{"id":"n-3faf35b58dcbb6d6","revision":0,"first_index":0,"fingerprint":"f6-765e1f4dee9a440f","name":"mobius_x_over_x_plus_one of reversal of Triangle T(n,k), 0 <= k <= n, read by rows, given by [1,0,-1,0,0,0,0,0,0,...] DELTA [3,-2,-1,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.","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":true,"rows":[["1"],["3","4"],["3","10","8"],["9","42","64","32"],["9","60","148","160","64"],["27","216","684","1072","832","256"],["27","270","1116","2440","2976","1920","512"],["81","918","4428","11784","18688","17664","9216","2048"],["81","1080","6264","20640","42256","55040","44544","20480","4096"],["243","3564","23112","86976","209328","334144","353792","239616","94208","16384"],["243","4050","30240","133200","383280","752800","1022080","947200","573440","204800","32768"],["729","13122","106920","520560","1682640","3791520","6077440","6929920","5509120","2908160","917504","131072"],["729","14580","133164","734400","2723760","7156800","13660480","19084800","19368960","13926400","6733824","1966080","262144"],["2187","46656","457812","2735856","11108880","32365440","69608640","111896320","134446080","119255040","75907072","32833536","8650752","1048576"],["2187","51030","551124","3651480","16580592","54583200","134339520","251113600","358238720","388147200","314417152","184647680","74317824","18350080","2097152"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","7","21","147","441","3087","9261","64827","194481","1361367","4084101","28588707","85766121","600362847","1801088541"],"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":[["1","2","1"]]},"terms":[{"coeff":{"coeffs":[["13","56","89","62","16"],["-10","-46","-78","-58","-16"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["6","38","94","114","68","16"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":173417,"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-8e0b9e070770fa78"]},{"id":173418,"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-8e0b9e070770fa78"]},{"id":173419,"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-8e0b9e070770fa78"]}],"region_exact":{},"incoming_edges":[{"id":"e-8e0b9e070770fa78","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-3faf35b58dcbb6d6","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":"05731a13a277cb20a3b0bc81f5e317bca1cf6e3c3ab38901585714c65ebd8fa2","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-5426b03c17935143","role":"sequence","name":"reversal of Triangle T(n,k), 0 <= k <= n, read by rows, given by [1,0,-1,0,0,0,0,0,0,...] DELTA [3,-2,-1,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.","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-5a32587fa12a3b22","kind":"fingerprint_related","node_a_id":"n-67db2effba549c79","node_b_id":"n-3faf35b58dcbb6d6","fingerprint":"f6-765e1f4dee9a440f","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-765e1f4dee9a440f"},"other_id":"n-67db2effba549c79","other":{"id":"n-67db2effba549c79","name":"basis_map_A007318 of Triangle T(n,k), 0 <= k <= n, read by rows, given by [1,0,-1,0,0,0,0,0,0,...] DELTA [3,-2,-1,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.","first_index":0,"fingerprint":"f6-765e1f4dee9a440f","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","3"],["8","10","3"]],"card_rows":[["1"],["4","3"],["8","10","3"],["32","64","42","9"]],"row_sums_preview":["1","7","21","147","441","3087","9261","64827"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7196be931c2ed996","steps":["x \u21a6 x+1"],"edges":["e-1c523c2967bf00ac"],"text":"A152842 \u2192 x \u21a6 x+1"},"headline":"A152842 \u2192 x \u21a6 x+1","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:   3,    4;\n  2:   3,   10,     8;\n  3:   9,   42,    64,    32;\n  4:   9,   60,   148,   160,     64;\n  5:  27,  216,   684,  1072,    832,    256;\n  6:  27,  270,  1116,  2440,   2976,   1920,    512;\n  7:  81,  918,  4428, 11784,  18688,  17664,   9216,   2048;\n  8:  81, 1080,  6264, 20640,  42256,  55040,  44544,  20480,  4096;\n  9: 243, 3564, 23112, 86976, 209328, 334144, 353792, 239616, 94208, 16384;\n","recurrence_sentence":"The row polynomials satisfy (1+2*x+x^2)*P(n,x) = -(10*n-13+(46*n-56)*x+(78*n-89)*x^2+(58*n-62)*x^3+(16*n-16)*x^4)*P(n-1,x) + (6+38*x+94*x^2+114*x^3+68*x^4+16*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[(-(10*n - 13 + (46*n - 56)*x + (78*n - 89)*x^2 + (58*n - 62)*x^3 + (16*n - 16)*x^4)*P[n - 1, x] + (6 + 38*x + 94*x^2 + 114*x^3 + 68*x^4 + 16*x^5)*D[P[n - 1, x], x])/(1 + 2*x + 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] = (-(10*n - 13 + (46*n - 56)*x + (78*n - 89)*x^2 + (58*n - 62)*x^3 + (16*n - 16)*x^4)*v[i - 1] + (6 + 38*x + 94*x^2 + 114*x^3 + 68*x^4 + 16*x^5)*deriv(v[i - 1], x))/(1 + 2*x + 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-7196be931c2ed996","steps":["reverse","x \u21a6 x/(x+1)"],"edges":["e-67f427bf8180eb6f","e-8e0b9e070770fa78"],"text":"A152842 \u2192 reverse \u2192 x \u21a6 x/(x+1)"},"headline":"A152842 \u2192 reverse \u2192 x \u21a6 x/(x+1)","chain":null}