{"id":"n-06589d5336ab7523","revision":1,"first_index":0,"fingerprint":"f6-0d853bc775b295f0","name":"mobius_x_over_x_plus_one of Riordan array (1, 2+2x).","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"],["0","2"],["0","2","6"],["0","0","8","16"],["0","0","4","32","44"],["0","0","0","24","112","120"],["0","0","0","8","120","376","328"],["0","0","0","0","64","512","1216","896"],["0","0","0","0","16","384","2016","3840","2448"],["0","0","0","0","0","160","1920","7488","11904","6688"],["0","0","0","0","0","32","1120","8640","26688","36384","18272"],["0","0","0","0","0","0","384","6400","36096","92160","109952","49920"],["0","0","0","0","0","0","64","3072","32320","142848","310464","329216","136384"],["0","0","0","0","0","0","0","896","19712","149632","542208","1025152","978176","372608"],["0","0","0","0","0","0","0","128","8064","110208","649600","1991040","3329664","2887552","1017984"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","2","8","24","80","256","832","2688","8704","28160","91136","294912","954368","3088384","9994240"],"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"],["1","2","1"]]},"terms":[{"coeff":{"coeffs":[["2","12","24","20","6"],["-2","-10","-20","-18","-6"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","4","18","30","22","6"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":163652,"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-a0f85d448207b69e"]},{"id":163653,"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-a0f85d448207b69e"]},{"id":163654,"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-a0f85d448207b69e"]}],"region_exact":{},"incoming_edges":[{"id":"e-a0f85d448207b69e","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-06589d5336ab7523","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":"b1ad326866b0cf0a822f439eb67967a5d6ef3a47dcfef8533e4cdd8d19e686a7","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-f5a001953ee28e8f","role":"sequence","name":"Riordan array (1, 2+2x).","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-66c7a4e35a8fb88b","kind":"oeis_match","node_a_id":"n-06589d5336ab7523","node_b_id":null,"fingerprint":null,"oeis_id":"A085449","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":"A085449","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-c45eb6b2b2b158cf","kind":"oeis_match","node_a_id":"n-06589d5336ab7523","node_b_id":null,"fingerprint":null,"oeis_id":"A063727","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":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A063727","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A063727","relation":"match_row_sums"},{"oeis_id":"A085449","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-06589d5336ab7523","oeis_id":"A063727","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":25,"oeis_terms_available":25,"term_count":200,"raw_size":11112}],"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: 0, 2;\n  2: 0, 2, 6;\n  3: 0, 0, 8, 16;\n  4: 0, 0, 4, 32,  44;\n  5: 0, 0, 0, 24, 112, 120;\n  6: 0, 0, 0,  8, 120, 376,  328;\n  7: 0, 0, 0,  0,  64, 512, 1216,  896;\n  8: 0, 0, 0,  0,  16, 384, 2016, 3840,  2448;\n  9: 0, 0, 0,  0,   0, 160, 1920, 7488, 11904, 6688;\n","recurrence_sentence":"The row polynomials satisfy (n+2*n*x+n*x^2)*P(n,x) = -(2*n-2+(10*n-12)*x+(20*n-24)*x^2+(18*n-20)*x^3+(6*n-6)*x^4)*P(n-1,x) + (4*x+18*x^2+30*x^3+22*x^4+6*x^5)*d/dx 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[(-(2*n - 2 + (10*n - 12)*x + (20*n - 24)*x^2 + (18*n - 20)*x^3 + (6*n - 6)*x^4)*P[n - 1, x] + (4*x + 18*x^2 + 30*x^3 + 22*x^4 + 6*x^5)*D[P[n - 1, x], x])/(n + 2*n*x + n*x^2)]];\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] = (-(2*n - 2 + (10*n - 12)*x + (20*n - 24)*x^2 + (18*n - 20)*x^3 + (6*n - 6)*x^4)*v[i - 1] + (4*x + 18*x^2 + 30*x^3 + 22*x^4 + 6*x^5)*deriv(v[i - 1], x))/(n + 2*n*x + n*x^2)); 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-f5a001953ee28e8f","steps":["x \u21a6 x/(x+1)"],"edges":["e-a0f85d448207b69e"],"text":"A099040 \u2192 x \u21a6 x/(x+1)"},"headline":"A099040 \u2192 x \u21a6 x/(x+1)","chain":null}