{"id":"n-97ab161dcdd2c9b4","revision":1,"first_index":1,"fingerprint":"f6-2cff9b1fa72c8bfd","name":"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":-2,"palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":false,"rows":[[],["-2"],["-4","6"],["-6","20","-12"],["-8","42","-60","20"],["-10","72","-168","140","-30"],["-12","110","-360","504","-280","42"],["-14","156","-660","1320","-1260","504","-56"],["-16","210","-1092","2860","-3960","2772","-840","72"],["-18","272","-1680","5460","-10010","10296","-5544","1320","-90"],["-20","342","-2448","9520","-21840","30030","-24024","10296","-1980","110"],["-22","420","-3420","15504","-42840","74256","-80080","51480","-18018","2860","-132"],["-24","506","-4620","23940","-77520","162792","-222768","194480","-102960","30030","-4004","156"],["-26","600","-6072","35420","-131670","325584","-542640","604656","-437580","194480","-48048","5460","-182"],["-28","702","-7800","50600","-212520","605682","-1193808","1627920","-1511640","923780","-350064","74256","-7280","210"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","-2","2","2","-6","4","4","-10","6","6","-14","8","8","-18","10"],"nonn":false,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["0"],["-2"]],"recurrence":{"denominator":{"coeffs":[["6"],["-7"],["2"]]},"terms":[{"coeff":{"coeffs":[["3","-12"],["-5","19"],["2","-6"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1","-4"],["0","-1","4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":163461,"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-e72fe32e447a59b1"]}],"region_exact":{},"incoming_edges":[{"id":"e-e72fe32e447a59b1","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"0\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-97ab161dcdd2c9b4","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":"ba2e0b92373c0a00117a2b5a8d0e8b27a841487328d6fe379a5c134a24b35136","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:   0;\n   2:  -2;\n   3:  -4,   6;\n   4:  -6,  20,   -12;\n   5:  -8,  42,   -60,   20;\n   6: -10,  72,  -168,  140,    -30;\n   7: -12, 110,  -360,  504,   -280,    42;\n   8: -14, 156,  -660, 1320,  -1260,   504,   -56;\n   9: -16, 210, -1092, 2860,  -3960,  2772,  -840,   72;\n  10: -18, 272, -1680, 5460, -10010, 10296, -5544, 1320, -90;\n","recurrence_sentence":"The row polynomials satisfy (2*n^2-7*n+6)*P(n,x) = (2*n^2-5*n+3-(6*n^2-19*n+12)*x)*P(n-1,x) - ((n-1)*x-(4*n-4)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = -2.","mathematica":"P[1, x_] := 0;\nP[2, x_] := -2;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n^2 - 5*n + 3 - (6*n^2 - 19*n + 12)*x)*P[n - 1, x] - ((n - 1)*x - (4*n - 4)*x^2)*D[P[n - 1, x], x])/(2*n^2 - 7*n + 6)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 0; v[2] = -2; for(n = 3, N, my(i = n); v[i] = ((2*n^2 - 5*n + 3 - (6*n^2 - 19*n + 12)*x)*v[i - 1] - ((n - 1)*x - (4*n - 4)*x^2)*deriv(v[i - 1], x))/(2*n^2 - 7*n + 6)); 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"],"edges":["e-e72fe32e447a59b1"],"text":"A053123 \u2192 D"},"headline":"A053123 \u2192 D","chain":null}