{"id":"n-0de906cd6e92b6be","revision":1,"first_index":1,"fingerprint":"f6-8ba428bd17294bc6","name":"one_plus_derivative of Triangle of coefficients of shifted Chebyshev's S(n,x-2) = U(n,x/2-1) polynomials (exponents of x in decreasing order).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":false,"rows":[["1"],["-1","-2"],["-3","2","3"],["-5","14","-2","-4"],["-7","34","-39","0","5"],["-9","62","-132","84","5","-6"],["-11","98","-305","384","-154","-14","7"],["-13","142","-582","1100","-930","252","28","-8"],["-15","194","-987","2496","-3245","1980","-378","-48","9"],["-17","254","-1544","4900","-8645","8294","-3828","528","75","-10"],["-19","322","-2277","8704","-19460","25662","-19019","6864","-693","-110","11"],["-21","398","-3210","14364","-38964","65688","-67704","40040","-11583","858","154","-12"],["-23","482","-4367","22400","-71535","147288","-195636","162656","-78650","18590","-1001","-208","13"],["-25","574","-5772","33396","-122815","299250","-488376","527136","-361998","145860","-28600","1092","273","-14"],["-27","674","-7449","48000","-199870","563178","-1092861","1457376","-1308150","755820","-257686","42432","-1092","-350","15"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","-3","2","3","-7","4","5","-11","6","7","-15","8","9","-19","10"],"nonn":false,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["-30","38","-8"],["62","-54","8"],["-40","16","0"],["8","0","0"]]},"terms":[{"coeff":{"coeffs":[["-18","102","-128","32"],["42","-206","162","-24"],["-32","128","-48","0"],["8","-24","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-6","35","-48","16"],["0","10","-48","32","0"],["0","-4","16","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":163462,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, +inf)","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)","region_exact":null,"edges":["e-d32ce5c030354d3d"]}],"region_exact":{},"incoming_edges":[{"id":"e-d32ce5c030354d3d","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-0de906cd6e92b6be","premise_count":1,"theorem":"Laguerre's theorem on differential operators: if phi is a real polynomial with only real roots, then phi(D) maps every real-rooted polynomial to a real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"fa5a09b6f83d4ffa109de372df949265b9a70fb2d9dfc2c4bd28143360f3a935","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{}}],"premises":[{"position":0,"node_id":"n-0b6e037cc63716ca","role":"sequence","name":"Triangle of coefficients of shifted Chebyshev's S(n,x-2) = U(n,x/2-1) polynomials (exponents of x in decreasing order).","uses":{"claim":"real_rooted","from_index":1,"region":{}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)"}]}],"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:  -1,  -2;\n   3:  -3,   2,     3;\n   4:  -5,  14,    -2,   -4;\n   5:  -7,  34,   -39,    0,     5;\n   6:  -9,  62,  -132,   84,     5,   -6;\n   7: -11,  98,  -305,  384,  -154,  -14,     7;\n   8: -13, 142,  -582, 1100,  -930,  252,    28,  -8;\n   9: -15, 194,  -987, 2496, -3245, 1980,  -378, -48,  9;\n  10: -17, 254, -1544, 4900, -8645, 8294, -3828, 528, 75, -10;\n","recurrence_sentence":"The row polynomials satisfy (8*n^3-40*n^2+62*n-30+(16*n^2-54*n+38)*x+(8*n-8)*x^2)*P(n,x) = (8*n^3-32*n^2+42*n-18-(24*n^3-128*n^2+206*n-102)*x-(48*n^2-162*n+128)*x^2-(24*n-32)*x^3)*P(n-1,x) - ((4*n^2-10*n+6)*x-(16*n^2-48*n+35)*x^2-(32*n-48)*x^3-16*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((8*n^3 - 32*n^2 + 42*n - 18 - (24*n^3 - 128*n^2 + 206*n - 102)*x - (48*n^2 - 162*n + 128)*x^2 - (24*n - 32)*x^3)*P[n - 1, x] - ((4*n^2 - 10*n + 6)*x - (16*n^2 - 48*n + 35)*x^2 - (32*n - 48)*x^3 - 16*x^4)*D[P[n - 1, x], x])/(8*n^3 - 40*n^2 + 62*n - 30 + (16*n^2 - 54*n + 38)*x + (8*n - 8)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((8*n^3 - 32*n^2 + 42*n - 18 - (24*n^3 - 128*n^2 + 206*n - 102)*x - (48*n^2 - 162*n + 128)*x^2 - (24*n - 32)*x^3)*v[i - 1] - ((4*n^2 - 10*n + 6)*x - (16*n^2 - 48*n + 35)*x^2 - (32*n - 48)*x^3 - 16*x^4)*deriv(v[i - 1], x))/(8*n^3 - 40*n^2 + 62*n - 30 + (16*n^2 - 54*n + 38)*x + (8*n - 8)*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-0b6e037cc63716ca","steps":["D+1"],"edges":["e-d32ce5c030354d3d"],"text":"A053123 \u2192 D+1"},"headline":"A053123 \u2192 D+1","chain":null}