{"id":"n-68458fe983bd99b5","revision":1,"first_index":1,"fingerprint":"f6-8abd41dc566391f2","name":"obreschkoff_combination of Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n and having k weak ascents (1<=k<=n). A Schroeder path of length 2n is a lattice path from (0,0) to (2n,0) consisting of U=(1,1), D=(1,-1) and H=(2,0) steps and never going below the x-axis. A weak ascent in a Schroeder path is a maximal sequence of consecutive U and H steps.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["6","2"],["12","14","2"],["24","60","26","2"],["48","208","184","42","2"],["96","640","960","440","62","2"],["192","1824","4160","3200","900","86","2"],["384","4928","15904","18480","8680","1652","114","2"],["768","12800","55552","90944","64288","20384","2800","146","2"],["1536","32256","181248","397824","395136","188160","43008","4464","182","2"],["3072","79360","560640","1589760","2112768","1411200","483840","83520","6780","222","2"],["6144","191488","1661440","5913600","10137600","9047808","4346496","1124640","151800","9900","266","2"],["12288","454656","4753408","20753920","44605440","51296256","32896512","11918016","2410320","261360","13992","314","2"],["24576","1064960","13205504","69408768","182820352","263357952","217451520","105101568","29748576","4832256","430144","19240","366","2"],["49152","2465792","35782656","222896128","706241536","1245916672","1287796224","800946432","302318016","68708640","9161152","681408","25844","422","2"],["98304","5652480","94904320","691425280","2594801664","5503353856","6962475520","5418167040","2630559360","797020224","148660512","16562000","1044680","34020","482","2"]],"degrees":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"row_sums":["8","28","112","484","2200","10364","50144","247684","1243816","6331164","32591184","169376484","887465784","4682990076","24864759744"],"nonn":true,"definitions":[{"position":0,"source":"derived:obreschkoff_combination","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["6","2"]],"recurrence":{"denominator":{"coeffs":[["-6","4","-2"],["15","-2","1"],["9","-2","1"]]},"terms":[{"coeff":{"coeffs":[["-12","10","-8","0"],["30","-21","12","-3"],["18","23","-4","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","20","-14","2"],["0","36","-26","8","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162734,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, +inf)","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)","region_exact":null,"edges":["e-798bfa3a83d1e4b4"]}],"region_exact":{},"incoming_edges":[{"id":"e-798bfa3a83d1e4b4","operation":{"operation":"obreschkoff_combination"},"operation_detail":null,"conclusion_id":"n-68458fe983bd99b5","premise_count":2,"theorem":"Hermite--Kakeya--Obreschkoff: real-rooted f, g with interlacing zeros give a zero or real-rooted a f + b g for all real a, b (Obreschkoff 1963; Fisk, Polynomials, roots, and interlacing, Thm 1.43)","lean_route":null,"lean_declarations":[],"certificate_hash":"f5a9e8c794a313105309dec21d6a16c65cb973371f26ca744596efffb5279c37","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{}}],"premises":[{"position":0,"node_id":"n-6a30765140629bd5","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n and having k weak ascents (1<=k<=n). A Schroeder path of length 2n is a lattice path from (0,0) to (2n,0) consisting of U=(1,1), D=(1,-1) and H=(2,0) steps and never going below the x-axis. A weak ascent in a Schroeder path is a maximal sequence of consecutive U and H steps.","uses":{"claim":"real_rooted","from_index":1,"region":{}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)"},{"position":1,"node_id":"n-6a30765140629bd5","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n and having k weak ascents (1<=k<=n). A Schroeder path of length 2n is a lattice path from (0,0) to (2n,0) consisting of U=(1,1), D=(1,-1) and H=(2,0) steps and never going below the x-axis. A weak ascent in a Schroeder path is a maximal sequence of consecutive U and H steps.","uses":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"uses_text":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:    6,     2;\n   2:   12,    14,      2;\n   3:   24,    60,     26,       2;\n   4:   48,   208,    184,      42,       2;\n   5:   96,   640,    960,     440,      62,       2;\n   6:  192,  1824,   4160,    3200,     900,      86,      2;\n   7:  384,  4928,  15904,   18480,    8680,    1652,    114,     2;\n   8:  768, 12800,  55552,   90944,   64288,   20384,   2800,   146,    2;\n   9: 1536, 32256, 181248,  397824,  395136,  188160,  43008,  4464,  182,   2;\n  10: 3072, 79360, 560640, 1589760, 2112768, 1411200, 483840, 83520, 6780, 222, 2;\n","recurrence_sentence":"The row polynomials satisfy (9*n^2+15*n-6-(2*n^2+2*n-4)*x+(n^2+n-2)*x^2)*P(n,x) = (18*n^2+30*n-12+(23*n^2-21*n+10)*x-(4*n^2-12*n+8)*x^2+(3*n^2-3*n)*x^3)*P(n-1,x) + (36*n*x-(26*n-20)*x^2+(8*n-14)*x^3-(2*n-2)*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 6 + 2*x.","mathematica":"P[1, x_] := 6 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[((18*n^2 + 30*n - 12 + (23*n^2 - 21*n + 10)*x - (4*n^2 - 12*n + 8)*x^2 + (3*n^2 - 3*n)*x^3)*P[n - 1, x] + (36*n*x - (26*n - 20)*x^2 + (8*n - 14)*x^3 - (2*n - 2)*x^4)*D[P[n - 1, x], x])/(9*n^2 + 15*n - 6 - (2*n^2 + 2*n - 4)*x + (n^2 + n - 2)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 6 + 2*x; for(n = 2, N, my(i = n); v[i] = ((18*n^2 + 30*n - 12 + (23*n^2 - 21*n + 10)*x - (4*n^2 - 12*n + 8)*x^2 + (3*n^2 - 3*n)*x^3)*v[i - 1] + (36*n*x - (26*n - 20)*x^2 + (8*n - 14)*x^3 - (2*n - 2)*x^4)*deriv(v[i - 1], x))/(9*n^2 + 15*n - 6 - (2*n^2 + 2*n - 4)*x + (n^2 + n - 2)*x^2)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-6a30765140629bd5","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-798bfa3a83d1e4b4"],"text":"A114655 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A114655 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}