{"id":"n-081b3f700db2d097","revision":1,"first_index":0,"fingerprint":"f6-7bfcf1196f8cac0f","name":"veronese_section_2_0 of Cube of lower triangular normalized binomial matrix.","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"],["3"],["9","1"],["27","9"],["81","54","1"],["243","270","15"],["729","1215","135","1"],["2187","5103","945","21"],["6561","20412","5670","252","1"],["19683","78732","30618","2268","27"],["59049","295245","153090","17010","405","1"],["177147","1082565","721710","112266","4455","33"],["531441","3897234","3247695","673596","40095","594","1"],["1594323","13817466","14073345","3752892","312741","7722","39"],["4782969","48361131","59108049","19702683","2189187","81081","819","1"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","3","10","36","136","528","2080","8256","32896","131328","524800","2098176","8390656","33558528","134225920"],"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"],["3"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["6","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-9","1"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":168517,"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-5d982f8f22366510"]},{"id":168518,"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-5d982f8f22366510"]},{"id":168519,"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-5d982f8f22366510"]}],"region_exact":{},"incoming_edges":[{"id":"e-5d982f8f22366510","operation":{"index_shift":0,"modulus":2,"operation":"veronese_section","residue":0},"operation_detail":"r=2, j=0, shift 0","conclusion_id":"n-081b3f700db2d097","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":"35ad2ccd007eaf2fd11f9c0201ab34c48ca3d8f2a9fee993716716bd2afe8308","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-8d887bfc40d036b9","role":"sequence","name":"Cube of lower triangular normalized binomial matrix.","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-c9109cf391ab591a","kind":"oeis_match","node_a_id":"n-081b3f700db2d097","node_b_id":null,"fingerprint":null,"oeis_id":"A007582","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A007582","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A007582","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-081b3f700db2d097","oeis_id":"A007582","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":26,"oeis_terms_available":26,"term_count":200,"raw_size":12912}],"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:     3;\n  2:     9,     1;\n  3:    27,     9;\n  4:    81,    54,     1;\n  5:   243,   270,    15;\n  6:   729,  1215,   135,    1;\n  7:  2187,  5103,   945,   21;\n  8:  6561, 20412,  5670,  252,  1;\n  9: 19683, 78732, 30618, 2268, 27;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = 6*P(n-1,x) - (9-x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3;\nP[n_, x_] := P[n, x] = Expand[6*P[n - 1, x] - (9 - x)*P[n - 2, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3; for(n = 2, N, my(i = n + 1); v[i] = 6*v[i - 1] - (9 - x)*v[i - 2]); 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-8d887bfc40d036b9","steps":["Veronese 2"],"edges":["e-5d982f8f22366510"],"text":"A027465 \u2192 Veronese 2"},"headline":"A027465 \u2192 Veronese 2","chain":null}
