cp's OEIS Frontend

A053297 Row sums of array T in A053199.

%I A053297 #21 May 07 2025 00:49:00
%S A053297 1,5,22,92,372,1468,5688,21728,82064,307088,1140320,4206912,15434048,
%T A053297 56350912,204875648,742104064,2679197952,9644109056,34623075840,
%U A053297 124001176576,443136848896,1580464036864,5626501838848,19996918849536,70960191213568,251445325991936
%N A053297 Row sums of array T in A053199.
%C A053297 The generating series is a power series composition G(F(t)) where F(t) = t + 3*t^2 + 7*t^3 + 15*t^4 + ... is generating series of A000225, and G(t) = t + 2*t^2 + 3*t^3 + 4*t^4 + ... is generating series of the natural numbers A000027. Proof follows as in reference below. - _Oboifeng Dira_, Nov 03 2016
%H A053297 G. C. Greubel, <a href="/A053297/b053297.txt">Table of n, a(n) for n = 1..1000</a>
%H A053297 O. 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, pp. 849-853.
%F A053297 G.f.: x * (1 - x) * (1 - 2*x) / (1 - 4*x + 2*x^2)^2. - _Michael Somos_, Nov 03 2016
%F A053297 a(n) = 8*a(n-1) + 20*a(n-2) - 16*a(n-3) + 4*a(n-4) for all n in Z. - _Michael Somos_, Nov 03 2016
%F A053297 a(n) = -a(-n) * 2^n for all n in Z. - _Michael Somos_, Nov 03 2016
%e A053297 G.f. = x + 5*x^2 + 22*x^3 + 92*x^4 + 372*x^5 + 1468*x^6 + 5688*x^7 + 21728*x^8 + ...
%t A053297 Drop[CoefficientList[Series[x*(1-x)*(1-2*x)/(1-4*x+2*x^2)^2, {x,0,50}], x], 1] (* _G. C. Greubel_, May 24 2018 *)
%o A053297 (PARI) my(x='x+O('x^30)); Vec(x*(1-x)*(1-2*x)/(1-4*x+2*x^2)^2) \\ _G. C. Greubel_, May 24 2018
%o A053297 (Magma) m:=25; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(x*(1-x)*(1-2*x)/(1-4*x+2*x^2)^2)); // _G. C. Greubel_, May 24 2018
%Y A053297 Cf. A000027, A000225, A053199.
%K A053297 nonn
%O A053297 1,2
%A A053297 _Clark Kimberling_, Mar 18 2000