{"id":"n-75a5395154fdfd6e","revision":1,"first_index":0,"fingerprint":"f6-95f614e8f81f764d","name":"veronese_section_2_0 of A133080 * A007318.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["2"],["1","1"],["2","4"],["1","6","1"],["2","16","6"],["1","15","15","1"],["2","36","50","8"],["1","28","70","28","1"],["2","64","196","112","10"],["1","45","210","210","45","1"],["2","100","540","672","210","12"],["1","66","495","924","495","66","1"],["2","144","1210","2640","1782","352","14"],["1","91","1001","3003","3003","1001","91","1"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","2","2","6","8","24","32","96","128","384","512","1536","2048","6144","8192"],"nonn":true,"definitions":[{"position":0,"source":"derived:veronese_section_2_0","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["2"],["1","1"],["2","4"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","2","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-1","2","-1"]]},"deriv_order":0,"offset":4,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":163070,"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-901d84379bd0989c"]},{"id":163071,"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-901d84379bd0989c"]},{"id":163072,"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-901d84379bd0989c"]}],"region_exact":{},"incoming_edges":[{"id":"e-901d84379bd0989c","operation":{"index_shift":0,"modulus":2,"operation":"veronese_section","residue":0},"operation_detail":"r=2, j=0, shift 0","conclusion_id":"n-75a5395154fdfd6e","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":"08aa0c5687c4546691242727f787ec21fc4293b77b14f8347b3c3f518c322962","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-d75669a8cd7f9393","role":"sequence","name":"A133080 * A007318.","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":[{"id":"l-6a191d872ce3a901","kind":"oeis_match","node_a_id":"n-75a5395154fdfd6e","node_b_id":null,"fingerprint":null,"oeis_id":"A122756","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":2,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":1,"target":"row_sums","terms_compared":13},"in_catalog":false,"oeis_id":"A122756","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A122756","relation":"match_row_sums"}],"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: 2;\n  2: 1,  1;\n  3: 2,  4;\n  4: 1,  6,   1;\n  5: 2, 16,   6;\n  6: 1, 15,  15,   1;\n  7: 2, 36,  50,   8;\n  8: 1, 28,  70,  28,  1;\n  9: 2, 64, 196, 112, 10;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2+2*x)*P(n-2,x) - (1-2*x+x^2)*P(n-4,x) for n >= 4, with P(0,x) = 1, P(1,x) = 2, P(2,x) = 1 + x, P(3,x) = 2 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2;\nP[2, x_] := 1 + x;\nP[3, x_] := 2 + 4*x;\nP[n_, x_] := P[n, x] = Expand[(2 + 2*x)*P[n - 2, x] - (1 - 2*x + x^2)*P[n - 4, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2; v[3] = 1 + x; v[4] = 2 + 4*x; for(n = 4, N, my(i = n + 1); v[i] = (2 + 2*x)*v[i - 2] - (1 - 2*x + x^2)*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-d75669a8cd7f9393","steps":["Veronese 2"],"edges":["e-901d84379bd0989c"],"text":"A133087 \u2192 Veronese 2"},"headline":"A133087 \u2192 Veronese 2","chain":null}