{"id":"n-d3d7c00ff700f502","revision":0,"first_index":0,"fingerprint":"f6-ab269ce9163b66d2","name":"theta_plus_one_squared of Coefficients of a generalized Jaco-Lucas polynomial (even indices) read by rows.","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":[["2"],["3","8"],["7","32","18"],["18","120","135","32"],["47","416","720","384","50"],["123","1360","3195","2720","875","72"],["322","4272","12690","14912","7875","1728","98"],["843","13048","46872","70336","51975","19152","3087","128"],["2207","39040","164520","300544","286000","149184","41160","5120","162"],["5778","114984","556065","1197408","1395225","944136","372498","80640","8019","200"],["15127","334480","1825560","4527680","6245375","5201568","2679810","837120","147015","12000","242"],["39603","963248","5856345","16439456","26203375","25934832","16521967","6779520","1732104","253000","17303","288"],["103682","2751072","18436788","57784448","104518050","119894688","91085708","46413312","15661107","3352800","415272","24192","338"],["271443","7802600","57142800","197791360","400286250","522306720","461156150","281132800","118262430","33605000","6142565","655200","32955","392"],["710647","21998816","174790350","662206720","1482649000","2168822880","2182982340","1550553600","782689635","278278000","67827760","10745280","999635","43904","450"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["2","11","57","305","1617","8345","41897","205441","987937","4674953","21825977","100741041","460441457","2086588665","9385299017"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one_squared","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["2"],["3","8"]],"recurrence":{"denominator":{"coeffs":[["-200","130","-82","-4","0"],["200","-630","562","28","0"],["0","500","-1080","160","-8"],["0","0","600","-424","40"],["0","0","0","240","-64"],["0","0","0","0","32"]]},"terms":[{"coeff":{"coeffs":[["-800","-830","782","1160","300","-24"],["800","-1870","-3662","-1244","176","88"],["0","2700","-120","-3140","-1100","-24"],["0","0","3000","2104","-448","-200"],["0","0","0","1120","944","64"],["0","0","0","0","128","96"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-125","-75","565","961","530","68","-16"],["0","-750","-1550","-760","260","268","48"],["0","0","-1200","-2600","-1732","-316","16"],["0","0","0","-600","-1240","-768","-128"],["0","0","0","0","-80","-144","-64"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":176400,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-124b64a5b19b8fe3"]},{"id":176401,"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-124b64a5b19b8fe3"]},{"id":176402,"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-124b64a5b19b8fe3"]}],"region_exact":{},"incoming_edges":[{"id":"e-124b64a5b19b8fe3","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"euler_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"2\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-d3d7c00ff700f502","premise_count":1,"theorem":"Laguerre's theorem on multiplier sequences: if f is a real polynomial whose roots are all real and <= 0, then (f(k))_(k >= 0) is a multiplier sequence, so x^k -> f(k) x^k maps every real-rooted polynomial to a real-rooted polynomial or zero (Laguerre; see T. Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. Branden, arXiv:1410.6601: sum_k f(k) x^k / k! = f(theta) e^x = C e^x x^m prod (1 + x/a_j), a_j > 0); f(k) has one sign for k >= 0, so coefficients of one sign stay of one sign; applied to f(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.isMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isMultiplierSequence_iff_preserves"],"certificate_hash":"ca08ea8ede1be9272dba24b65e671e30887fd806ef2aad7c50f343c2fd86456d","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-9b40f045d129e237","role":"sequence","name":"Coefficients of a generalized Jaco-Lucas polynomial (even indices) read by rows.","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:    2;\n  1:    3,      8;\n  2:    7,     32,     18;\n  3:   18,    120,    135,      32;\n  4:   47,    416,    720,     384,      50;\n  5:  123,   1360,   3195,    2720,     875,     72;\n  6:  322,   4272,  12690,   14912,    7875,   1728,     98;\n  7:  843,  13048,  46872,   70336,   51975,  19152,   3087,   128;\n  8: 2207,  39040, 164520,  300544,  286000, 149184,  41160,  5120,  162;\n  9: 5778, 114984, 556065, 1197408, 1395225, 944136, 372498, 80640, 8019, 200;\n","recurrence_sentence":"The row polynomials satisfy (200*n-200+(500*n^2-630*n+130)*x+(600*n^3-1080*n^2+562*n-82)*x^2+(240*n^4-424*n^3+160*n^2+28*n-4)*x^3+(32*n^5-64*n^4+40*n^3-8*n^2)*x^4)*P(n,x) = (800*n-800+(2700*n^2-1870*n-830)*x+(3000*n^3-120*n^2-3662*n+782)*x^2+(1120*n^4+2104*n^3-3140*n^2-1244*n+1160)*x^3+(128*n^5+944*n^4-448*n^3-1100*n^2+176*n+300)*x^4+(96*n^5+64*n^4-200*n^3-24*n^2+88*n-24)*x^5)*P(n-1,x) - (125+(750*n+75)*x+(1200*n^2+1550*n-565)*x^2+(600*n^3+2600*n^2+760*n-961)*x^3+(80*n^4+1240*n^3+1732*n^2-260*n-530)*x^4+(144*n^4+768*n^3+316*n^2-268*n-68)*x^5+(64*n^4+128*n^3-16*n^2-48*n+16)*x^6)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 2, P(1,x) = 3 + 8*x.","mathematica":"P[0, x_] := 2;\nP[1, x_] := 3 + 8*x;\nP[n_, x_] := P[n, x] = Expand[Together[((800*n - 800 + (2700*n^2 - 1870*n - 830)*x + (3000*n^3 - 120*n^2 - 3662*n + 782)*x^2 + (1120*n^4 + 2104*n^3 - 3140*n^2 - 1244*n + 1160)*x^3 + (128*n^5 + 944*n^4 - 448*n^3 - 1100*n^2 + 176*n + 300)*x^4 + (96*n^5 + 64*n^4 - 200*n^3 - 24*n^2 + 88*n - 24)*x^5)*P[n - 1, x] - (125 + (750*n + 75)*x + (1200*n^2 + 1550*n - 565)*x^2 + (600*n^3 + 2600*n^2 + 760*n - 961)*x^3 + (80*n^4 + 1240*n^3 + 1732*n^2 - 260*n - 530)*x^4 + (144*n^4 + 768*n^3 + 316*n^2 - 268*n - 68)*x^5 + (64*n^4 + 128*n^3 - 16*n^2 - 48*n + 16)*x^6)*D[P[n - 1, x], x])/(200*n - 200 + (500*n^2 - 630*n + 130)*x + (600*n^3 - 1080*n^2 + 562*n - 82)*x^2 + (240*n^4 - 424*n^3 + 160*n^2 + 28*n - 4)*x^3 + (32*n^5 - 64*n^4 + 40*n^3 - 8*n^2)*x^4)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 2; v[2] = 3 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = ((800*n - 800 + (2700*n^2 - 1870*n - 830)*x + (3000*n^3 - 120*n^2 - 3662*n + 782)*x^2 + (1120*n^4 + 2104*n^3 - 3140*n^2 - 1244*n + 1160)*x^3 + (128*n^5 + 944*n^4 - 448*n^3 - 1100*n^2 + 176*n + 300)*x^4 + (96*n^5 + 64*n^4 - 200*n^3 - 24*n^2 + 88*n - 24)*x^5)*v[i - 1] - (125 + (750*n + 75)*x + (1200*n^2 + 1550*n - 565)*x^2 + (600*n^3 + 2600*n^2 + 760*n - 961)*x^3 + (80*n^4 + 1240*n^3 + 1732*n^2 - 260*n - 530)*x^4 + (144*n^4 + 768*n^3 + 316*n^2 - 268*n - 68)*x^5 + (64*n^4 + 128*n^3 - 16*n^2 - 48*n + 16)*x^6)*deriv(v[i - 1], x))/(200*n - 200 + (500*n^2 - 630*n + 130)*x + (600*n^3 - 1080*n^2 + 562*n - 82)*x^2 + (240*n^4 - 424*n^3 + 160*n^2 + 28*n - 4)*x^3 + (32*n^5 - 64*n^4 + 40*n^3 - 8*n^2)*x^4)); 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-9b40f045d129e237","steps":["(\u03b8+1)\u00b2"],"edges":["e-124b64a5b19b8fe3"],"text":"A122076 \u2192 (\u03b8+1)\u00b2"},"headline":"A122076 \u2192 (\u03b8+1)\u00b2","chain":null}