# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a238846 Showing 1-1 of 1 %I A238846 #37 Mar 06 2023 20:36:26 %S A238846 1,4,13,40,120,354,1031,2972,8495,24110,68016,190884,533293,1484020, %T A238846 4115185,11375764,31358376,86223942,236540915,647556620,1769374931, %U A238846 4826148314,13142564448,35736448200,97037995225,263156279524,712795854421,1928547574912,5212430732760 %N A238846 Expansion of (1-2*x)/(1-3*x+x^2)^2. %C A238846 Convolution of 1, 1, 2, 5, 13, ... (A001519(n)) with 1, 3, 8, 21, 55, ... (A001906(n+1)). %H A238846 Michael De Vlieger, Table of n, a(n) for n = 0..2385 %H A238846 Sergi Elizalde, Rigoberto Flórez, and José Luis Ramírez, Enumerating symmetric peaks in non-decreasing Dyck paths, Ars Mathematica Contemporanea (2021). %H A238846 David Eppstein, Non-crossing Hamiltonian Paths and Cycles in Output-Polynomial Time, arXiv:2303.00147 [cs.CG], 2023, p. 20. %H A238846 Valentin Ovsienko, Shadow sequences of integers, from Fibonacci to Markov and back, arXiv:2111.02553 [math.CO], 2021. %H A238846 Index entries for linear recurrences with constant coefficients, signature (6,-11,6,-1). %F A238846 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 A238846 a(n) = 3*a(n-1) - a(n-2) + A001519(n) for n>1, a(0)=1, a(1)=4. %F A238846 a(n) = A238731(n+1, 1). %F A238846 a(n) = -A124037(n+1, 1). %F A238846 a(n) = (-1)^n*A126126(n+1, 1). %F A238846 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 A238846 a(n) = 2*A001870(n) - A001871(n). - _Philippe Deléham_, Mar 06 2014 %F A238846 a(n) = A197649(n+1) - 3*A001871(n-1). - _Philippe Deléham_, Mar 06 2014 %F A238846 a(n) = A001871(n) - 2*A001871(n-1). - _Philippe Deléham_, Mar 06 2014 %F A238846 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 A238846 a(0) = 1*1 = 1; %e A238846 a(1) = 1*3 + 1*1 = 4; %e A238846 a(2) = 1*8 + 1*3 + 2*1 = 13; %e A238846 a(3) = 1*21 + 1*8 + 2*3 + 5*1 = 40; %e A238846 a(4) = 1*55 + 1*21 + 2*8 + 5*3 + 13*1 = 120; etc. (from first recurrence formula). %e A238846 a(0) = 3*0 - 0 + 1 = 1; %e A238846 a(1) = 3*1 - 0 + 1 = 4; %e A238846 a(2) = 3*4 - 1 + 2 = 13; %e A238846 a(3) = 3*13 - 4 + 5 = 40; %e A238846 a(4) = 3*40 - 13 + 13 = 120; etc (from second recurrence formula). %e A238846 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 A238846 LinearRecurrence[{6, -11, 6, -1}, {1, 4, 13, 40}, 30] (* _Bruno Berselli_, Mar 06 2014 *) %t A238846 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 A238846 (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 A238846 Cf. A000045, A001519, A001906. %Y A238846 Cf. A001870, A001871, A197649. %K A238846 nonn,easy %O A238846 0,2 %A A238846 _Philippe Deléham_, Mar 05 2014 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE