{"id":"n-632d96a251a43047","revision":1,"first_index":0,"fingerprint":"f6-e38e65f0a12d686c","name":"veronese_section_2_1 of Pascal-(1,7,1) array.","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"],["9"],["17","1"],["25","25"],["33","241","1"],["41","1161","41"],["49","3297","721","1"],["57","7161","7161","57"],["65","13265","44961","1457","1"],["73","22121","192969","22121","73"],["81","34241","616049","223329","2449","1"],["89","50137","1597849","1597849","50137","89"],["97","70321","3573569","8493073","702241","3697","1"],["105","95305","7162729","35066313","7162729","95305","105"]],"degrees":[-1,0,0,1,1,2,2,3,3,4,4,5,5,6,6],"row_sums":["0","1","9","18","50","275","1243","4068","14436","59749","237357","876150","3296150","12842999","49582591"],"nonn":true,"definitions":[{"position":0,"source":"derived:veronese_section_2_1","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["1"],["9"],["17","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-1","1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","14"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["0","49"]]},"deriv_order":0,"offset":4,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":168469,"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-bace45a12b7c6e7a"]},{"id":168470,"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-bace45a12b7c6e7a"]},{"id":168471,"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-bace45a12b7c6e7a"]},{"id":168472,"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-bace45a12b7c6e7a"]}],"region_exact":{},"incoming_edges":[{"id":"e-bace45a12b7c6e7a","operation":{"index_shift":0,"modulus":2,"operation":"veronese_section","residue":1},"operation_detail":"r=2, j=1, shift 0","conclusion_id":"n-632d96a251a43047","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":"b1cacdf039acdc5d964fd8c77892654183b5284178bec4b61ca2a351d0ac7c62","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-1e2f1b7f566b371c","role":"sequence","name":"Pascal-(1,7,1) array.","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:  0;\n  1:  1;\n  2:  9;\n  3: 17,     1;\n  4: 25,    25;\n  5: 33,   241,     1;\n  6: 41,  1161,    41;\n  7: 49,  3297,   721,    1;\n  8: 57,  7161,  7161,   57;\n  9: 65, 13265, 44961, 1457, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = 2*P(n-1,x) - (1-x)*P(n-2,x) + 14*x*P(n-3,x) + 49*x*P(n-4,x) for n >= 4, with P(0,x) = 0, P(1,x) = 1, P(2,x) = 9, P(3,x) = 17 + x.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[2, x_] := 9;\nP[3, x_] := 17 + x;\nP[n_, x_] := P[n, x] = Expand[2*P[n - 1, x] - (1 - x)*P[n - 2, x] + 14*x*P[n - 3, x] + 49*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] = 0; v[2] = 1; v[3] = 9; v[4] = 17 + x; for(n = 4, N, my(i = n + 1); v[i] = 2*v[i - 1] - (1 - x)*v[i - 2] + 14*x*v[i - 3] + 49*x*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-1e2f1b7f566b371c","steps":["Veronese 2 (j=1)"],"edges":["e-bace45a12b7c6e7a"],"text":"A081582 \u2192 Veronese 2 (j=1)"},"headline":"A081582 \u2192 Veronese 2 (j=1)","chain":null}
