{"id":"n-3454f35390d57b05","revision":1,"first_index":0,"fingerprint":"f6-09159489a3fc41c2","name":"veronese_section_2_0 of Riordan array (1, 2+2x).","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"],[],["0","4"],["0","8"],["0","4","16"],["0","0","64"],["0","0","96","64"],["0","0","64","384"],["0","0","16","960","256"],["0","0","0","1280","2048"],["0","0","0","960","7168","1024"],["0","0","0","384","14336","10240"],["0","0","0","64","17920","46080","4096"],["0","0","0","0","14336","122880","49152"],["0","0","0","0","7168","215040","270336","16384"]],"degrees":[0,-1,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","0","4","8","20","64","160","448","1232","3328","9152","24960","68160","186368","508928"],"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"],["0"],["0","4"],["0","8"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","4"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","8"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["0","4"]]},"deriv_order":0,"offset":4,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":163272,"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-07e54b52255e06bd"]},{"id":163273,"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-07e54b52255e06bd"]},{"id":163274,"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-07e54b52255e06bd"]}],"region_exact":{},"incoming_edges":[{"id":"e-07e54b52255e06bd","operation":{"index_shift":0,"modulus":2,"operation":"veronese_section","residue":0},"operation_detail":"r=2, j=0, shift 0","conclusion_id":"n-3454f35390d57b05","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":"a86aa4247ba517edfdad2a18909a98283af43665d84ba776658c469334735e0c","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-f5a001953ee28e8f","role":"sequence","name":"Riordan array (1, 2+2x).","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: 0;\n  2: 0, 4;\n  3: 0, 8;\n  4: 0, 4, 16;\n  5: 0, 0, 64;\n  6: 0, 0, 96,   64;\n  7: 0, 0, 64,  384;\n  8: 0, 0, 16,  960,  256;\n  9: 0, 0,  0, 1280, 2048;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = 4*x*P(n-2,x) + 8*x*P(n-3,x) + 4*x*P(n-4,x) for n >= 4, with P(0,x) = 1, P(1,x) = 0, P(2,x) = 4*x, P(3,x) = 8*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 0;\nP[2, x_] := 4*x;\nP[3, x_] := 8*x;\nP[n_, x_] := P[n, x] = Expand[4*x*P[n - 2, x] + 8*x*P[n - 3, x] + 4*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] = 0; v[3] = 4*x; v[4] = 8*x; for(n = 4, N, my(i = n + 1); v[i] = 4*x*v[i - 2] + 8*x*v[i - 3] + 4*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-f5a001953ee28e8f","steps":["Veronese 2"],"edges":["e-07e54b52255e06bd"],"text":"A099040 \u2192 Veronese 2"},"headline":"A099040 \u2192 Veronese 2","chain":null}