{"id":"n-b2994d0cc7b611de","revision":1,"first_index":0,"fingerprint":"f6-27cc38aba11af89b","name":"obreschkoff_combination of Triangle of increasing mobiles (circular rooted trees) with n nodes and k leaves.","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":[["2"],["2","1"],["2","5","2"],["2","15","20","6"],["2","37","116","102","24"],["2","83","522","970","624","120"],["2","177","2038","6914","8844","4440","720"],["2","367","7296","41498","91744","87948","36000","5040"],["2","749","24704","222746","784664","1253396","951264","327600","40320"],["2","1515","80606","1107638","5892576","14498852","17855784","11146896","3306240","362880"],["2","3049","256466","5217590","40351972","145217708","268809240","266817024","140902560","36650880","3628800"],["2","6119","801924","23629902","258233000","1309986528","3465730384","5074498032","4192285968","1913171040","442713600","39916800"],["2","12261","2477164","103947814","1570426584","10923554608","39848062400","81932879424","98368557408","69301505376","27791614080","5787936000","479001600"],["2","24547","7586802","447387346","9184121840","85733800800","419764548800","1170273985648","1945625943840","1967751650736","1204750441728","430307130240","81430272000","6227020800"],["2","49121","23095886","1893894490","52099416892","641633156416","4128879596096","15202261366928","33817354919552","46821965813040","40733353346976","22002839217792","7077145340160","1226723097600","87178291200"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["2","3","9","43","281","2321","23135","269895","3605445","54252989","907855291","16720973299","336109974721","7321503915129","171793350602151"],"nonn":true,"definitions":[{"position":0,"source":"derived:obreschkoff_combination","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["2"],["2","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","-2"],["0","1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","1"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0"],["0","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162732,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, +inf)","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)","region_exact":null,"edges":["e-fd8a72d8822c546e"]}],"region_exact":{},"incoming_edges":[{"id":"e-fd8a72d8822c546e","operation":{"operation":"obreschkoff_combination"},"operation_detail":null,"conclusion_id":"n-b2994d0cc7b611de","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":"e322fede305d6c3d2a9fcc2c3c3fe07aa666c9055debe5bf9679428096070ae4","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-50226352e9cd0ce4","role":"sequence","name":"Triangle of increasing mobiles (circular rooted trees) with n nodes and k leaves.","uses":{"claim":"real_rooted","from_index":0,"region":{}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)"},{"position":1,"node_id":"n-50226352e9cd0ce4","role":"sequence","name":"Triangle of increasing mobiles (circular rooted trees) with n nodes and k leaves.","uses":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"uses_text":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0: 2;\n  1: 2,    1;\n  2: 2,    5,     2;\n  3: 2,   15,    20,       6;\n  4: 2,   37,   116,     102,      24;\n  5: 2,   83,   522,     970,     624,      120;\n  6: 2,  177,  2038,    6914,    8844,     4440,      720;\n  7: 2,  367,  7296,   41498,   91744,    87948,    36000,     5040;\n  8: 2,  749, 24704,  222746,  784664,  1253396,   951264,   327600,   40320;\n  9: 2, 1515, 80606, 1107638, 5892576, 14498852, 17855784, 11146896, 3306240, 362880;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+(n-2)*x)*P(n-2,x) + x*d/dx P(n-2,x) + n*x*P(n-1,x) + x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 2, P(1,x) = 2 + x.","mathematica":"P[0, x_] := 2;\nP[1, x_] := 2 + x;\nP[n_, x_] := P[n, x] = Expand[(1 + (n - 2)*x)*P[n - 2, x] + x*D[P[n - 2, x], x] + n*x*P[n - 1, x] + x*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 2; v[2] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = (1 + (n - 2)*x)*v[i - 2] + x*deriv(v[i - 2], x) + n*x*v[i - 1] + x*deriv(v[i - 1], x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-50226352e9cd0ce4","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-fd8a72d8822c546e"],"text":"A055356 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A055356 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}