{"id":"n-b8a5d3147cb58db8","revision":1,"first_index":0,"fingerprint":"f6-504f8620b4722641","name":"veronese_section_2_1 of Riordan array (1, 2+2x).","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":[[],["2"],["2"],["0","8"],["0","24"],["0","24","32"],["0","8","160"],["0","0","320","128"],["0","0","320","896"],["0","0","160","2688","512"],["0","0","32","4480","4608"],["0","0","0","4480","18432","2048"],["0","0","0","2688","43008","22528"],["0","0","0","896","64512","112640","8192"],["0","0","0","128","64512","337920","106496"]],"degrees":[-1,0,0,1,1,2,2,3,3,4,4,5,5,6,6],"row_sums":["0","2","2","8","24","56","168","448","1216","3360","9120","24960","68224","186240","509056"],"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"],["2"],["2"],["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":163275,"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-3006d34f722f9e52"]},{"id":163276,"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-3006d34f722f9e52"]},{"id":163277,"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-3006d34f722f9e52"]}],"region_exact":{},"incoming_edges":[{"id":"e-3006d34f722f9e52","operation":{"index_shift":0,"modulus":2,"operation":"veronese_section","residue":1},"operation_detail":"r=2, j=1, shift 0","conclusion_id":"n-b8a5d3147cb58db8","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":"a435da14a37941a8e9e82051f08564d4eb69d32fe580d1d185a541aec7175f6f","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: 0;\n  1: 2;\n  2: 2;\n  3: 0,  8;\n  4: 0, 24;\n  5: 0, 24,  32;\n  6: 0,  8, 160;\n  7: 0,  0, 320,  128;\n  8: 0,  0, 320,  896;\n  9: 0,  0, 160, 2688, 512;\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) = 0, P(1,x) = 2, P(2,x) = 2, P(3,x) = 8*x.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 2;\nP[2, x_] := 2;\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] = 0; v[2] = 2; v[3] = 2; 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 (j=1)"],"edges":["e-3006d34f722f9e52"],"text":"A099040 \u2192 Veronese 2 (j=1)"},"headline":"A099040 \u2192 Veronese 2 (j=1)","chain":null}