of A335436.
0

%I #29 Sep 14 2020 00:33:44

%S 1,7,34,150,628,2540,10024,38840,148368,560368,2096928,7786592,

%T 28726592,105390272,384788096,1398978432,5067403520,18294707968,

%U 65854095872,236421150208,846732997632,3025927678976,10792083499008,38420157773824,136547503083520,484546494459904,1716976084393984

%N Row sums of A335436.

%C This sequence is also a composition of generating functions H(x) = G(F(x)), where G(x) = x/(1-4*x)^2 is the generating function of A002697 and F(x) = x*(1-x)/(1-2*x^2) is the generating function of 0, A016116*(-1)^n.

%H Oboifeng Dira, <a href="http://www.seams-bull-math.ynu.edu.cn/downloadfile.jsp?filemenu=_201706&amp;filename=07_41(6).pdf">A Note on Composition and Recursion</a>, Southeast Asian Bulletin of Mathematics (2017), Vol. 41, Issue 6, 849-853.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (8,-20,16,-4).

%F a(n) = 8*a(n-1)-20*a(n-2)+16*a(n-3)-4*a(n-4), a(0)=1, a(1)=7, a(2)=34, a(3)=150 for n>=4.

%F G.f.: (1-x)*(1-2*x^2)/(1-4*x+2*x^2)^2.

%F a(0)=1; a(n) = 2*n+1+Sum_{k=1..n}[(2+sqrt(2))^(k+1)-(2-sqrt(2))^(k+1)]*(2n-k+1)/(4*sqrt(2)), n>=1.

%F G.f.: G(F(x))/x where G(x) is g.f of A002697 and F(x) is g.f of 0,A016116*(-1)^n.

%e For n = 4, a(4) = 8*a(3)-20*a(2)+16*a(1)-4*a(0) = 8*150-20*34+16*7-4*1 = 628.

%p f:=x->x*(1-x)/(1-2*x^2):g:=x->(x)/(1-4*x)^2:

%p C:=n->coeff(series(g(f(x))/x,x,n+1),x,n): seq(C(n),n=0..30);

%Y Composition of g.fs of A002697 and A016116.

%Y Cf. A335436.

%K nonn

%O 0,2

%A _Oboifeng Dira_, Sep 11 2020