{"id":"n-a2065984d69a8de6","revision":1,"first_index":0,"fingerprint":"f6-e40c17441104e88c","name":"veronese_section_2_0 of Triangle of coefficients of polynomials u(n,x) jointly generated with A207607; see the Formula section.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["4","2"],["5","11"],["6","36","2"],["7","91","19"],["8","196","100","2"],["9","378","385","27"],["10","672","1210","196","2"],["11","1122","3289","1015","35"],["12","1782","8008","4200","324","2"],["13","2717","17875","14756","2109","43"],["14","4004","37180","45696","10830","484","2"],["15","5733","72930","127908","46683","3795","51"],["16","8008","136136","329460","175560","23276","676","2"],["17","10948","243542","791350","591261","118910","6201","59"],["18","14688","419900","1790712","1817046","526240","44226","900","2"]],"degrees":[1,1,2,2,3,3,4,4,5,5,6,6,7,7,8],"row_sums":["6","16","44","117","306","799","2090","5472","14328","37513","98210","257115","673134","1762288","4613732"],"nonn":true,"definitions":[{"position":0,"source":"derived:veronese_section_2_0","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["4","2"],["5","11"],["6","36","2"],["7","91","19"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["4","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-6","1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["4","0"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["-1","0"]]},"deriv_order":0,"offset":4,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":168423,"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-958000d578c323c8"]},{"id":168424,"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-958000d578c323c8"]},{"id":168425,"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-958000d578c323c8"]},{"id":168426,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"previous_interlaces_next"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 1","region_exact":null,"edges":["e-958000d578c323c8"]}],"region_exact":{},"incoming_edges":[{"id":"e-958000d578c323c8","operation":{"index_shift":0,"modulus":2,"operation":"veronese_section","residue":0},"operation_detail":"r=2, j=0, shift 0","conclusion_id":"n-a2065984d69a8de6","premise_count":1,"theorem":"Veronese sections of Polya frequency sequences (Aissen--Schoenberg--Whitney--Edrei): if P is real-rooted with coefficients of one sign, equivalently nonzero with all roots in (-inf, 0], every section V_(r,j)(P) is zero or real-rooted with roots in (-inf, 0]","lean_route":null,"lean_declarations":[],"certificate_hash":"47df4cba8c5966937c80b12d195618d3d53440cc887be8d88b378abb995088b2","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"previous_interlaces_next"}],"premises":[{"position":0,"node_id":"n-dfab35176bd08e72","role":"sequence","name":"Triangle of coefficients of polynomials u(n,x) jointly generated with A207607; see the Formula section.","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:  4,    2;\n  1:  5,   11;\n  2:  6,   36,     2;\n  3:  7,   91,    19;\n  4:  8,  196,   100,     2;\n  5:  9,  378,   385,    27;\n  6: 10,  672,  1210,   196,    2;\n  7: 11, 1122,  3289,  1015,   35;\n  8: 12, 1782,  8008,  4200,  324,  2;\n  9: 13, 2717, 17875, 14756, 2109, 43;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = 4*P(n-1,x) - (6-x)*P(n-2,x) + 4*P(n-3,x) - P(n-4,x) for n >= 4, with P(0,x) = 4 + 2*x, P(1,x) = 5 + 11*x, P(2,x) = 6 + 36*x + 2*x^2, P(3,x) = 7 + 91*x + 19*x^2.","mathematica":"P[0, x_] := 4 + 2*x;\nP[1, x_] := 5 + 11*x;\nP[2, x_] := 6 + 36*x + 2*x^2;\nP[3, x_] := 7 + 91*x + 19*x^2;\nP[n_, x_] := P[n, x] = Expand[4*P[n - 1, x] - (6 - x)*P[n - 2, x] + 4*P[n - 3, x] - P[n - 4, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 4 + 2*x; v[2] = 5 + 11*x; v[3] = 6 + 36*x + 2*x^2; v[4] = 7 + 91*x + 19*x^2; for(n = 4, N, my(i = n + 1); v[i] = 4*v[i - 1] - (6 - x)*v[i - 2] + 4*v[i - 3] - v[i - 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-dfab35176bd08e72","steps":["Veronese 2"],"edges":["e-958000d578c323c8"],"text":"A207606 \u2192 Veronese 2"},"headline":"A207606 \u2192 Veronese 2","chain":null}
