{"id":"n-e7e87c6a9f61cef2","revision":1,"first_index":0,"fingerprint":"f6-bb22eb5bdf1c795b","name":"veronese_section_2_0 of Array T(m,n) read by ascending antidiagonals: T(m,n) = m*binomial(n+m-2, n-1) for m, n >= 1.","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"],["2"],["3","1"],["4","6"],["5","18","1"],["6","40","10"],["7","75","45","1"],["8","126","140","14"],["9","196","350","84","1"],["10","288","756","336","18"],["11","405","1470","1050","135","1"],["12","550","2640","2772","660","22"],["13","726","4455","6468","2475","198","1"],["14","936","7150","13728","7722","1144","26"],["15","1183","11011","27027","21021","5005","273","1"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","2","4","10","24","56","128","288","640","1408","3072","6656","14336","30720","65536"],"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"],["2"],["3","1"],["4","6"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["4","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-6","2","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["4","-4","0"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["-1","2","-1"]]},"deriv_order":0,"offset":4,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":163202,"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-d1549cff19275cfe"]},{"id":163203,"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-d1549cff19275cfe"]},{"id":163204,"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-d1549cff19275cfe"]}],"region_exact":{},"incoming_edges":[{"id":"e-d1549cff19275cfe","operation":{"index_shift":0,"modulus":2,"operation":"veronese_section","residue":0},"operation_detail":"r=2, j=0, shift 0","conclusion_id":"n-e7e87c6a9f61cef2","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":"2547fdca30d205aa6df607b15f5266123eff3fc931899de9e565ab9071330603","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-f24dc5f77f658331","role":"sequence","name":"Array T(m,n) read by ascending antidiagonals: T(m,n) = m*binomial(n+m-2, n-1) for m, n >= 1.","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":[{"id":"l-1259a422c47fdc2b","kind":"oeis_match","node_a_id":"n-e7e87c6a9f61cef2","node_b_id":null,"fingerprint":null,"oeis_id":"A087447","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":2,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":2,"target":"row_sums","terms_compared":13},"in_catalog":false,"oeis_id":"A087447","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-1eef61edbee23a8d","kind":"oeis_match","node_a_id":"n-e7e87c6a9f61cef2","node_b_id":null,"fingerprint":null,"oeis_id":"A079859","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":2,"low_zeros_dropped":false,"oeis_offset":4,"oeis_term_position":0,"target":"row_sums","terms_compared":13},"in_catalog":false,"oeis_id":"A079859","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-d9059e8f8a440f86","kind":"oeis_match","node_a_id":"n-e7e87c6a9f61cef2","node_b_id":null,"fingerprint":null,"oeis_id":"A129953","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":2,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":2,"target":"row_sums","terms_compared":13},"in_catalog":false,"oeis_id":"A129953","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A079859","relation":"match_row_sums"},{"oeis_id":"A087447","relation":"match_row_sums"},{"oeis_id":"A129953","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-e7e87c6a9f61cef2","oeis_id":"A079859","target":"row_sums","oeis_offset":4,"first_row":2,"rows_used":200,"terms_matched":13,"oeis_terms_agreeing":31,"oeis_terms_available":31,"term_count":200,"raw_size":7369}],"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:  2;\n  2:  3,   1;\n  3:  4,   6;\n  4:  5,  18,   1;\n  5:  6,  40,  10;\n  6:  7,  75,  45,   1;\n  7:  8, 126, 140,  14;\n  8:  9, 196, 350,  84,  1;\n  9: 10, 288, 756, 336, 18;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = 4*P(n-1,x) - (6-2*x)*P(n-2,x) + (4-4*x)*P(n-3,x) - (1-2*x+x^2)*P(n-4,x) for n >= 4, with P(0,x) = 1, P(1,x) = 2, P(2,x) = 3 + x, P(3,x) = 4 + 6*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2;\nP[2, x_] := 3 + x;\nP[3, x_] := 4 + 6*x;\nP[n_, x_] := P[n, x] = Expand[4*P[n - 1, x] - (6 - 2*x)*P[n - 2, x] + (4 - 4*x)*P[n - 3, x] - (1 - 2*x + x^2)*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] = 2; v[3] = 3 + x; v[4] = 4 + 6*x; for(n = 4, N, my(i = n + 1); v[i] = 4*v[i - 1] - (6 - 2*x)*v[i - 2] + (4 - 4*x)*v[i - 3] - (1 - 2*x + x^2)*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-c8c19eba9d68881f","steps":["D+1","Veronese 2"],"edges":["e-5c65a1a70531c253","e-d1549cff19275cfe"],"text":"A007318 \u2192 D+1 \u2192 Veronese 2"},"headline":"A007318 \u2192 D+1 \u2192 Veronese 2","chain":null}