{"id":"n-ad2dfc31d01b8299","revision":0,"first_index":1,"fingerprint":"f6-b761c8caa7513a1c","name":"mobius_x_over_x_plus_one of basis_map_A013610 of Coefficients of a special case of Poisson-Charlier polynomials.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["2","5"],["5","25","29"],["16","123","297","217"],["65","692","2649","4238","2297"],["326","4513","24278","62813","76828","34349"],["1957","33681","236646","864131","1715196","1730223","674965"],["13700","283475","2476329","11800312","32974687","53658582","46551947","16276481"],["109601","2658016","27879794","164848300","599280419","1366146700","1895981534","1451119564","461527793"],["986410","27490797","337437900","2390863026","10756203714","31784753943","61479918480","74684677446","51306357798","14993138773"],["9864101","311028161","4380861735","36260713014","195066333213","711454979979","1777731238401","2995912238262","3244786011423","2026293977909","548258687501"],["108505112","3822401707","60835139705","576965144463","3619706940966","15754562833623","48469676865387","105198051703449","157428858128406","154126909846013","88353864906397","22272738733865"],["1302061345","50718622020","900858460779","9642344091394","69219778235310","350801229351456","1285584621591507","3427977378472326","6589066068086994","8882552919593728","7945391772608847","4214010479341626","994870668959209"],["16926797486","722784648329","14182300196130","169240659427823","1369615648154822","7932541529318706","33801591970830342","107187326579388279","252635388793024914","436415523257381534","535928019070243478","441760300072796667","218168619790055840","48451779617935997"],["236975164805","11011840715497","236676502456736","3117411776107457","28100912847302203","183291334395554324","891585104927435877","3283269478757393865","9189639339383219331","19432694656305133460","30517003772880642133","34445627700237571601","26349034503749608622","12182407293027406891","2554818339078836357"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","7","59","653","9941","203107","5256799","164035513","5969551721","247761828287","11540465933699","595866101149093","33770070139476541","2083834869000200347","139060837772684549159"],"nonn":true,"definitions":[{"position":0,"source":"derived:mobius_x_over_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["2","5"]],"recurrence":{"denominator":{"coeffs":[["1","2","1"]]},"terms":[{"coeff":{"coeffs":[["2","20","66","80","32"],["-1","-10","-33","-40","-16"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","-3","-6","-3"],["1","6","9","4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":187671,"position":0,"claim":"real_rooted","from_index":1,"region_key":"[-1, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in [-1, 0]","region_exact":null,"edges":["e-391a3622190e4b26"]},{"id":187672,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-391a3622190e4b26"]},{"id":187673,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-391a3622190e4b26"]}],"region_exact":{},"incoming_edges":[{"id":"e-391a3622190e4b26","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"mobius\",\"a\":\"1\",\"b\":\"0\",\"c\":\"1\",\"e\":\"1\",\"schedule\":{\"first_index\":1,\"first_degree\":0,\"step\":1}}"},"operation_detail":"mobius, shift 0","conclusion_id":"n-ad2dfc31d01b8299","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":"3fbeeac860518aae0af140b8b746353eef7e089770e038a3d5d6297f6651ecf7","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-6c2910cdb87c65c2","role":"sequence","name":"basis_map_A013610 of Coefficients of a special case of Poisson-Charlier polynomials.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:      1;\n   2:      2,        5;\n   3:      5,       25,        29;\n   4:     16,      123,       297,        217;\n   5:     65,      692,      2649,       4238,        2297;\n   6:    326,     4513,     24278,      62813,       76828,       34349;\n   7:   1957,    33681,    236646,     864131,     1715196,     1730223,      674965;\n   8:  13700,   283475,   2476329,   11800312,    32974687,    53658582,    46551947,    16276481;\n   9: 109601,  2658016,  27879794,  164848300,   599280419,  1366146700,  1895981534,  1451119564,   461527793;\n  10: 986410, 27490797, 337437900, 2390863026, 10756203714, 31784753943, 61479918480, 74684677446, 51306357798, 14993138773;\n","recurrence_sentence":"The row polynomials satisfy (1+2*x+x^2)*P(n,x) = -(n-2+(10*n-20)*x+(33*n-66)*x^2+(40*n-80)*x^3+(16*n-32)*x^4)*P(n-2,x) + (n+(6*n-3)*x+(9*n-6)*x^2+(4*n-3)*x^3)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + 5*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 2 + 5*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n - 2 + (10*n - 20)*x + (33*n - 66)*x^2 + (40*n - 80)*x^3 + (16*n - 32)*x^4)*P[n - 2, x] + (n + (6*n - 3)*x + (9*n - 6)*x^2 + (4*n - 3)*x^3)*P[n - 1, x])/(1 + 2*x + x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + 5*x; for(n = 3, N, my(i = n); v[i] = (-(n - 2 + (10*n - 20)*x + (33*n - 66)*x^2 + (40*n - 80)*x^3 + (16*n - 32)*x^4)*v[i - 2] + (n + (6*n - 3)*x + (9*n - 6)*x^2 + (4*n - 3)*x^3)*v[i - 1])/(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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-c9dce8c25621fdea","steps":["basis A013610","x \u21a6 x/(x+1)"],"edges":["e-3699a0e7eb346ca6","e-391a3622190e4b26"],"text":"A046716 \u2192 basis A013610 \u2192 x \u21a6 x/(x+1)"},"headline":"A046716 \u2192 basis A013610 \u2192 x \u21a6 x/(x+1)","chain":null}