{"id":"n-fbe62a60256a6a00","revision":0,"first_index":0,"fingerprint":"f6-487bd17ac511de6a","name":"shift_x_plus_one of basis_map_A007318 of Pascal-(1,2,1) array read by antidiagonals.","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"],["13","8","1"],["51","47","13","1"],["205","250","106","18","1"],["819","1245","714","190","23","1"],["3277","5964","4311","1568","299","28","1"],["13107","27787","24243","11203","2937","433","33","1"],["52429","126878","129688","72746","24346","4946","592","38","1"],["209715","570425","668488","441224","179938","46790","7720","776","43","1"],["838861","2533360","3348397","2542520","1223914","388784","82210","11384","985","48","1"],["3355443","11140071","16393353","14077829","7816462","2937302","759874","134906","16063","1219","53","1"],["13421773","48586818","78780438","75493714","47507911","20631332","6323332","1374548","209803","21882","1478","58","1"],["53687091","210453397","372781686","394359602","277438953","136784039","48515428","12506348","2338021","312451","28966","1762","63","1"],["214748365","906238100","1741093843","2015396224","1567695533","865332220","348886315","104179328","23108719","3782508","449025","37440","2071","68","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","22","112","580","2992","15448","79744","411664","2125120","10970464","56632576","292353088","1509207808","7790949760"],"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"]],"recurrence":{"denominator":{"coeffs":[["0"],["1"]]},"terms":[{"coeff":{"coeffs":[["9","2"],["-9","-1"],["3","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["67","41","4"],["-25","-14","-1"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["50","53","16","1"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}},{"position":1,"source":"derived:basis_map_A007318","source_kind":"derived","equality_edge_id":"e-40666f6b4adccd85","recurrence":{"first_index":0,"initial_polynomials":[["1"],["3","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["4","2"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["3","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":175387,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-d097a651a53f0215"]},{"id":175388,"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-d097a651a53f0215","e-eab81ad3df331f44"]},{"id":175389,"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-d097a651a53f0215","e-eab81ad3df331f44"]},{"id":175390,"position":3,"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-eab81ad3df331f44"]},{"id":175391,"position":4,"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-eab81ad3df331f44"]}],"region_exact":{},"incoming_edges":[{"id":"e-eab81ad3df331f44","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-fbe62a60256a6a00","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":"06fe29f87286c3e37205d0e999e392a6e574af02d9916467726373896f6c5428","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-3bdf31527eaa2fb3","role":"sequence","name":"basis_map_A007318 of Pascal-(1,2,1) array read by antidiagonals.","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-40666f6b4adccd85","operation":{"operation":"definition_equality"},"operation_detail":null,"conclusion_id":"n-fbe62a60256a6a00","premise_count":0,"theorem":"all-index identity Q_n = P_n certified by exact Ore-algebra division and identifying rows (polytool preserver-image certificate of the identity map)","lean_route":null,"lean_declarations":[],"certificate_hash":"d48e6efd39116ad8f384a3c444da6b2c46c6bc6ff9fefaf8de16d5ae2781ff8a","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[],"premises":[]},{"id":"e-d097a651a53f0215","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-fbe62a60256a6a00","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":"16638fe5878193c4ebff3f1f6e925e66d0ae971a19d2eb25e3fcb6cda971b55a","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}}],"premises":[{"position":0,"node_id":"n-3bdf31527eaa2fb3","role":"sequence","name":"basis_map_A007318 of Pascal-(1,2,1) array read by antidiagonals.","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]"}]}],"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:     13,      8,      1;\n  3:     51,     47,     13,      1;\n  4:    205,    250,    106,     18,      1;\n  5:    819,   1245,    714,    190,     23,     1;\n  6:   3277,   5964,   4311,   1568,    299,    28,    1;\n  7:  13107,  27787,  24243,  11203,   2937,   433,   33,   1;\n  8:  52429, 126878, 129688,  72746,  24346,  4946,  592,  38,  1;\n  9: 209715, 570425, 668488, 441224, 179938, 46790, 7720, 776, 43, 1;\n","recurrence_sentence":"The row polynomials satisfy n*P(n,x) = (3*n^2-9*n+9-(n-2)*x)*P(n-1,x) - (25*n-67+(14*n-41)*x+(n-4)*x^2)*d/dx P(n-1,x) + (50+53*x+16*x^2+x^3)*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((3*n^2 - 9*n + 9 - (n - 2)*x)*P[n - 1, x] - (25*n - 67 + (14*n - 41)*x + (n - 4)*x^2)*D[P[n - 1, x], x] + (50 + 53*x + 16*x^2 + x^3)*D[P[n - 1, x], {x, 2}])/(n)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((3*n^2 - 9*n + 9 - (n - 2)*x)*v[i - 1] - (25*n - 67 + (14*n - 41)*x + (n - 4)*x^2)*deriv(v[i - 1], x) + (50 + 53*x + 16*x^2 + x^3)*deriv(deriv(v[i - 1], x), x))/(n)); 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-f25cd955a876cde3","steps":["x \u21a6 x+1","x \u21a6 x+1"],"edges":["e-6f8e74d475c3b6cc","e-d097a651a53f0215"],"text":"A081577 \u2192 x \u21a6 x+1 \u2192 x \u21a6 x+1"},"headline":"A081577 \u2192 x \u21a6 x+1 \u2192 x \u21a6 x+1","chain":null}