{"id":"n-41688b0638635402","revision":0,"first_index":0,"fingerprint":"f6-23b2bc8195b15215","name":"mobius_x_over_x_plus_one of Triangle of coefficients of polynomials u(n,x) jointly generated with A112351; see the Formula section.","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"],["1","2"],["1","6","6"],["1","10","26","18"],["1","14","62","102","54"],["1","18","114","314","378","162"],["1","22","182","718","1422","1350","486"],["1","26","266","1378","3890","5994","4698","1458"],["1","30","366","2358","8742","19086","24030","16038","4374"],["1","34","482","3722","17194","49202","87282","92826","53946","13122"],["1","38","614","5534","30718","109526","253974","378702","348462","179334","39366"],["1","42","762","7858","51042","218874","631754","1227906","1577394","1278666","590490","118098"],["1","46","926","10758","80150","402718","1397902","3377142","5639814","6359310","4605822","1928934","354294"],["1","50","1106","14298","120282","694210","2824930","8202602","16999146","24856146","24962418","16336890","6259194","1062882"],["1","54","1302","18542","173934","1135206","5310278","18075294","44984094","81493398","105890166","95829966","57198798","20194758","3188646"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","13","55","233","987","4181","17711","75025","317811","1346269","5702887","24157817","102334155","433494437"],"nonn":true,"definitions":[{"position":0,"source":"derived:mobius_x_over_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["-3","-9","-9","-3"],["4","12","12","4"]]},"terms":[{"coeff":{"coeffs":[["3","9","9","3","0"],["-6","-20","-24","-12","-2"],["4","16","24","16","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["1","27","116","214","201","95","18"],["-1","-14","-55","-100","-95","-46","-9"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1","14","55","100","95","46","9"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":173727,"position":0,"claim":"real_rooted","from_index":0,"region_key":"[-1, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in [-1, 0]","region_exact":null,"edges":["e-f78e6d71c7943272"]},{"id":173728,"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-f78e6d71c7943272"]},{"id":173729,"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-f78e6d71c7943272"]}],"region_exact":{},"incoming_edges":[{"id":"e-f78e6d71c7943272","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"mobius\",\"a\":\"1\",\"b\":\"0\",\"c\":\"1\",\"e\":\"1\",\"schedule\":{\"first_index\":0,\"first_degree\":0,\"step\":1}}"},"operation_detail":"mobius, shift 0","conclusion_id":"n-41688b0638635402","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":"fTransform","lean_declarations":["RealRooted.splits_fPolynomial"],"certificate_hash":"11a380b96b69ab621fa63749ed7d93f276cc2e60dd43a388e09d12fe76dd46df","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-522def1e8ebf49cd","role":"sequence","name":"Triangle of coefficients of polynomials u(n,x) jointly generated with A112351; see the Formula section.","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: 1,  2;\n  2: 1,  6,   6;\n  3: 1, 10,  26,   18;\n  4: 1, 14,  62,  102,    54;\n  5: 1, 18, 114,  314,   378,   162;\n  6: 1, 22, 182,  718,  1422,  1350,   486;\n  7: 1, 26, 266, 1378,  3890,  5994,  4698,  1458;\n  8: 1, 30, 366, 2358,  8742, 19086, 24030, 16038,  4374;\n  9: 1, 34, 482, 3722, 17194, 49202, 87282, 92826, 53946, 13122;\n","recurrence_sentence":"The row polynomials satisfy (4*n-3+(12*n-9)*x+(12*n-9)*x^2+(4*n-3)*x^3)*P(n,x) = (4*n^2-6*n+3+(16*n^2-20*n+9)*x+(24*n^2-24*n+9)*x^2+(16*n^2-12*n+3)*x^3+(4*n^2-2*n)*x^4)*P(n-1,x) - (n-1+(14*n-27)*x+(55*n-116)*x^2+(100*n-214)*x^3+(95*n-201)*x^4+(46*n-95)*x^5+(9*n-18)*x^6)*d/dx P(n-1,x) + (x+14*x^2+55*x^3+100*x^4+95*x^5+46*x^6+9*x^7)*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[((4*n^2 - 6*n + 3 + (16*n^2 - 20*n + 9)*x + (24*n^2 - 24*n + 9)*x^2 + (16*n^2 - 12*n + 3)*x^3 + (4*n^2 - 2*n)*x^4)*P[n - 1, x] - (n - 1 + (14*n - 27)*x + (55*n - 116)*x^2 + (100*n - 214)*x^3 + (95*n - 201)*x^4 + (46*n - 95)*x^5 + (9*n - 18)*x^6)*D[P[n - 1, x], x] + (x + 14*x^2 + 55*x^3 + 100*x^4 + 95*x^5 + 46*x^6 + 9*x^7)*D[P[n - 1, x], {x, 2}])/(4*n - 3 + (12*n - 9)*x + (12*n - 9)*x^2 + (4*n - 3)*x^3)]];\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] = ((4*n^2 - 6*n + 3 + (16*n^2 - 20*n + 9)*x + (24*n^2 - 24*n + 9)*x^2 + (16*n^2 - 12*n + 3)*x^3 + (4*n^2 - 2*n)*x^4)*v[i - 1] - (n - 1 + (14*n - 27)*x + (55*n - 116)*x^2 + (100*n - 214)*x^3 + (95*n - 201)*x^4 + (46*n - 95)*x^5 + (9*n - 18)*x^6)*deriv(v[i - 1], x) + (x + 14*x^2 + 55*x^3 + 100*x^4 + 95*x^5 + 46*x^6 + 9*x^7)*deriv(deriv(v[i - 1], x), x))/(4*n - 3 + (12*n - 9)*x + (12*n - 9)*x^2 + (4*n - 3)*x^3)); 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-522def1e8ebf49cd","steps":["x \u21a6 x/(x+1)"],"edges":["e-f78e6d71c7943272"],"text":"A209414 \u2192 x \u21a6 x/(x+1)"},"headline":"A209414 \u2192 x \u21a6 x/(x+1)","chain":null}
