{"id":"n-55aa638526f289df","revision":1,"first_index":1,"fingerprint":"f6-87e173019ea380b5","name":"veronese_section_2_0 of Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["2"],["3","1"],["4","6"],["5","21","1"],["6","56","10"],["7","126","55","1"],["8","252","220","14"],["9","462","715","105","1"],["10","792","2002","560","18"],["11","1287","5005","2380","171","1"],["12","2002","11440","8568","1140","22"],["13","3003","24310","27132","5985","253","1"],["14","4368","48620","77520","26334","2024","26"],["15","6188","92378","203490","100947","12650","351","1"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","2","4","10","27","72","189","494","1292","3382","8855","23184","60697","158906","416020"],"nonn":true,"definitions":[{"position":0,"source":"derived:veronese_section_2_0","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["2"],["3","1"],["4","6"]],"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":163250,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-b8f275a63993d6a1"]},{"id":163251,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-b8f275a63993d6a1"]},{"id":163252,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-b8f275a63993d6a1"]},{"id":163253,"position":3,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"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 >= 2","region_exact":null,"edges":["e-b8f275a63993d6a1"]}],"region_exact":{},"incoming_edges":[{"id":"e-b8f275a63993d6a1","operation":{"index_shift":0,"modulus":2,"operation":"veronese_section","residue":0},"operation_detail":"r=2, j=0, shift 0","conclusion_id":"n-55aa638526f289df","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":"fe0e623e20728c84e271d7c44e6d46a382050b6aa354cf5b33d73bf8ecafca12","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"previous_interlaces_next"}],"premises":[{"position":0,"node_id":"n-976cdb6e4a693ec8","role":"sequence","name":"Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:  1;\n   2:  2;\n   3:  3,   1;\n   4:  4,   6;\n   5:  5,  21,    1;\n   6:  6,  56,   10;\n   7:  7, 126,   55,   1;\n   8:  8, 252,  220,  14;\n   9:  9, 462,  715, 105,  1;\n  10: 10, 792, 2002, 560, 18;\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 >= 5, with P(1,x) = 1, P(2,x) = 2, P(3,x) = 3 + x, P(4,x) = 4 + 6*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 2;\nP[3, x_] := 3 + x;\nP[4, x_] := 4 + 6*x;\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, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2; v[3] = 3 + x; v[4] = 4 + 6*x; for(n = 5, N, my(i = n); 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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-976cdb6e4a693ec8","steps":["Veronese 2"],"edges":["e-b8f275a63993d6a1"],"text":"A078812 \u2192 Veronese 2"},"headline":"A078812 \u2192 Veronese 2","chain":null}
