{"id":"n-b59b1769d0dc40fc","revision":1,"first_index":0,"fingerprint":"f6-d773d6147703cd7e","name":"basis-map coefficients of Riordan array (1, 2+2x).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":false,"rows":[["1"],["-6","2"],["30","-22","4"],["-144","168","-64","8"],["684","-1104","652","-168","16"],["-3240","6696","-5400","2136","-416","32"],["15336","-38664","39672","-21304","6336","-992","64"],["-72576","216000","-269568","183552","-73856","17600","-2304","128"],["343440","-1178496","1734048","-1433280","729616","-234688","46656","-5248","256"],["-1625184","6316704","-10711872","10427328","-6434016","2614048","-700288","119424","-11776","512"],["7690464","-33392736","64143360","-71919936","52214496","-25684960","8680512","-1992320","297472","-26112","1024"],["-36391680","174586752","-374740992","475818624","-397668096","229986432","-94023168","27197824","-5459968","724992","-57344","2048"],["172207296","-904566528","2145973824","-3044587392","2880518976","-1916716032","920658880","-321918336","81385984","-14518272","1735680","-124928","4096"],["-814893696","4651509888","-12087356544","18955078272","-20034642816","15086079360","-8333273472","3421594496","-1044997120","234611712","-37656576","4093952","-270336","8192"],["3856118400","-23767032960","67142183040","-115345710720","134745724800","-113324427648","70814414208","-33423286144","11981019136","-3247782912","655712256","-95655936","9535488","-581632","16384"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","-4","12","-32","80","-192","448","-1024","2304","-5120","11264","-24576","53248","-114688","245760"],"nonn":false,"definitions":[{"position":0,"source":"derived:basis_map_coefficients_of:oeis_catalog:A099040","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["-6","2"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-6","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-6","2"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":160113,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(1, 3]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"1"},"upper":{"closed":true,"value":"3"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (1, 3]","region_exact":null,"edges":["e-319b2a76ea6c1ab0"]},{"id":160114,"position":1,"claim":"one_signed_coefficients","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":true,"claim":"one_signed_coefficients","from_index":0},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 0: the coefficients of P_n(-x) have one sign","region_exact":null,"edges":["e-319b2a76ea6c1ab0"]},{"id":160115,"position":2,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-319b2a76ea6c1ab0"]}],"region_exact":{},"incoming_edges":[{"id":"e-319b2a76ea6c1ab0","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"-3\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-b59b1769d0dc40fc","premise_count":1,"theorem":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":null,"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"certificate_hash":"645609b74b314a982fdd6e74564ef145804c84f619b0afd737007dfe914e36ab","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"1"},"upper":{"closed":true,"value":"3"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-f5a001953ee28e8f","role":"sequence","name":"Riordan array (1, 2+2x).","uses":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"-2"},"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-2, 0]"}]}],"used_by":[{"id":"e-af79ef336fd9f8da","operation":"basis_map","operation_detail":"A027465","conclusion_id":"n-f5a001953ee28e8f","conclusion_name":"Riordan array (1, 2+2x)."}],"used_by_total":1,"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:        1;\n  1:       -6,        2;\n  2:       30,      -22,         4;\n  3:     -144,      168,       -64,        8;\n  4:      684,    -1104,       652,     -168,       16;\n  5:    -3240,     6696,     -5400,     2136,     -416,      32;\n  6:    15336,   -38664,     39672,   -21304,     6336,    -992,      64;\n  7:   -72576,   216000,   -269568,   183552,   -73856,   17600,   -2304,    128;\n  8:   343440, -1178496,   1734048, -1433280,   729616, -234688,   46656,  -5248,    256;\n  9: -1625184,  6316704, -10711872, 10427328, -6434016, 2614048, -700288, 119424, -11776, 512;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(6-2*x)*P(n-1,x) - (6-2*x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = -6 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := -6 + 2*x;\nP[n_, x_] := P[n, x] = Expand[-(6 - 2*x)*P[n - 1, x] - (6 - 2*x)*P[n - 2, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = -6 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(6 - 2*x)*v[i - 1] - (6 - 2*x)*v[i - 2]); 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-f5a001953ee28e8f","steps":["x \u21a6 x\u22123"],"edges":["e-319b2a76ea6c1ab0"],"text":"A099040 \u2192 x \u21a6 x\u22123"},"headline":"A099040 \u2192 x \u21a6 x\u22123","chain":null}