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A054447 Row sums of triangle A054446 (partial row sums triangle of Fibonacci convolution triangle).

%I A054447 #55 Mar 03 2024 11:14:04
%S A054447 1,3,9,26,73,201,545,1460,3873,10191,26633,69198,178889,460437,
%T A054447 1180545,3016552,7684481,19522203,49473097,125093506,315654537,
%U A054447 795016545,1998909985,5017895196,12578040097,31485713511,78716283081,196563649718,490301138569,1221726409005
%N A054447 Row sums of triangle A054446 (partial row sums triangle of Fibonacci convolution triangle).
%H A054447 Michael De Vlieger, <a href="/A054447/b054447.txt">Table of n, a(n) for n = 0..2604</a>
%H A054447 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 A054447 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4, -2, -4, -1).
%F A054447 a(n) = Sum_{m=0..n} A054446(n,m) = ((n+1)*P(n+2)+(2-n)*P(n+1))/4, with P(n)=A000129(n) (Pell numbers).
%F A054447 G.f.: Pell(x)/(1-x*Fib(x)) = (Pell(x)^2)/Fib(x), with Pell(x)= 1/(1-2*x-x^2) = g.f. A000129(n+1) (Pell numbers without 0) and Fib(x)=1/(1-x-x^2) = g.f. A000045(n+1) (Fibonacci numbers without 0).
%F A054447 a(n) = Sum_(k*Sum_(binomial(i,n-k-i)*binomial(k+i-1,k-1),i,ceiling((n-k)/2),n-k),k,1,n), n>0. - _Vladimir Kruchinin_, Sep 06 2010
%F A054447 a(n) = 4*a(n-1) - 2*a(n-2) - 4*a(n-3) - a(n-4), a(0)=1, a(1)=3, a(2)=9, a(3)=26. - _Philippe Deléham_, Jan 22 2014
%F A054447 G.f.: (1-x-x^2)/(1-2*x-x^2)^2 = g(f(x))/x, where g is g.f. of A001477 and f is g.f. of A000045. - _Oboifeng Dira_, Jun 21 2020
%t A054447 LinearRecurrence[{4, -2, -4, -1}, {1, 3, 9, 26}, 30] (* _Michael De Vlieger_, Jun 23 2020 *)
%o A054447 (Maxima) a(n):=sum(k*sum(binomial(i,n-k-i)*binomial(k+i-1,k-1),i,ceiling((n-k)/2),n-k),k,1,n); /* _Vladimir Kruchinin_, Sep 06 2010 */
%Y A054447 Cf. A000129, A000045, A054446, A001477.
%K A054447 easy,nonn
%O A054447 0,2
%A A054447 _Wolfdieter Lang_, Apr 27 2000