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%I #37 Mar 06 2023 20:36:26
%S 1,4,13,40,120,354,1031,2972,8495,24110,68016,190884,533293,1484020,
%T 4115185,11375764,31358376,86223942,236540915,647556620,1769374931,
%U 4826148314,13142564448,35736448200,97037995225,263156279524,712795854421,1928547574912,5212430732760
%N Expansion of (1-2*x)/(1-3*x+x^2)^2.
%C Convolution of 1, 1, 2, 5, 13, ... (A001519(n)) with 1, 3, 8, 21, 55, ... (A001906(n+1)).
%H Michael De Vlieger, <a href="/A238846/b238846.txt">Table of n, a(n) for n = 0..2385</a>
%H Sergi Elizalde, Rigoberto Flórez, and José Luis Ramírez, <a href="https://doi.org/10.26493/1855-3974.2478.d1b">Enumerating symmetric peaks in non-decreasing Dyck paths</a>, Ars Mathematica Contemporanea (2021).
%H David Eppstein, <a href="https://arxiv.org/abs/2303.00147">Non-crossing Hamiltonian Paths and Cycles in Output-Polynomial Time</a>, arXiv:2303.00147 [cs.CG], 2023, p. 20.
%H Valentin Ovsienko, <a href="https://arxiv.org/abs/2111.02553">Shadow sequences of integers, from Fibonacci to Markov and back</a>, arXiv:2111.02553 [math.CO], 2021.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (6,-11,6,-1).
%F a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) - a(n-4) for n>3, a(0)=1, a(1)=4, a(2)=13, a(3)=40.
%F a(n) = 3*a(n-1) - a(n-2) + A001519(n) for n>1, a(0)=1, a(1)=4.
%F a(n) = A238731(n+1, 1).
%F a(n) = -A124037(n+1, 1).
%F a(n) = (-1)^n*A126126(n+1, 1).
%F a(n) = ( (3 + sqrt(5))^(1+n)*(8 - (1 - sqrt(5))*(13 + 5*n)) + (3 - sqrt(5))^(1+n)*(8 - (1 + sqrt(5))*(13 + 5*n)) ) / (25*2^(2+n)). - _Bruno Berselli_, Mar 06 2014
%F a(n) = 2*A001870(n) - A001871(n). - _Philippe Deléham_, Mar 06 2014
%F a(n) = A197649(n+1) - 3*A001871(n-1). - _Philippe Deléham_, Mar 06 2014
%F a(n) = A001871(n) - 2*A001871(n-1). - _Philippe Deléham_, Mar 06 2014
%F 0 = 2 + a(n)*(a(n+1) - a(n+3)) + a(n+1)*(-6*a(n+1) + 12*a(n+2)) + a(n+2)*(-6*a(n+2) + a(n+3)) for all n in Z. - _Michael Somos_, Nov 23 2021
%e a(0) = 1*1 = 1;
%e a(1) = 1*3 + 1*1 = 4;
%e a(2) = 1*8 + 1*3 + 2*1 = 13;
%e a(3) = 1*21 + 1*8 + 2*3 + 5*1 = 40;
%e a(4) = 1*55 + 1*21 + 2*8 + 5*3 + 13*1 = 120; etc. (from first recurrence formula).
%e a(0) = 3*0 - 0 + 1 = 1;
%e a(1) = 3*1 - 0 + 1 = 4;
%e a(2) = 3*4 - 1 + 2 = 13;
%e a(3) = 3*13 - 4 + 5 = 40;
%e a(4) = 3*40 - 13 + 13 = 120; etc (from second recurrence formula).
%e G.f. = 1 + 4*x + 13*x^2 + 40*x^3 + 120*x^4 + 354*x^5 + 1031*x^6 + ... - _Michael Somos_, Nov 23 2021
%t LinearRecurrence[{6, -11, 6, -1}, {1, 4, 13, 40}, 30] (* _Bruno Berselli_, Mar 06 2014 *)
%t a[ n_] := If[n < 0, SeriesCoefficient[ x^3*(2 - x)/(1 - 3*x + x^2)^2, {x, 0, -n}], SeriesCoefficient[ (1 - 2*x)/(1 - 3*x + x^2)^2, {x, 0, n}]]; (* _Michael Somos_, Nov 23 2021 *)
%o (PARI) {a(n) = if(n<0, polcoeff( x^3*(2-x)/(1-3*x+x^2)^2 + x*O(x^-n), -n), polcoeff( (1-2*x)/(1-3*x+x^2)^2 + x*O(x^n), n))}; /* _Michael Somos_, Nov 23 2021 */
%Y Cf. A000045, A001519, A001906.
%Y Cf. A001870, A001871, A197649.
%K nonn,easy
%O 0,2
%A _Philippe Deléham_, Mar 05 2014
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