{"id":"n-cc04537b3d998791","revision":1,"first_index":1,"fingerprint":"f6-93cd8b7280a9a4ce","name":"basis-map coefficients of Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":false,"rows":[["1"],["-5","1"],["20","-9","1"],["-75","55","-13","1"],["275","-285","106","-17","1"],["-1000","1350","-695","173","-21","1"],["3625","-6050","4010","-1369","256","-25","1"],["-13125","26125","-21275","9295","-2371","355","-29","1"],["47500","-109875","106400","-56895","18501","-3765","470","-33","1"],["-171875","453125","-509375","323125","-128250","33180","-5615","601","-37","1"],["621875","-1840625","2358125","-1734125","814975","-256824","55140","-7985","748","-41","1"],["-2250000","7387500","-10631250","8903750","-4844625","1804945","-471269","86445","-10939","911","-45","1"],["8140625","-29362500","46912500","-44121250","27317875","-11770255","3628766","-808429","129415","-14541","1090","-49","1"],["-29453125","115765625","-203381250","212368750","-147583750","72299800","-25752795","6767417","-1314364","186626","-18855","1285","-53","1"],["106562500","-453328125","868746875","-997706250","769576875","-422913600","171149690","-51918969","11883733","-2045374","260910","-23945","1496","-57","1"]],"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:A063967","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["-5","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-5","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-5","1"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":160516,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(1, 5]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":false,"value":"1"},"upper":{"closed":true,"value":"5"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (1, 5]","region_exact":null,"edges":["e-cc7a67409dceb768"]},{"id":160517,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":true,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n(-x) have one sign","region_exact":null,"edges":["e-cc7a67409dceb768"]},{"id":160518,"position":2,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2","region_exact":null,"edges":["e-cc7a67409dceb768"]}],"region_exact":{},"incoming_edges":[{"id":"e-cc7a67409dceb768","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"-6\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-cc04537b3d998791","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":"1188dfe55a5f9f98c86a8ef2ca3ac145e52c7a5ca4fe623580b995b8afadffe2","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":false,"value":"1"},"upper":{"closed":true,"value":"5"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-c72653d9ab0ee8be","role":"sequence","name":"Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.","uses":{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":false,"value":"-5"},"upper":{"closed":true,"value":"-1"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-5, -1]"}]}],"used_by":[{"id":"e-20a66a93533ac8e0","operation":"basis_map","operation_detail":"A038255","conclusion_id":"n-c72653d9ab0ee8be","conclusion_name":"Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1."}],"used_by_total":1,"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:       1;\n   2:      -5,       1;\n   3:      20,      -9,       1;\n   4:     -75,      55,     -13,      1;\n   5:     275,    -285,     106,    -17,       1;\n   6:   -1000,    1350,    -695,    173,     -21,     1;\n   7:    3625,   -6050,    4010,  -1369,     256,   -25,     1;\n   8:  -13125,   26125,  -21275,   9295,   -2371,   355,   -29,   1;\n   9:   47500, -109875,  106400, -56895,   18501, -3765,   470, -33,   1;\n  10: -171875,  453125, -509375, 323125, -128250, 33180, -5615, 601, -37, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(5-x)*P(n-1,x) - (5-x)*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = -5 + x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := -5 + x;\nP[n_, x_] := P[n, x] = Expand[-(5 - x)*P[n - 1, x] - (5 - x)*P[n - 2, x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = -5 + x; for(n = 3, N, my(i = n); v[i] = -(5 - x)*v[i - 1] - (5 - 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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-c72653d9ab0ee8be","steps":["x \u21a6 x\u22126"],"edges":["e-cc7a67409dceb768"],"text":"A063967 \u2192 x \u21a6 x\u22126"},"headline":"A063967 \u2192 x \u21a6 x\u22126","chain":null}