{"id":"n-944e86db7073f824","revision":0,"first_index":0,"fingerprint":"f6-60dee4d68f98f7ab","name":"mobius_x_over_x_plus_one of basis_map_A013609 of Riordan array (1, 3+3x).","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"],["3","9"],["12","66","90"],["45","369","999","891"],["171","1854","7488","13338","8829"],["648","8748","46980","125388","166212","87480"],["2457","39690","265923","945540","1881063","1984338","866781"],["9315","175203","1406727","6248583","16579161","26268057","23005053","8588349"],["35316","757998","7093170","37791198","125357706","265052250","348769638","261070938","85096170"],["133893","3229227","34507944","214415424","853574922","2257367454","3965223456","4460377752","2915032617","843160671"],["507627","13590504","163280205","1159131384","5383869606","17093846664","37567132722","56422063080","55415564127","32135568048","8354311569"],["1924560","56634552","755634744","6033320640","32028072624","118681106688","313212272256","588650505552","772001902080","672800054472","350636268744","82777250160"],["7296561","234087732","3434110506","30459745764","181914810903","770616131400","2374049161548","5358778368840","8795275900047","10235609323524","8016539124906","3793530020436","820184055561"],["27663363","960930621","15372961002","149963521662","995166922833","4743761769639","16709755417884","44035406304372","86813555242869","126444588609339","132243627865050","94040609651118","40751375771223","8126651751489"],["104879772","3921584058","67944311946","722942566956","5277340839888","27956751451326","110829896539974","333994487901768","768816933048108","1345134907104534","1760766593436630","1672014023210700","1088751378709800","435126316046130","80521522263450"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","12","168","2304","31680","435456","5985792","82280448","1131024384","15547023360","213708865536","2937634947072","40380632137728","555070824382464","7629985063895040"],"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"],["2","4","2"]]},"terms":[{"coeff":{"coeffs":[["17","90","168","134","39"],["-11","-60","-126","-116","-39"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["7","55","162","226","151","39"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":185742,"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-35d09cef4999594c"]},{"id":185743,"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-35d09cef4999594c"]},{"id":185744,"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-35d09cef4999594c"]}],"region_exact":{},"incoming_edges":[{"id":"e-35d09cef4999594c","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-944e86db7073f824","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":"f15d30c132289fb9e0df1ab1386a590f42dc7395f871aae808d8a19898cba365","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-f8c44f01f6cd4a34","role":"sequence","name":"basis_map_A013609 of Riordan array (1, 3+3x).","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:      3,       9;\n  2:     12,      66,       90;\n  3:     45,     369,      999,       891;\n  4:    171,    1854,     7488,     13338,      8829;\n  5:    648,    8748,    46980,    125388,    166212,      87480;\n  6:   2457,   39690,   265923,    945540,   1881063,    1984338,     866781;\n  7:   9315,  175203,  1406727,   6248583,  16579161,   26268057,   23005053,    8588349;\n  8:  35316,  757998,  7093170,  37791198, 125357706,  265052250,  348769638,  261070938,   85096170;\n  9: 133893, 3229227, 34507944, 214415424, 853574922, 2257367454, 3965223456, 4460377752, 2915032617, 843160671;\n","recurrence_sentence":"The row polynomials satisfy (2*n+4*n*x+2*n*x^2)*P(n,x) = -(11*n-17+(60*n-90)*x+(126*n-168)*x^2+(116*n-134)*x^3+(39*n-39)*x^4)*P(n-1,x) + (7+55*x+162*x^2+226*x^3+151*x^4+39*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[(-(11*n - 17 + (60*n - 90)*x + (126*n - 168)*x^2 + (116*n - 134)*x^3 + (39*n - 39)*x^4)*P[n - 1, x] + (7 + 55*x + 162*x^2 + 226*x^3 + 151*x^4 + 39*x^5)*D[P[n - 1, x], x])/(2*n + 4*n*x + 2*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] = (-(11*n - 17 + (60*n - 90)*x + (126*n - 168)*x^2 + (116*n - 134)*x^3 + (39*n - 39)*x^4)*v[i - 1] + (7 + 55*x + 162*x^2 + 226*x^3 + 151*x^4 + 39*x^5)*deriv(v[i - 1], x))/(2*n + 4*n*x + 2*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-016d7724d51f6416","steps":["basis A013609","x \u21a6 x/(x+1)"],"edges":["e-2339e629d7f9e0b9","e-35d09cef4999594c"],"text":"A099093 \u2192 basis A013609 \u2192 x \u21a6 x/(x+1)"},"headline":"A099093 \u2192 basis A013609 \u2192 x \u21a6 x/(x+1)","chain":null}