{"id":"n-a793cd06bbb8aaa5","revision":1,"first_index":0,"fingerprint":"f6-ea9a5170d08e2562","name":"mobius_x_over_x_plus_one of Pascal-(1,9,1) array.","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","13","13"],["1","24","66","44"],["1","35","240","410","205"],["1","46","535","1700","2015","806"],["1","57","951","5245","11265","10371","3457"],["1","68","1488","12376","41870","66336","49588","14168"],["1","79","2146","24424","122386","298666","376984","237796","59449"],["1","90","2925","42720","296010","1031652","1948170","2035080","1108845","246410"],["1","101","3825","68595","620580","2944962","7800942","12012270","10712025","5139305","1027861"],["1","112","4846","103380","1168575","7234692","25639824","54432552","70585815","54858680","23503766","4273412"],["1","123","5988","148406","2027115","15774417","72199086","202425336","357769827","400548965","275883096","106785438","17797573"],["1","134","7251","205004","3297961","31276242","179377323","642694152","1483271883","2241253922","2205981481","1363813644","481363051","74055854"],["1","145","8635","274505","5097515","57451853","402143595","1793210325","5240279985","10249554155","13513358453","11863665155","6653007145","2158029055","308289865"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","27","135","891","5103","31347","185895","1121931","6711903","40330467","241805655","1451365371","8706597903","52244370387"],"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":[["0","0","0","0"],["1","3","3","1"]]},"terms":[{"coeff":{"coeffs":[["20","180","600","1000","900","420","80"],["-30","-259","-857","-1437","-1309","-620","-120"],["11","84","266","444","411","200","40"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["1","105","670","1810","2605","2101","900","160"],["-1","-46","-294","-816","-1209","-1002","-440","-80"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1","45","250","610","805","601","240","40"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":164942,"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-54eec0a056ff5a9d"]},{"id":164943,"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-54eec0a056ff5a9d"]},{"id":164944,"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-54eec0a056ff5a9d"]}],"region_exact":{},"incoming_edges":[{"id":"e-54eec0a056ff5a9d","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-a793cd06bbb8aaa5","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":"94c36c267c9ca578be1c1e290e672bfec45a0c6f98fcba72392257a4d56395a8","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-7ac21ea1b3b7e282","role":"sequence","name":"Pascal-(1,9,1) array.","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":[{"id":"l-c434a4d29001ca67","kind":"fingerprint_related","node_a_id":"n-2ee89feef605b3fb","node_b_id":"n-a793cd06bbb8aaa5","fingerprint":"f6-ea9a5170d08e2562","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-ea9a5170d08e2562"},"other_id":"n-2ee89feef605b3fb","other":{"id":"n-2ee89feef605b3fb","name":"basis_map_A007318 of Pascal-(1,9,1) array.","first_index":0,"fingerprint":"f6-ea9a5170d08e2562","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","1"],["13","13","1"]],"card_rows":[["1"],["2","1"],["13","13","1"],["44","66","24","1"]],"row_sums_preview":["1","3","27","135","891","5103","31347","185895"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7ac21ea1b3b7e282","steps":["basis A007318"],"edges":["e-a3fef8dca23d39bb"],"text":"A143685 \u2192 basis A007318"},"headline":"A143685 \u2192 basis A007318","chain":null}},{"id":"l-450ae00fef97128f","kind":"oeis_match","node_a_id":"n-a793cd06bbb8aaa5","node_b_id":null,"fingerprint":null,"oeis_id":"A080424","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A080424","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A080424","relation":"match_row_sums"}],"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, 13,   13;\n  3: 1, 24,   66,    44;\n  4: 1, 35,  240,   410,    205;\n  5: 1, 46,  535,  1700,   2015,     806;\n  6: 1, 57,  951,  5245,  11265,   10371,    3457;\n  7: 1, 68, 1488, 12376,  41870,   66336,   49588,   14168;\n  8: 1, 79, 2146, 24424, 122386,  298666,  376984,  237796,   59449;\n  9: 1, 90, 2925, 42720, 296010, 1031652, 1948170, 2035080, 1108845, 246410;\n","recurrence_sentence":"The row polynomials satisfy (n+3*n*x+3*n*x^2+n*x^3)*P(n,x) = (11*n^2-30*n+20+(84*n^2-259*n+180)*x+(266*n^2-857*n+600)*x^2+(444*n^2-1437*n+1000)*x^3+(411*n^2-1309*n+900)*x^4+(200*n^2-620*n+420)*x^5+(40*n^2-120*n+80)*x^6)*P(n-1,x) - (n-1+(46*n-105)*x+(294*n-670)*x^2+(816*n-1810)*x^3+(1209*n-2605)*x^4+(1002*n-2101)*x^5+(440*n-900)*x^6+(80*n-160)*x^7)*d/dx P(n-1,x) + (x+45*x^2+250*x^3+610*x^4+805*x^5+601*x^6+240*x^7+40*x^8)*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[((11*n^2 - 30*n + 20 + (84*n^2 - 259*n + 180)*x + (266*n^2 - 857*n + 600)*x^2 + (444*n^2 - 1437*n + 1000)*x^3 + (411*n^2 - 1309*n + 900)*x^4 + (200*n^2 - 620*n + 420)*x^5 + (40*n^2 - 120*n + 80)*x^6)*P[n - 1, x] - (n - 1 + (46*n - 105)*x + (294*n - 670)*x^2 + (816*n - 1810)*x^3 + (1209*n - 2605)*x^4 + (1002*n - 2101)*x^5 + (440*n - 900)*x^6 + (80*n - 160)*x^7)*D[P[n - 1, x], x] + (x + 45*x^2 + 250*x^3 + 610*x^4 + 805*x^5 + 601*x^6 + 240*x^7 + 40*x^8)*D[P[n - 1, x], {x, 2}])/(n + 3*n*x + 3*n*x^2 + n*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] = ((11*n^2 - 30*n + 20 + (84*n^2 - 259*n + 180)*x + (266*n^2 - 857*n + 600)*x^2 + (444*n^2 - 1437*n + 1000)*x^3 + (411*n^2 - 1309*n + 900)*x^4 + (200*n^2 - 620*n + 420)*x^5 + (40*n^2 - 120*n + 80)*x^6)*v[i - 1] - (n - 1 + (46*n - 105)*x + (294*n - 670)*x^2 + (816*n - 1810)*x^3 + (1209*n - 2605)*x^4 + (1002*n - 2101)*x^5 + (440*n - 900)*x^6 + (80*n - 160)*x^7)*deriv(v[i - 1], x) + (x + 45*x^2 + 250*x^3 + 610*x^4 + 805*x^5 + 601*x^6 + 240*x^7 + 40*x^8)*deriv(deriv(v[i - 1], x), x))/(n + 3*n*x + 3*n*x^2 + n*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-7ac21ea1b3b7e282","steps":["x \u21a6 x/(x+1)"],"edges":["e-54eec0a056ff5a9d"],"text":"A143685 \u2192 x \u21a6 x/(x+1)"},"headline":"A143685 \u2192 x \u21a6 x/(x+1)","chain":null}
