{"id":"n-3bb53a141e5d1684","revision":0,"first_index":0,"fingerprint":"f6-010ab161ddea77e9","name":"reversal of basis_map_A007318 of Triangular array formed by the Mersenne numbers.","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","4"],["1","8","14"],["1","12","44","48"],["1","16","90","208","164"],["1","20","152","544","908","560"],["1","24","230","1120","2904","3776","1912"],["1","28","324","2000","7080","14304","15200","6528"],["1","32","434","3248","14620","40384","66608","59776","22288"],["1","36","560","4928","26964","94864","213984","297600","230992","76096"],["1","40","702","7104","45808","196224","564200","1072768","1288176","880512","259808"],["1","44","860","9840","73104","369600","1295168","3139840","5151280","5438016","3319872","887040"],["1","48","1034","13200","111060","647808","2681952","7928064","16582240","23897600","22495584","12405504","3028544"],["1","52","1224","17248","162140","1072368","5126976","17916672","45704160","83946880","107783424","91511808","46010816","10340096"],["1","56","1430","22048","229064","1694528","9194328","37128960","112006944","250907392","410406464","474850304","367066880","169572352","35303296"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","23","105","479","2185","9967","45465","207391","946025","4315343","19684665","89792639","409593865","1868384047"],"nonn":true,"definitions":[{"position":0,"source":"derived:reversal","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0"],["2"]]},"terms":[{"coeff":{"coeffs":[["0","-4","-8"],["2","12","8"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-1","-8","-8"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":178048,"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":false,"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-303645006a016b02"]},{"id":178049,"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-303645006a016b02"]},{"id":178050,"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-303645006a016b02"]}],"region_exact":{},"incoming_edges":[{"id":"e-303645006a016b02","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"reversal\",\"schedule\":{\"first_index\":0,\"first_degree\":0,\"step\":1}}"},"operation_detail":"reversal, shift 0","conclusion_id":"n-3bb53a141e5d1684","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":"reversed","lean_declarations":["RealRooted.X_pow_mul_reverse_eq_zero_or_splits","RealRooted.reflect_eq_zero_or_splits"],"certificate_hash":"331f14cdb654a2fdbdfda043d2a25eb6b9cd15c39ffb05285e4ec1ec9af5b008","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"-1"},"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-6bdb5d7e28f56300","role":"sequence","name":"basis_map_A007318 of Triangular array formed by the Mersenne numbers.","uses":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"-7"},"upper":{"closed":false,"value":"-1"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-7, -1)"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-2df0a7744c535de5","kind":"candidate_equal","node_a_id":"n-5e3fe73521ba8587","node_b_id":"n-3bb53a141e5d1684","fingerprint":null,"oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"reason":"no all-index equality certificate was found","rows_compared":15},"other_id":"n-5e3fe73521ba8587","other":{"id":"n-5e3fe73521ba8587","name":"mobius_x_over_x_plus_one of reversal of Triangular array formed by the Mersenne numbers.","first_index":0,"fingerprint":"f6-010ab161ddea77e9","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","4"],["1","8","14"]],"card_rows":[["1"],["1","4"],["1","8","14"],["1","12","44","48"]],"row_sums_preview":["1","5","23","105","479","2185","9967","45465"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-9b5c7e97e5191da6","steps":["reverse","x \u21a6 x/(x+1)"],"edges":["e-5faf1c18373ba45e","e-3b1d7eca2f6e8e60"],"text":"A110441 \u2192 reverse \u2192 x \u21a6 x/(x+1)"},"headline":"A110441 \u2192 reverse \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-7d9e7c3ea6fcbc5e","kind":"fingerprint_related","node_a_id":"n-6bdb5d7e28f56300","node_b_id":"n-3bb53a141e5d1684","fingerprint":"f6-010ab161ddea77e9","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-010ab161ddea77e9"},"other_id":"n-6bdb5d7e28f56300","other":{"id":"n-6bdb5d7e28f56300","name":"basis_map_A007318 of Triangular array formed by the Mersenne numbers.","first_index":0,"fingerprint":"f6-010ab161ddea77e9","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["4","1"],["14","8","1"]],"card_rows":[["1"],["4","1"],["14","8","1"],["48","44","12","1"]],"row_sums_preview":["1","5","23","105","479","2185","9967","45465"],"regions":["(-inf, 0]","(-7, -1)"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-9b5c7e97e5191da6","steps":["basis A007318"],"edges":["e-1d7f8f614531604f"],"text":"A110441 \u2192 basis A007318"},"headline":"A110441 \u2192 basis A007318","chain":null}}],"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,  4;\n  2: 1,  8,  14;\n  3: 1, 12,  44,   48;\n  4: 1, 16,  90,  208,   164;\n  5: 1, 20, 152,  544,   908,   560;\n  6: 1, 24, 230, 1120,  2904,  3776,   1912;\n  7: 1, 28, 324, 2000,  7080, 14304,  15200,   6528;\n  8: 1, 32, 434, 3248, 14620, 40384,  66608,  59776,  22288;\n  9: 1, 36, 560, 4928, 26964, 94864, 213984, 297600, 230992, 76096;\n","recurrence_sentence":"The row polynomials satisfy 2*n*P(n,x) = (2*n+(12*n-4)*x+(8*n-8)*x^2)*P(n-1,x) - (x+8*x^2+8*x^3)*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 + (12*n - 4)*x + (8*n - 8)*x^2)*P[n - 1, x] - (x + 8*x^2 + 8*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] = ((2*n + (12*n - 4)*x + (8*n - 8)*x^2)*v[i - 1] - (x + 8*x^2 + 8*x^3)*deriv(v[i - 1], x))/(2*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-9b5c7e97e5191da6","steps":["basis A007318","reverse"],"edges":["e-1d7f8f614531604f","e-303645006a016b02"],"text":"A110441 \u2192 basis A007318 \u2192 reverse"},"headline":"A110441 \u2192 basis A007318 \u2192 reverse","chain":null}