{"id":"n-7d3534761470650a","revision":0,"first_index":0,"fingerprint":"f6-efea276bec11ceea","name":"shift_x_plus_one of basis_map_A007318 of Triangular array read by rows. T(n, k) is the coefficient of x^k in a(n+3) where a(1) = a(2) = a(3) = 1 and a(m+2) = (m*x + 2)*a(m+1) - a(m) for all m in Z.","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":[["1"],["3","1"],["17","12","2"],["133","146","52","6"],["1313","1980","1102","268","24"],["15623","30179","23072","8720","1628","120"],["217409","514264","502980","260244","75088","11448","720"],["3462921","9719908","11624456","7676044","3021488","708664","91656","5040"],["62115169","202147448","286496492","230904196","115720048","36916408","7318400","823968","40320"],["1238840459","4592265573","7537632416","7188876304","4389517236","1779099928","478524016","82339920","8222112","362880"],["27192374929","113216099748","211464072390","233300698652","168342422184","82998454368","28311209232","6595894432","1004245344","90204480","3628800"],["651378157837","3011710252598","6312977202172","7918132687634","6602136300352","3841950448928","1591973495600","469642428000","96648504896","13211243424","1079340480","39916800"],["16908639728833","86007788361844","200066466215258","281393875605896","266504793638576","179033348821984","87466405063504","31307789180768","8147566017952","1503184183296","186594144768","13989922560","479001600"],["472790534249487","2625018680353863","6713649325529024","10471974445075808","11113652468456424","8473654133868112","4774900902968304","2013211720459056","635036459347744","147994304122240","24764951095872","2817401796864","195281038080","6227020800"],["14166807387755777","85283620101746864","237959674824609544","407868930034074680","479750711491115456","409621725225611312","261790718558139184","127213656466147168","47227910300841152","13328836370352320","2814682196442752","431217379325568","45301577296896","2920745157120","87178291200"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","31","337","4687","79342","1582153","36310177","942482449","27295680844","872519304559","30510879978721","1158540919886839","47469666835381678","2087506799836906993"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["3","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["2","0"],["2","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":185586,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-4, -2)","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"-4"},"upper":{"closed":false,"value":"-2"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4, -2)","region_exact":null,"edges":["e-2ab62ca339238a64"]},{"id":185587,"position":1,"claim":"one_signed_coefficients","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":0},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 0: the coefficients of P_n have one sign","region_exact":null,"edges":["e-2ab62ca339238a64","e-65e96d3dc6394d47"]},{"id":185588,"position":2,"claim":"hadamard_kernel","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":0},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-2ab62ca339238a64","e-65e96d3dc6394d47"]},{"id":185589,"position":3,"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-2ab62ca339238a64","e-65e96d3dc6394d47"]},{"id":185590,"position":4,"claim":"real_rooted","from_index":0,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-65e96d3dc6394d47"]}],"region_exact":{},"incoming_edges":[{"id":"e-65e96d3dc6394d47","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-7d3534761470650a","premise_count":1,"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"0d44b292a613bc27e15ccb3b8b26088451f6798e6312f0676a4073d74dd03b95","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-4a57e8264fe984f4","role":"sequence","name":"basis_map_A007318 of Triangular array read by rows. T(n, k) is the coefficient of x^k in a(n+3) where a(1) = a(2) = a(3) = 1 and a(m+2) = (m*x + 2)*a(m+1) - a(m) for all m in Z.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]},{"id":"e-2ab62ca339238a64","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-7d3534761470650a","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":"b4a14af7bca4afa7fced0eeb6c6e7617ab75df8a1816109cdf65d3b5bcd1c476","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"-4"},"upper":{"closed":false,"value":"-2"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-4a57e8264fe984f4","role":"sequence","name":"basis_map_A007318 of Triangular array read by rows. T(n, k) is the coefficient of x^k in a(n+3) where a(1) = a(2) = a(3) = 1 and a(m+2) = (m*x + 2)*a(m+1) - a(m) for all m in Z.","uses":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"-3"},"upper":{"closed":false,"value":"-1"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-3, -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:          1;\n  1:          3,          1;\n  2:         17,         12,          2;\n  3:        133,        146,         52,          6;\n  4:       1313,       1980,       1102,        268,         24;\n  5:      15623,      30179,      23072,       8720,       1628,        120;\n  6:     217409,     514264,     502980,     260244,      75088,      11448,       720;\n  7:    3462921,    9719908,   11624456,    7676044,    3021488,     708664,     91656,     5040;\n  8:   62115169,  202147448,  286496492,  230904196,  115720048,   36916408,   7318400,   823968,   40320;\n  9: 1238840459, 4592265573, 7537632416, 7188876304, 4389517236, 1779099928, 478524016, 82339920, 8222112, 362880;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -P(n-2,x) + (2*n+2+n*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + x;\nP[n_, x_] := P[n, x] = Expand[-P[n - 2, x] + (2*n + 2 + n*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + x; for(n = 2, N, my(i = n + 1); v[i] = -v[i - 2] + (2*n + 2 + n*x)*v[i - 1]); 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-f3e0631eb8740f76","steps":["basis A007318","x \u21a6 x+1"],"edges":["e-3f96abc359582f71","e-2ab62ca339238a64"],"text":"A358735 \u2192 basis A007318 \u2192 x \u21a6 x+1"},"headline":"A358735 \u2192 basis A007318 \u2192 x \u21a6 x+1","chain":null}
