{"id":"n-68623ae8738f73d7","revision":0,"first_index":0,"fingerprint":"f6-e4990967de006696","name":"mobius_x_over_x_plus_one of basis_map_A007318 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"],["2","3"],["13","39","27"],["44","198","288","135"],["205","1230","2700","2565","891"],["806","6045","17820","25785","18306","5103"],["3457","31113","114975","223155","239841","135351","31347"],["14168","148764","661392","1613250","2330856","1994544","936036","185895"],["59449","713388","3706128","10883430","19754766","22692312","16109442","6462585","1121931"],["246410","3326535","19776600","67939830","148600332","214571430","204528240","124101315","43499430","6711903"],["1027861","15417915","103219515","406066770","1039375962","1808455086","2166033960","1763367165","933859935","290540763","40330467"],["4273412","70511298","524934000","2327096385","6824709432","13901306832","20066421924","20526296265","14581560060","6851376738","1916494344","241805655"],["17797573","320356314","2625162732","12948045075","42806845737","99925190568","168868124622","208157271255","185746273065","117017127738","49404172536","12551750685","1451365371"],["74055854","1444089153","12916526868","70157867175","258160625982","679537569429","1316289672504","1899672497685","2042603994222","1616110919703","914563430388","350562398445","81583633674","8706597903"],["308289865","6474087165","62761762575","372243528045","1508906611323","4421733378165","9659508543495","15979722249195","20115814506165","19175168227059","13624992536805","6998001801975","2456089125165","527282474175","52244370387"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","79","665","7591","73865","779239","7884905","81503431","833302025","8567695399","87836786345","901839483271","9252383878985","94961251491559"],"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":[["201","1329","3630","5250","4245","1821","324"],["-289","-1920","-5277","-7687","-6264","-2709","-486"],["90","600","1662","2448","2022","888","162"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["260","2143","7463","14270","16210","10955","4083","648"],["-120","-1000","-3523","-6816","-7834","-5356","-2019","-324"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["40","400","1721","4167","6220","5870","3425","1131","162"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":179864,"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-1322f1e2ddff0985"]},{"id":179865,"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-1322f1e2ddff0985"]},{"id":179866,"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-1322f1e2ddff0985"]}],"region_exact":{},"incoming_edges":[{"id":"e-1322f1e2ddff0985","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-68623ae8738f73d7","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":"cd9d696225e3847d3d836a1b7691cdc4fcd9db272a49df30be9a6bef35123005","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-2ee89feef605b3fb","role":"sequence","name":"basis_map_A007318 of 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":[],"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:      2,       3;\n  2:     13,      39,       27;\n  3:     44,     198,      288,      135;\n  4:    205,    1230,     2700,     2565,       891;\n  5:    806,    6045,    17820,    25785,     18306,      5103;\n  6:   3457,   31113,   114975,   223155,    239841,    135351,     31347;\n  7:  14168,  148764,   661392,  1613250,   2330856,   1994544,    936036,    185895;\n  8:  59449,  713388,  3706128, 10883430,  19754766,  22692312,  16109442,   6462585,  1121931;\n  9: 246410, 3326535, 19776600, 67939830, 148600332, 214571430, 204528240, 124101315, 43499430, 6711903;\n","recurrence_sentence":"The row polynomials satisfy (n+3*n*x+3*n*x^2+n*x^3)*P(n,x) = (90*n^2-289*n+201+(600*n^2-1920*n+1329)*x+(1662*n^2-5277*n+3630)*x^2+(2448*n^2-7687*n+5250)*x^3+(2022*n^2-6264*n+4245)*x^4+(888*n^2-2709*n+1821)*x^5+(162*n^2-486*n+324)*x^6)*P(n-1,x) - (120*n-260+(1000*n-2143)*x+(3523*n-7463)*x^2+(6816*n-14270)*x^3+(7834*n-16210)*x^4+(5356*n-10955)*x^5+(2019*n-4083)*x^6+(324*n-648)*x^7)*d/dx P(n-1,x) + (40+400*x+1721*x^2+4167*x^3+6220*x^4+5870*x^5+3425*x^6+1131*x^7+162*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[((90*n^2 - 289*n + 201 + (600*n^2 - 1920*n + 1329)*x + (1662*n^2 - 5277*n + 3630)*x^2 + (2448*n^2 - 7687*n + 5250)*x^3 + (2022*n^2 - 6264*n + 4245)*x^4 + (888*n^2 - 2709*n + 1821)*x^5 + (162*n^2 - 486*n + 324)*x^6)*P[n - 1, x] - (120*n - 260 + (1000*n - 2143)*x + (3523*n - 7463)*x^2 + (6816*n - 14270)*x^3 + (7834*n - 16210)*x^4 + (5356*n - 10955)*x^5 + (2019*n - 4083)*x^6 + (324*n - 648)*x^7)*D[P[n - 1, x], x] + (40 + 400*x + 1721*x^2 + 4167*x^3 + 6220*x^4 + 5870*x^5 + 3425*x^6 + 1131*x^7 + 162*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] = ((90*n^2 - 289*n + 201 + (600*n^2 - 1920*n + 1329)*x + (1662*n^2 - 5277*n + 3630)*x^2 + (2448*n^2 - 7687*n + 5250)*x^3 + (2022*n^2 - 6264*n + 4245)*x^4 + (888*n^2 - 2709*n + 1821)*x^5 + (162*n^2 - 486*n + 324)*x^6)*v[i - 1] - (120*n - 260 + (1000*n - 2143)*x + (3523*n - 7463)*x^2 + (6816*n - 14270)*x^3 + (7834*n - 16210)*x^4 + (5356*n - 10955)*x^5 + (2019*n - 4083)*x^6 + (324*n - 648)*x^7)*deriv(v[i - 1], x) + (40 + 400*x + 1721*x^2 + 4167*x^3 + 6220*x^4 + 5870*x^5 + 3425*x^6 + 1131*x^7 + 162*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":["basis A007318","x \u21a6 x/(x+1)"],"edges":["e-a3fef8dca23d39bb","e-1322f1e2ddff0985"],"text":"A143685 \u2192 basis A007318 \u2192 x \u21a6 x/(x+1)"},"headline":"A143685 \u2192 basis A007318 \u2192 x \u21a6 x/(x+1)","chain":null}