{"id":"n-d9b08839e64d1348","revision":1,"first_index":0,"fingerprint":"f6-2b8dd76c465a8917","name":"veronese_section_2_0 of Pascal-(1,4,1) array.","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"],["1"],["1","1"],["1","11"],["1","46","1"],["1","106","21"],["1","191","191","1"],["1","301","1011","31"],["1","436","3606","436","1"],["1","596","9726","3716","41"],["1","781","21746","21746","781","1"],["1","991","42666","94526","9221","51"],["1","1226","76111","324556","76111","1226","1"],["1","1486","126331","926836","470951","18526","61"],["1","1771","198201","2294491","2294491","198201","1771","1"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","1","2","12","48","128","384","1344","4480","14080","45056","147456","479232","1544192","4988928"],"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"],["1"],["1","1"],["1","11"]],"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","8"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["0","16"]]},"deriv_order":0,"offset":4,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":168367,"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-e0d3eb2fed64a69c"]},{"id":168368,"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-e0d3eb2fed64a69c"]},{"id":168369,"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-e0d3eb2fed64a69c"]},{"id":168370,"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-e0d3eb2fed64a69c"]}],"region_exact":{},"incoming_edges":[{"id":"e-e0d3eb2fed64a69c","operation":{"index_shift":0,"modulus":2,"operation":"veronese_section","residue":0},"operation_detail":"r=2, j=0, shift 0","conclusion_id":"n-d9b08839e64d1348","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":"15df76b0798332712b2fd2e4b5298579d48159dbb232f89ac5621df3035d5aec","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-ed9434bfd889c409","role":"sequence","name":"Pascal-(1,4,1) 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: 1;\n  1: 1;\n  2: 1,   1;\n  3: 1,  11;\n  4: 1,  46,    1;\n  5: 1, 106,   21;\n  6: 1, 191,  191,    1;\n  7: 1, 301, 1011,   31;\n  8: 1, 436, 3606,  436,  1;\n  9: 1, 596, 9726, 3716, 41;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = 2*P(n-1,x) - (1-x)*P(n-2,x) + 8*x*P(n-3,x) + 16*x*P(n-4,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1, P(2,x) = 1 + x, P(3,x) = 1 + 11*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[2, x_] := 1 + x;\nP[3, x_] := 1 + 11*x;\nP[n_, x_] := P[n, x] = Expand[2*P[n - 1, x] - (1 - x)*P[n - 2, x] + 8*x*P[n - 3, x] + 16*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] = 1; v[2] = 1; v[3] = 1 + x; v[4] = 1 + 11*x; for(n = 4, N, my(i = n + 1); v[i] = 2*v[i - 1] - (1 - x)*v[i - 2] + 8*x*v[i - 3] + 16*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-ed9434bfd889c409","steps":["Veronese 2"],"edges":["e-e0d3eb2fed64a69c"],"text":"A081579 \u2192 Veronese 2"},"headline":"A081579 \u2192 Veronese 2","chain":null}