{"id":"n-8a6cb57f8c79f2fa","revision":0,"first_index":0,"fingerprint":"f6-132406c0e640de7f","name":"mobius_x_over_x_plus_one of Triangle of coefficients of Chebyshev's S(n,x+5) polynomials (exponents of x in increasing order).","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"],["5","6"],["24","58","35"],["115","418","506","204"],["551","2674","4863","3928","1189"],["2640","16028","38902","47184","28599","6930"],["12649","92204","279916","453004","412191","199938","40391"],["60605","515606","1879206","3803500","4617110","3361524","1359124","235416"],["290376","2824158","12012693","29187696","44308435","43032956","26112306","9051104","1372105"],["1391275","15226230","74037390","209937120","382565035","464616146","376059984","195614168","59337193","7997214"],["6665999","81073890","443592945","1437872700","3057751375","4457633418","4511510143","3130122072","1424784354","384213914","46611179"],["31938720","427356664","2598526940","9477683040","23039516410","39194992004","47615493912","41307320936","25078028689","10147489926","2463007758","271669860"],["153027601","2234009752","14944367960","60573444360","165686590570","322201049612","456762893230","475618792376","361037530420","194841717070","70960154985","15658959576","1583407981"],["733199285","11596980262","84640719324","377431335976","1147280209580","2510373083616","4067975769492","4942938020496","4503592241883","3038822899630","1476015061046","488710552488","98864449883","9228778026"],["3512968824","59844032066","473220063127","2302304805792","7699211216506","18721399000448","34135265766132","47411199090984","50406551323989","40823132379162","24791331583465","10947222142100","3322724082225","620504813890","53789260175"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","11","117","1243","13205","140283","1490293","15832091","168191829","1786781755","18981831989","201653024859","2142255945493","22758203300987","241771212528885"],"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":[["26","110","174","122","32"],["-16","-78","-140","-110","-32"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["21","115","251","273","148","32"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":180318,"position":0,"claim":"real_rooted","from_index":0,"region_key":"[-1, -3/4)","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":false,"value":"-3/4"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in [-1, -3/4)","region_exact":null,"edges":["e-176dcc9e47e34f2d"]},{"id":180319,"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-176dcc9e47e34f2d"]},{"id":180320,"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-176dcc9e47e34f2d"]}],"region_exact":{},"incoming_edges":[{"id":"e-176dcc9e47e34f2d","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-8a6cb57f8c79f2fa","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":"c0f7ec4a330d6ca51d6555360f4399b98f40cae9f047741f802fbd6fb82932e4","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":false,"value":"-3/4"}}}],"premises":[{"position":0,"node_id":"n-c2b7c74f8491f01b","role":"sequence","name":"Triangle of coefficients of Chebyshev's S(n,x+5) polynomials (exponents of x in increasing order).","uses":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"-7"},"upper":{"closed":false,"value":"-3"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-7, -3)"}]}],"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:       5,        6;\n  2:      24,       58,       35;\n  3:     115,      418,      506,       204;\n  4:     551,     2674,     4863,      3928,      1189;\n  5:    2640,    16028,    38902,     47184,     28599,      6930;\n  6:   12649,    92204,   279916,    453004,    412191,    199938,     40391;\n  7:   60605,   515606,  1879206,   3803500,   4617110,   3361524,   1359124,    235416;\n  8:  290376,  2824158, 12012693,  29187696,  44308435,  43032956,  26112306,   9051104,  1372105;\n  9: 1391275, 15226230, 74037390, 209937120, 382565035, 464616146, 376059984, 195614168, 59337193, 7997214;\n","recurrence_sentence":"The row polynomials satisfy (2*n+4*n*x+2*n*x^2)*P(n,x) = -(16*n-26+(78*n-110)*x+(140*n-174)*x^2+(110*n-122)*x^3+(32*n-32)*x^4)*P(n-1,x) + (21+115*x+251*x^2+273*x^3+148*x^4+32*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[(-(16*n - 26 + (78*n - 110)*x + (140*n - 174)*x^2 + (110*n - 122)*x^3 + (32*n - 32)*x^4)*P[n - 1, x] + (21 + 115*x + 251*x^2 + 273*x^3 + 148*x^4 + 32*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] = (-(16*n - 26 + (78*n - 110)*x + (140*n - 174)*x^2 + (110*n - 122)*x^3 + (32*n - 32)*x^4)*v[i - 1] + (21 + 115*x + 251*x^2 + 273*x^3 + 148*x^4 + 32*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-c2b7c74f8491f01b","steps":["x \u21a6 x/(x+1)"],"edges":["e-176dcc9e47e34f2d"],"text":"A207824 \u2192 x \u21a6 x/(x+1)"},"headline":"A207824 \u2192 x \u21a6 x/(x+1)","chain":null}
